what's hypothesis maths

A hypothesis is a proposition that is consistent with known data, but has been neither verified nor shown to be false.

In statistics, a hypothesis (sometimes called a statistical hypothesis) refers to a statement on which hypothesis testing will be based. Particularly important statistical hypotheses include the null hypothesis and alternative hypothesis .

In symbolic logic , a hypothesis is the first part of an implication (with the second part being known as the predicate ).

In general mathematical usage, "hypothesis" is roughly synonymous with " conjecture ."

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Weisstein, Eric W. "Hypothesis." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Hypothesis.html

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A statement that could be true, which might then be tested.

Example: Sam has a hypothesis that "large dogs are better at catching tennis balls than small dogs". We can test that hypothesis by having hundreds of different sized dogs try to catch tennis balls.

Sometimes the hypothesis won't be tested, it is simply a good explanation (which could be wrong). Conjecture is a better word for this.

Example: you notice the temperature drops just as the sun rises. Your hypothesis is that the sun warms the air high above you, which rises up and then cooler air comes from the sides.

Note: when someone says "I have a theory" they should say "I have a hypothesis", because in mathematics a theory is actually well proven.

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Explanation of Hypothesis

Contradiction, simple hypothesis, complex hypothesis, null hypothesis, alternative hypothesis, empirical hypothesis, statistical hypothesis, special example of hypothesis, solution part (a), solution part (b), hypothesis|definition & meaning.

A hypothesis is a claim or statement  that makes sense in the context of some information or data at hand but hasn’t been established as true or false through experimentation or proof.

In mathematics, any statement or equation that describes some relationship between certain variables can be termed as hypothesis if it is consistent with some initial supporting data or information, however, its yet   to be proven true or false by some definite and trustworthy experiment or mathematical law. 

Following example illustrates one such hypothesis to shed some light on this very fundamental concept which is often used in different areas of mathematics.

Figure 1: Example of Hypothesis

Here we have considered an example of a young startup company that manufactures state of the art batteries. The hypothesis or the claim of the company is that their batteries have a mean life of more than 1000 hours. Now its very easy to understand that they can prove their claim on some testing experiment in their lab.

However, the statement can only be proven if and only if at least one batch of their production batteries have actually been deployed in the real world for more than 1000 hours . After 1000 hours, data needs to be collected and it needs to be seen what is the probability of this statement being true .

The following paragraphs further explain this concept.

As explained with the help of an example earlier, a hypothesis in mathematics is an untested claim that is backed up by all the known data or some other discoveries or some weak experiments.

In any mathematical discovery, we first start by assuming something or some relationship . This supposed statement is called a supposition. A supposition, however, becomes a hypothesis when it is supported by all available data and a large number of contradictory findings.

The hypothesis is an important part of the scientific method that is widely known today for making new discoveries. The field of mathematics inherited this process. Following figure shows this cycle as a graphic:

Figure 2: Role of Hypothesis in the Scientific Method 

The above figure shows a simplified version of the scientific method. It shows that whenever a supposition is supported by some data, its termed as hypothesis. Once a hypothesis is proven by some well known and widely acceptable experiment or proof, its becomes a law. If the hypothesis is rejected by some contradictory results then the supposition is changed and the cycle continues.

Lets try to understand the scientific method and the hypothesis concept with the help of an example. Lets say that a teacher wanted to analyze the relationship between the students performance in a certain subject, lets call it A, based on whether or not they studied a minor course, lets call it B.

Now the teacher puts forth a supposition that the students taking the course B prior to course A must perform better in the latter due to the obvious linkages in the key concepts. Due to this linkage, this supposition can be termed as a hypothesis.

However to test the hypothesis, the teacher has to collect data from all of his/her students such that he/she knows which students have taken course B and which ones haven’t. Then at the end of the semester, the performance of the students must be measured and compared with their course B enrollments.

If the students that took course B prior to course A perform better, then the hypothesis concludes successful . Otherwise, the supposition may need revision.

The following figure explains this problem graphically.

Figure 3: Teacher and Course Example of Hypothesis

Important Terms Related to Hypothesis

To further elaborate the concept of hypothesis, we first need to understand a few key terms that are widely used in this area such as conjecture, contradiction and some special types of hypothesis (simple, complex, null, alternative, empirical, statistical). These terms are briefly explained below:

A conjecture is a term used to describe a mathematical assertion that has notbeenproved. While testing   may occasionally turn up millions of examples in favour of a conjecture, most experts in the area will typically only accept a proof . In mathematics, this term is synonymous to the term hypothesis.

In mathematics, a contradiction occurs if the results of an experiment or proof are against some hypothesis.  In other words, a contradiction discredits a hypothesis.

A simple hypothesis is such a type of hypothesis that claims there is a correlation between two variables. The first is known as a dependent variable while the second is known as an independent variable.

A complex hypothesis is such a type of hypothesis that claims there is a correlation between more than two variables.  Both the dependent and independent variables in this hypothesis may be more than one in numbers.

A null hypothesis, usually denoted by H0, is such a type of hypothesis that claims there is no statistical relationship and significance between two sets of observed data and measured occurrences for each set of defined, single observable variables. In short the variables are independent.

An alternative hypothesis, usually denoted by H1 or Ha, is such a type of hypothesis where the variables may be statistically influenced by some unknown factors or variables. In short the variables are dependent on some unknown phenomena .

An Empirical hypothesis is such a type of hypothesis that is built on top of some empirical data or experiment or formulation.

A statistical hypothesis is such a type of hypothesis that is built on top of some statistical data or experiment or formulation. It may be logical or illogical in nature.

According to the Riemann hypothesis, only negative even integers and complex numbers with real part 1/2 have zeros in the Riemann zeta function . It is regarded by many as the most significant open issue in pure mathematics.

Figure 4: Riemann Hypothesis

The Riemann hypothesis is the most well-known mathematical conjecture, and it has been the subject of innumerable proof efforts.

Numerical Examples

Identify the conclusions and hypothesis in the following given statements. Also state if the conclusion supports the hypothesis or not.

Part (a): If 30x = 30, then x = 1

Part (b): if 10x + 2 = 50, then x = 24

Hypothesis: 30x = 30

Conclusion: x = 10

Supports Hypothesis: Yes

Hypothesis: 10x + 2 = 50

Conclusion: x = 24

All images/mathematical drawings were created with GeoGebra.

Hour Hand Definition < Glossary Index > Identity Definition

Professor: Erika L.C. King Email: [email protected] Office: Lansing 304 Phone: (315)781-3355

The majority of statements in mathematics can be written in the form: "If A, then B." For example: "If a function is differentiable, then it is continuous". In this example, the "A" part is "a function is differentiable" and the "B" part is "a function is continuous." The "A" part of the statement is called the "hypothesis", and the "B" part of the statement is called the "conclusion". Thus the hypothesis is what we must assume in order to be positive that the conclusion will hold.

Whenever you are asked to state a theorem, be sure to include the hypothesis. In order to know when you may apply the theorem, you need to know what constraints you have. So in the example above, if we know that a function is differentiable, we may assume that it is continuous. However, if we do not know that a function is differentiable, continuity may not hold. Some theorems have MANY hypotheses, some of which are written in sentences before the ultimate "if, then" statement. For example, there might be a sentence that says: "Assume n is even." which is then followed by an if,then statement. Include all hypotheses and assumptions when asked to state theorems and definitions!

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Hypothesis Testing

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A hypothesis test is a statistical inference method used to test the significance of a proposed (hypothesized) relation between population statistics (parameters) and their corresponding sample estimators . In other words, hypothesis tests are used to determine if there is enough evidence in a sample to prove a hypothesis true for the entire population.

The test considers two hypotheses: the null hypothesis , which is a statement meant to be tested, usually something like "there is no effect" with the intention of proving this false, and the alternate hypothesis , which is the statement meant to stand after the test is performed. The two hypotheses must be mutually exclusive ; moreover, in most applications, the two are complementary (one being the negation of the other). The test works by comparing the \(p\)-value to the level of significance (a chosen target). If the \(p\)-value is less than or equal to the level of significance, then the null hypothesis is rejected.

When analyzing data, only samples of a certain size might be manageable as efficient computations. In some situations the error terms follow a continuous or infinite distribution, hence the use of samples to suggest accuracy of the chosen test statistics. The method of hypothesis testing gives an advantage over guessing what distribution or which parameters the data follows.

Definitions and Methodology

Hypothesis test and confidence intervals.

In statistical inference, properties (parameters) of a population are analyzed by sampling data sets. Given assumptions on the distribution, i.e. a statistical model of the data, certain hypotheses can be deduced from the known behavior of the model. These hypotheses must be tested against sampled data from the population.

The null hypothesis \((\)denoted \(H_0)\) is a statement that is assumed to be true. If the null hypothesis is rejected, then there is enough evidence (statistical significance) to accept the alternate hypothesis \((\)denoted \(H_1).\) Before doing any test for significance, both hypotheses must be clearly stated and non-conflictive, i.e. mutually exclusive, statements. Rejecting the null hypothesis, given that it is true, is called a type I error and it is denoted \(\alpha\), which is also its probability of occurrence. Failing to reject the null hypothesis, given that it is false, is called a type II error and it is denoted \(\beta\), which is also its probability of occurrence. Also, \(\alpha\) is known as the significance level , and \(1-\beta\) is known as the power of the test. \(H_0\) \(\textbf{is true}\)\(\hspace{15mm}\) \(H_0\) \(\textbf{is false}\) \(\textbf{Reject}\) \(H_0\)\(\hspace{10mm}\) Type I error Correct Decision \(\textbf{Reject}\) \(H_1\) Correct Decision Type II error The test statistic is the standardized value following the sampled data under the assumption that the null hypothesis is true, and a chosen particular test. These tests depend on the statistic to be studied and the assumed distribution it follows, e.g. the population mean following a normal distribution. The \(p\)-value is the probability of observing an extreme test statistic in the direction of the alternate hypothesis, given that the null hypothesis is true. The critical value is the value of the assumed distribution of the test statistic such that the probability of making a type I error is small.

Methodologies: Given an estimator \(\hat \theta\) of a population statistic \(\theta\), following a probability distribution \(P(T)\), computed from a sample \(\mathcal{S},\) and given a significance level \(\alpha\) and test statistic \(t^*,\) define \(H_0\) and \(H_1;\) compute the test statistic \(t^*.\) \(p\)-value Approach (most prevalent): Find the \(p\)-value using \(t^*\) (right-tailed). If the \(p\)-value is at most \(\alpha,\) reject \(H_0\). Otherwise, reject \(H_1\). Critical Value Approach: Find the critical value solving the equation \(P(T\geq t_\alpha)=\alpha\) (right-tailed). If \(t^*>t_\alpha\), reject \(H_0\). Otherwise, reject \(H_1\). Note: Failing to reject \(H_0\) only means inability to accept \(H_1\), and it does not mean to accept \(H_0\).

Assume a normally distributed population has recorded cholesterol levels with various statistics computed. From a sample of 100 subjects in the population, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is larger than 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the \(p\)-value approach with significance level \(\alpha=0.05:\) Define \(H_0\): \(\mu=200\). Define \(H_1\): \(\mu>200\). Since our values are normally distributed, the test statistic is \(z^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{100}}}\approx 3.09\). Using a standard normal distribution, we find that our \(p\)-value is approximately \(0.001\). Since the \(p\)-value is at most \(\alpha=0.05,\) we reject \(H_0\). Therefore, we can conclude that the test shows sufficient evidence to support the claim that \(\mu\) is larger than \(200\) mg/dL.

If the sample size was smaller, the normal and \(t\)-distributions behave differently. Also, the question itself must be managed by a double-tail test instead.

Assume a population's cholesterol levels are recorded and various statistics are computed. From a sample of 25 subjects, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is not equal to 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the \(p\)-value approach with significance level \(\alpha=0.05\) and the \(t\)-distribution with 24 degrees of freedom: Define \(H_0\): \(\mu=200\). Define \(H_1\): \(\mu\neq 200\). Using the \(t\)-distribution, the test statistic is \(t^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{25}}}\approx 1.54\). Using a \(t\)-distribution with 24 degrees of freedom, we find that our \(p\)-value is approximately \(2(0.068)=0.136\). We have multiplied by two since this is a two-tailed argument, i.e. the mean can be smaller than or larger than. Since the \(p\)-value is larger than \(\alpha=0.05,\) we fail to reject \(H_0\). Therefore, the test does not show sufficient evidence to support the claim that \(\mu\) is not equal to \(200\) mg/dL.

The complement of the rejection on a two-tailed hypothesis test (with significance level \(\alpha\)) for a population parameter \(\theta\) is equivalent to finding a confidence interval \((\)with confidence level \(1-\alpha)\) for the population parameter \(\theta\). If the assumption on the parameter \(\theta\) falls inside the confidence interval, then the test has failed to reject the null hypothesis \((\)with \(p\)-value greater than \(\alpha).\) Otherwise, if \(\theta\) does not fall in the confidence interval, then the null hypothesis is rejected in favor of the alternate \((\)with \(p\)-value at most \(\alpha).\)

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What Is a Hypothesis Test?

The null hypothesis significance testing (nhst) framework, our simple scenario.

• Our null hypothesis is $H_0\colon \pi=\frac{1}{2}$.  This says that the proportion is what we believe it should be.
• Our alternative hypothesis is $H_1\colon \pi\ne\frac{1}{2}$.  This says that the proportion has changed.

Testing our hypotheses

• We can work out the critical region for $X$, that is, those extreme values of $X$ which would lead us to reject the null hypothesis at 5% significance.  (This can be done even before performing the experiment.)  The probability of $X$ taking a value in this critical region, assuming that the null hypothesis is true, should be 5%, or as close at we can get to 5% without going over it.  In symbols, we can say: $$\mathrm{P}(\text{$X$ in critical region} | \text{$H_0$ is true}) \le 0.05.$$ Then we reject the null hypothesis if $X$ lies in that region.
• We can work out the probability of $X$ taking the value it did or a more extreme value, assuming that the null hypothesis is true.  This is known as the p-value .  If the p-value is less than 0.05, then we will reject the null hypothesis at 5% significance. [ note 1 ]  In symbols, we can write $$\text{p-value} = \mathrm{P}(\text{$X$ taking this or a more extreme value} | \text{$H_0$ is true}).$$

Other types of scenario

• Does this drug/treatment/intervention/... have any effect?
• Which of these drugs/... is more effective, or are they equally effective?
• Is the mean height/mass/intelligence/test score/... of this population equal to some predicted value?
• Is the standard deviation of the height/mass/... equal to some predicted value?
• For two distinct groups of people, is their mean height/mass/... of each group the same?
• Does this group of people's heights/masses/... appear to be following the probability distribution we expect?
• Do these two populations' heights/masses/... appear to have the same distribution as each other?
• Do this population's heights and weights appear to be correlated?

Interpreting the results

The key question that hypothesis testing (nhst) answers, what a hypothesis test does not tell us, a non-significant result.

• It could be that the null hypothesis is true.  In this case, we would have to be unlucky to get a significant p-value, so most of the time, we will end up accepting the null hypothesis.  (If the null hypothesis is true, we would reject it with a probability of only 0.05.)  
• On the other hand, it could be that the alternative hypothesis is true, but we did not use a large enough sample to obtain a significant result (or we were just unlucky).  In such a case, we could say that our test was insensitive .  In this situation (the alternative hypothesis is true but we do not reject the null hypothesis), we say that we have made a Type II error .  The probability of this happening depends on the sample size and on how different the true $\pi$ is from $\frac{1}{2}$ (or whatever our null hypothesis says), as is explored in Powerful Hypothesis Testing .

A significant result

• It could be that the null hypothesis is true.  In this case, we reject the null hypothesis with a probability of $0.05=\frac{1}{20}$, that is, one time in 20 (at a significance level of 5%), so we were just unlucky.  
• On the other hand, the alternative hypothesis could indeed be true.  Either the sample was large enough to obtain a significant result, or the sample size wasn't that large, but we were just lucky.

Using this tree diagram, we can work out the probabilities of $H_0$ being true or $H_1$ being true given our experimental results.  To avoid the expressions becoming unwieldy, we will write $H_0$ for "$\text{$H_0$ true}$", $H_1$ for "$\text{$H_1$ true}$" and "$\text{p}^+$" for "observed p-value or more extreme".  Then we can write (conditional) probabilities on the branches of the tree diagram leading to our observed p-value: [ note 2 ]

The two routes which give our observed p-value (or more extreme) have the following probabilities: $$\begin{align*} \mathrm{P}(H_0\cap \text{p}^+) &= \mathrm{P}(H_0) \times \mathrm{P}(\text{p}^+ | H_0) \\ \mathrm{P}(H_1\cap \text{p}^+) &= \mathrm{P}(H_1) \times \mathrm{P}(\text{p}^+ | H_1) \end{align*}$$ (Recall that $\mathrm{P}(H_0\cap \text{p}^+)$ means "the probability of $H_0$ being true and the p-value being that observed or more extreme".) We can therefore work out the probability of the alternative hypothesis being true given the observed p-value, using conditional probability: $$\begin{align*} \mathrm{P}(H_1|\text{p}^+) &= \frac{\mathrm{P}(H_1\cap \text{p}^+)}{\mathrm{P}(\text{p}^+)} \\ &= \frac{\mathrm{P}(H_1\cap \text{p}^+)}{\mathrm{P}(H_0\cap\text{p}^+)+\mathrm{P}(H_1\cap\text{p}^+)} \\ &= \frac{\mathrm{P}(H_1) \times \mathrm{P}(\text{p}^+ | H_1)}{\mathrm{P}(H_0) \times \mathrm{P}(\text{p}^+ | H_0) + \mathrm{P}(H_1) \times \mathrm{P}(\text{p}^+ | H_1)} \end{align*}$$ Though this is a mouthful, it is a calculation which only involves the four probabilities on the above tree diagram.  (This is an example of Bayes' Theorem , discussed further in this resource .) However, we immediately hit a big difficulty if we try to calculate this for a given experiment.  We know $\mathrm{P}(\text{p}^+ | H_0)$: this is just the p-value itself.  (The p-value tells us the probability of obtaining a result at least this extreme given that the null hypothesis is true.)  But we don't know the probability of the null hypothesis being true or false (that is, $\mathrm{P}(H_0)$ and $\mathrm{P}(H_1)=1-\mathrm{P}(H_0)$), nor do we know the probability of the observed result if the alternative hypothesis is true ($P(\text{p}^+|H_1)$), as knowing that the proportion of greens is not $\frac{1}{2}$ does not tell us what it actually is.  (Similar issues apply to all the other contexts of hypothesis testing listed above.)  So we are quite stuck: in the null hypothesis significance testing model, it is impossible to give a numerical answer to our key question: "Given our results, what is the probability that the alternative hypothesis is true?"  This is because we don't know two of the three probabilities that we need in order to answer the question. An example might highlight the issue a little better.  Let us suppose that we are trying to work out whether a coin is biased (alternative hypothesis), or whether the probability of heads is exactly $\frac{1}{2}$ (null hypothesis).  We toss the coin 50 times and obtain a p-value of 0.02.  Do we now believe that the coin is biased?  Most people believe that coins are not biased, and so are much more likely to attribute this result to chance or poor coin-tossing technique than to the coin being biased. On the other hand, consider a case of a road planner who introduces a traffic-calming feature to reduce the number of fatalities along a certain stretch of road.  The null hypothesis is that there is no change in fatality rate, while the alternative hypothesis is that the fatality rate has decreased.  A hypothesis test is performed on data collected for 24 months before and 24 months after the feature is built.  Again, the p-value was 0.02.  Do we believe that the alternative hypothesis is true?  In this case, we are more likely to believe that the alternative hypothesis is true, because it makes a lot of sense that this feature will reduce the number of fatalities. Our "instinctive" responses to these results are tied up with assigning values to the unknown probabilities in the formula above.  For the coin, we would probably take $\mathrm{P}(H_0)$ to be close to 1, say $0.99$, as we think it is very unlikely that the coin is biased, and $\mathrm{P}(\text{p}^+|H_1)$ will be, say, $0.1$: if the coin is biased, the bias is not likely to be very large, and so it is only a bit more likely that the result will be significant in this case.  Putting these figures into the formula above gives: $$\mathrm{P}(H_1|\text{p}^+) = \frac{0.01 \times 0.1}{0.99 \times 0.02 + 0.01 \times 0.1} \approx 0.05,$$ that is, we are still very doubtful that this coin is biased, even after performing the experiment.  Note that in this case, the probability of these results given that the null hypothesis is true is 0.02, whereas the probability that the null hypothesis is true given these results is $1-0.05=0.95$, which is very different.  This shows how dramatically different the answers to the two questions can be. On the other hand, for the fatalities situation, we might assume quite the opposite: we are pretty confident that the traffic-calming feature will help, so we might take $\mathrm{P}(H_0)$ to be $0.4$, and $\mathrm{P}(\text{p}^+|H_1)$ will be, say, $0.25$ (though the traffic-calming may help, the impact may be relatively small).  Putting these figures into the formula gives: $$\mathrm{P}(H_1|\text{p}^+) = \frac{0.6 \times 0.25}{0.4 \times 0.02 + 0.6 \times 0.25} \approx 0.95,$$ so we are now much more convinced that the traffic-calming feature is helping than we were before we had the data.  This time, the probability of these results given that the null hypothesis is true is still 0.02, whereas the probability that the null hypothesis is true given these results is $1-0.95=0.05$, which is not that different. This approach may seem very disturbing, as we have to make assumptions about what we believe before we do the hypothesis test.  But as we have seen, we cannot answer our key question without making such assumptions.  

Other approaches and some warnings

• Because our test is two-tailed (in the alternative hypothesis, the true proportion could be less than $\frac{1}{2}$ or more than $\frac{1}{2}$), we must be careful when calculating the p-value: we calculate the probability of the observed outcome or more extreme occurring, and then double the answer to account for the other tail.  We could also compare the probability of the value or more extreme to 0.025 instead of 0.05, but that would not be called a p-value. Likewise, when we determine the critical region, we will have two parts: a tail with large values of $X$ and a tail with small values of $X$; we require that the probability of $X$ lying in the large-value tail is as close as possible to 0.025 without going over it, and the same for the probability of $X$ lying in the small-value tail.  
• There are complications here when working with two-tail tests as opposed to one-tail tests.  We will ignore this problem, as it does not significantly affect the overall discussion.  
• "Likelihood" is a technical term.  For a discrete test statistic $X$, the likelihood of $H_0$ given the data $X=x$ means $P(X=x|H_0)$, in other words, how likely would this data be if $H_0$ were true.  It is not the probability of $H_0$ being true given the data.

Further reading

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17 Introduction to Hypothesis Testing

Jenna Lehmann

What is Hypothesis Testing?

Hypothesis testing is a big part of what we would actually consider testing for inferential statistics. It’s a procedure and set of rules that allow us to move from descriptive statistics to make inferences about a population based on sample data. It is a statistical method that uses sample data to evaluate a hypothesis about a population.

This type of test is usually used within the context of research. If we expect to see a difference between a treated and untreated group (in some cases the untreated group is the parameters we know about the population), we expect there to be a difference in the means between the two groups, but that the standard deviation remains the same, as if each individual score has had a value added or subtracted from it.

Steps of Hypothesis Testing

The following steps will be tailored to fit the first kind of hypothesis testing we will learn first: single-sample z-tests. There are many other kinds of tests, so keep this in mind.

• Null Hypothesis (H0): states that in the general population there is no change, no difference, or no relationship, or in the context of an experiment, it predicts that the independent variable has no effect on the dependent variable.
• Alternative Hypothesis (H1): states that there is a change, a difference, or a relationship for the general population, or in the context of an experiment, it predicts that the independent variable has an effect on the dependent variable.

• Critical Region: Composed of the extreme sample values that are very unlikely to be obtained if the null hypothesis is true. Determined by alpha level. If sample data fall in the critical region, the null hypothesis is rejected, because it’s very unlikely they’ve fallen there by chance.
• After collecting the data, we find the sample mean. Now we can compare the sample mean with the null hypothesis by computing a z-score that describes where the sample mean is located relative to the hypothesized population mean. We use the z-score formula.
• We decided previously what the two z-score boundaries are for a critical score. If the z-score we get after plugging the numbers in the aforementioned equation is outside of that critical region, we reject the null hypothesis. Otherwise, we would say that we failed to reject the null hypothesis.

Regions of the Distribution

Because we’re making judgments based on probability and proportion, our normal distributions and certain regions within them come into play.

The Critical Region is composed of the extreme sample values that are very unlikely to be obtained if the null hypothesis is true. Determined by alpha level. If sample data fall in the critical region, the null hypothesis is rejected, because it’s very unlikely they’ve fallen there by chance.

These regions come into play when talking about different errors.

A Type I Error occurs when a researcher rejects a null hypothesis that is actually true; the researcher concludes that a treatment has an effect when it actually doesn’t. This happens when a researcher unknowingly obtains an extreme, non-representative sample. This goes back to alpha level: it’s the probability that the test will lead to a Type I error if the null hypothesis is true.

A result is said to be significant or statistically significant if it is very unlikely to occur when the null hypothesis is true. That is, the result is sufficient to reject the null hypothesis. For instance, two means can be significantly different from one another.

Factors that Influence and Assumptions of Hypothesis Testing

Assumptions of Hypothesis Testing:

• Random sampling: it is assumed that the participants used in the study were selected randomly so that we can confidently generalize our findings from the sample to the population.
• Independent observation: two observations are independent if there is no consistent, predictable relationship between the first observation and the second. The value of σ is unchanged by the treatment; if the population standard deviation is unknown, we assume that the standard deviation for the unknown population (after treatment) is the same as it was for the population before treatment. There are ways of checking to see if this is true in SPSS or Excel.
• Normal sampling distribution: in order to use the unit normal table to identify the critical region, we need the distribution of sample means to be normal (which means we need the population to be distributed normally and/or each sample size needs to be 30 or greater based on what we know about the central limit theorem).

Factors that influence hypothesis testing:

• The variability of the scores, which is measured by either the standard deviation or the variance. The variability influences the size of the standard error in the denominator of the z-score.
• The number of scores in the sample. This value also influences the size of the standard error in the denominator.

Test statistic: indicates that the sample data are converted into a single, specific statistic that is used to test the hypothesis (in this case, the z-score statistic).

Directional Hypotheses and Tailed Tests

In a directional hypothesis test , also known as a one-tailed test, the statistical hypotheses specify with an increase or decrease in the population mean. That is, they make a statement about the direction of the effect.

The Hypotheses for a Directional Test:

• H0: The test scores are not increased/decreased (the treatment doesn’t work)
• H1: The test scores are increased/decreased (the treatment works as predicted)

Because we’re only worried about scores that are either greater or less than the scores predicted by the null hypothesis, we only worry about what’s going on in one tail meaning that the critical region only exists within one tail. This means that all of the alpha is contained in one tail rather than split up into both (so the whole 5% is located in the tail we care about, rather than 2.5% in each tail). So before, we cared about what’s going on at the 0.025 mark of the unit normal table to look at both tails, but now we care about 0.05 because we’re only looking at one tail.

A one-tailed test allows you to reject the null hypothesis when the difference between the sample and the population is relatively small, as long as that difference is in the direction that you predicted. A two-tailed test, on the other hand, requires a relatively large difference independent of direction. In practice, researchers hypothesize using a one-tailed method but base their findings off of whether the results fall into the critical region of a two-tailed method. For the purposes of this class, make sure to calculate your results using the test that is specified in the problem.

Effect Size

A measure of effect size is intended to provide a measurement of the absolute magnitude of a treatment effect, independent of the size of the sample(s) being used. Usually done with Cohen’s d. If you imagine the two distributions, they’re layered over one another. The more they overlap, the smaller the effect size (the means of the two distributions are close). The more they are spread apart, the greater the effect size (the means of the two distributions are farther apart).

Statistical Power

The power of a statistical test is the probability that the test will correctly reject a false null hypothesis. It’s usually what we’re hoping to get when we run an experiment. It’s displayed in the table posted above. Power and effect size are connected. So, we know that the greater the distance between the means, the greater the effect size. If the two distributions overlapped very little, there would be a greater chance of selecting a sample that leads to rejecting the null hypothesis.

This chapter was originally posted to the Math Support Center blog at the University of Baltimore on June 11, 2019.

Math and Statistics Guides from UB's Math & Statistics Center Copyright © by Jenna Lehmann is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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Hypotheses and Proofs

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What is a hypothesis?

A hypothesis is basically a theory that somebody states that needs to be tested in order to see if it is true. Most of the time a hypothesis is a statement which someone claims is true and then a series of tests are made to see if the person is correct.

Hypothesis – a proposed true statement that acts as a starting point for further investigation.

Devising theories is how all scientists progress, not just mathematicians, and the evidence that is found must be collected and interpreted to see if it gives any light on the truth in the statement. Statistics can either prove or disprove a theory, which is why we need the evidence that we gather to be as close to the truth as possible: so that we can give an answer to the question with a high level of confidence.

Hypotheses are just the plural of a single hypothesis. A hypothesis is the first thing that someone must come up with when doing a test, as we must initially know what it is we wish to find out rather than blindly going into carrying out certain surveys and tests.

Some examples of hypotheses are shown below:

• Britain is colder than Spain
• A dog is faster than a cat
• Blondes have more fun
• The square of the hypotenuse of a triangle is equal to the sum of the squares of the other two sides

Obviously, some of these hypotheses are correct and others are not. Even though some may look wrong or right we still need to test the hypothesis either way to find out if it is true or false.

Some hypotheses may be easier to test than others, for example it is easy to test the last hypothesis above as this is very mathematical. However, when it comes to measuring something like ‘fun’ which is shown in the hypothesis ‘Blondes have more fun’ we will begin to struggle! How do you measure something like fun and in what units? This is why it is much easier to test certain hypotheses when compared with others.

Another way to come up with a hypothesis is by doing some ‘trial and error’ type testing. When finding data you may realise that there is in fact a pattern and then state this as a hypothesis of your findings. This pattern should then be tested using mathematical skills to test its authenticity. There is still a big difference between finding a pattern in something and finding that something will always happen no matter what. The pattern that is found at any point may just be a coincidence as it is much harder to prove something using mathematics rather than simply noticing a pattern. However, once something is proved with mathematics it is a very strong indication that the hypothesis is not only a guess but is scientific fact.

A hypothesis must always:

• Be a statement that needs to be proven or disproven, never a question
• Be applied to a certain population
• Be testable, otherwise the hypothesis is rather pointless as we can never know any information about it!

There are also two different types of hypothesis which are explained here:

An Experimental Hypothesis –  This is a statement which should state a difference between two things that should be tested. For example, ‘Cheetahs are faster than lions’.

A Null Hypothesis –  This kind of hypothesis does not say something is more than another, instead it states that they are the same. For example, ‘There is no difference between the number of late buses on Tuesday and on Wednesday’.

Subjects and samples

We have already talked in an earlier lesson of different types of samples and how these are formed, so we will not dwell for too long on this. The main thing to make sure of when choosing subjects for a test is to link them to the hypothesis that we are looking into. This will then give a much better data set that will be a lot more relevant to the questions we are asking. There is no point in us gathering data from people that live in Ireland if our original hypothesis states something about Scottish people, so we need to also make sure that the sample taken is as relevant to the hypothesis as possible. As with all samples that are taken, there should never be any bias towards one subject or another (unless we are using something like quota sampling as outlined in an earlier lesson). This will then mean that a random collection of subjects is taken into account and will mean that the information that is acquired will be more useful to the hypothesis that we wish to look at.

The experimental method

By treating the hypothesis and the data collection as an experiment, we should use as many scientific methods as possible to ensure that the data we are collecting is very accurate.

The most important and best way of doing this is the  control of variables . A variable is basically anything that can change in a situation, which means there are a lot in the vast majority as lots of different things can be altered. By keeping all variables the same and only changing the ones which we wish to test, we will get data that is as reliable as possible. However, if variables are changed that can affect an outcome we may end up getting false data.

For example, when testing ‘A cheetah is faster than a lion’ we could simply make the two animals run against each other and see which is quickest. However, if we allowed the cheetah to run on flat ground and made the lion run up hill, then the times would not be accurate to the truth as it is much harder to run up a slope than on flat ground. It is for this reason that any variables should be the same for all subjects.

The only variable that is mentioned in the hypothesis ‘A cheetah runs faster than a lion’ is the animal that runs. Therefore, this is called the  independent variable  and is the only thing that we wish to change between experiments as it is the thing we wish to  prove has an effect on other results.

A  dependent variable  is something that we wish to measure in experiments to see if there is an effect. This is the speed at which something runs in our example, as we are changing the animal and measuring the speed.

Independent variable – something that stands alone and is not changed by other variables in the experiment. This variable is changed by the person carrying out the investigation to see if it influences the dependent variables. This can also be seen as an input when an experiment is created.

Dependent variable – this variable is measured in an experiment to see if it changes when the independent variable is changed. These represent an output after the experiment is carried out.

Standardised instructions

Another thing that is essential to carrying out experiments is to give both of the participants the same instructions in what you wish them to do. Although this may seem a little picky, there will be a definite difference in how a subject performs if they are given clear and concise instructions as opposed to given misleading and rushed ones.

Turning data into information

Experiments are carried out to produce a set of data but this is not the end of the problem! We will then need to interpret and change this information into something that will tell us what we need to know. This means we need to turn data in the form of numbers into actual information that can be useful to our investigation. Figures that are found through experiments are first shown as ‘raw data’ before we can use different tables and charts to show the patterns that have been found in the surveys and experiments that have been carried out. Once all the data is collected and in tables we can move on to using these to find patterns.

Once a hypothesis has been stated, we can look to prove or disprove it. In mathematics, a proof is a little different to what people usually think. A mathematical proof must show that something is the case without any doubt. We do this by working through step-by-step to build a proof that shows the hypothesis as being either right or wrong. Each small step in the proof must be correct so that the entire thing cannot be argued.

Setting out a proof

Being able to write a proof does not mean that you must work any differently to how you would usually answer a question. It simply means that you must show that something is the case. Questions on proofs may ask you to ‘prove’, ‘verify’ or ‘check’ a statement.

When doing this you will need to first understand the hypothesis that has been stated. Look at the example below to see how we would go about writing a simple proof.

Prove that 81 is not a prime number.

Here we have a hypothesis that 81 is not prime. So, to prove this, we can try to find a factor of 81 that is not 1 as we know the definition of a prime number is that it is only divisible by itself and 1. Therefore, we could simply show that:

The fact that 81 divided by 9 gives us 9 proves the hypothesis that 81 is not prime.

A proof for a hypothesis does not have to be very complex – it simply has to show that a statement is either true or false. Doing this will use your problem-solving skills though, as you may need to think outside the box and ensure that all of the information that you have is fully understood.

Harder examples

Being able to prove something can be very challenging. It is true that some mathematical equations are still yet to be proved and many mathematicians work on solving extremely complex proofs every day.

When looking at harder examples of proofs you will need to find like terms in equations and then think about how you can work through the proof to get the desired result.

Here we need to use the left-hand side to get to the right-hand side in order to prove that they are equal. We can do this by expanding the brackets on the left and collecting the like terms:

We have now expanded the brackets and collected the like terms. It is now that we will need to look at our hypothesis again and try to make the above equation into the right-hand side by moving terms around. We can see from the right-hand side of our hypothesis that we have a double bracket and then 2 added to this so we can begin by bringing 2 out of the above:

So we have now worked through an entire proof from start to finish. Here it is again using only mathematics and no writing:

In the above we have shown that the hypothesis is true by working through step-by-step and rearranging the equation on the left to get the one on the right.

The step-by-step approach to proofs

To prove something is correct we have used a step-by-step approach so far. This method is a very good way to get from the left-hand side of an equation to the right-hand side through different steps. To do this we can use specific rules:

1) Try to multiply out brackets early on where possible.  This will help you to cancel out certain terms in order to simplify the equation.

3) Take small steps each time.  A proof is about working through a problem slowly so that it is easy to spot what has been done in each step. Do not take big leaps in your work such as multiplying out brackets and collecting like terms all at once. Remember that the person marking your paper needs to see your working, so it is good to work in small stages.

4) Go back and check your work.  Once you have finished your proof you can go back and check each individual stage. One of the good things about carrying out a proof is that you will know if a mistake has been made in your arithmetic because you will not be able to get to the final solution. If this happens, go back and check your working throughout.

Harder proofs

When working through a proof that is more difficult it can be quite tricky. Sometimes we may have to carry out a lot of different steps or even prove something using another piece of knowledge. For example, it might be that we are asked to prove that an expression will always be even or that it will always be positive.

In the above equation we have worked through to get an answer that is completely multiplied by 4. This must therefore be even as any number (whether even or odd) will be even when multiplied by 4.

In this example we have had to use our knowledge that anything multiplied by 4 must be even. This information was not included in the question but is something that we know from previous lessons. Some examples of information that you may need to know in order to solve more difficult proofs are:

Any number that is multiplied by an even number must be even

A number multiplied by an even number and then added to an odd number will be odd

Any number multiplied by a number will give an answer that is divisible by the same number (e.g. 3 n  must be divisible by 3)

Any number that is squared must be positive

Above we have come to an answer that is multiplied by 3. This means that the answer has to be divisible by 3 also.

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Hypothesis Definition

In Statistics, the determination of the variation between the group of data due to true variation is done by hypothesis testing. The sample data are taken from the population parameter based on the assumptions. The hypothesis can be classified into various types. In this article, let us discuss the hypothesis definition, various types of hypothesis and the significance of hypothesis testing, which are explained in detail.

Hypothesis Definition in Statistics

In Statistics, a hypothesis is defined as a formal statement, which gives the explanation about the relationship between the two or more variables of the specified population. It helps the researcher to translate the given problem to a clear explanation for the outcome of the study. It clearly explains and predicts the expected outcome. It indicates the types of experimental design and directs the study of the research process.

Types of Hypothesis

The hypothesis can be broadly classified into different types. They are:

Simple Hypothesis

A simple hypothesis is a hypothesis that there exists a relationship between two variables. One is called a dependent variable, and the other is called an independent variable.

Complex Hypothesis

A complex hypothesis is used when there is a relationship between the existing variables. In this hypothesis, the dependent and independent variables are more than two.

Null Hypothesis

In the null hypothesis, there is no significant difference between the populations specified in the experiments, due to any experimental or sampling error. The null hypothesis is denoted by H 0 .

Alternative Hypothesis

In an alternative hypothesis, the simple observations are easily influenced by some random cause. It is denoted by the H a or H 1 .

Empirical Hypothesis

An empirical hypothesis is formed by the experiments and based on the evidence.

Statistical Hypothesis

In a statistical hypothesis, the statement should be logical or illogical, and the hypothesis is verified statistically.

Apart from these types of hypothesis, some other hypotheses are directional and non-directional hypothesis, associated hypothesis, casual hypothesis.

Characteristics of Hypothesis

The important characteristics of the hypothesis are:

• The hypothesis should be short and precise
• It should be specific
• A hypothesis must be related to the existing body of knowledge
• It should be capable of verification

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Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

• z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the size of the sample.
• t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). s is the sample standard deviation.
• \(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\). \(O_{i}\) is the observed value and \(E_{i}\) is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

• One sample: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
• Two samples: z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

• One sample: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\).
• Two samples: t = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

\(H_{0}\): The population parameter is ≤ some value

\(H_{1}\): The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

\(H_{0}\): The population parameter is ≥ some value

\(H_{1}\): The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

\(H_{0}\): the population parameter = some value

\(H_{1}\): the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

• Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
• Step 2: Set up the alternative hypothesis.
• Step 3: Choose the correct significance level, \(\alpha\), and find the critical value.
• Step 4: Calculate the correct test statistic (z, t or \(\chi\)) and p-value.
• Step 5: Compare the test statistic with the critical value or compare the p-value with \(\alpha\) to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.

Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.

Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.

1 - \(\alpha\) = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).

\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15

z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

Related Articles:

• Probability and Statistics
• Data Handling

Important Notes on Hypothesis Testing

• Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
• It involves the setting up of a null hypothesis and an alternate hypothesis.
• There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
• Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

• Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) > 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 5, s = 18. \(\alpha\) = 0.05 Using the t-distribution table, the critical value is 2.132 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
• Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. \(H_{0}\): \(\mu\) = 80, \(H_{1}\): \(\mu\) ≠ 80 \(\overline{x}\) = 88, \(\mu\) = 80, n = 36, \(\sigma\) = 10. \(\alpha\) = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) z = \(\frac{88-80}{\frac{10}{\sqrt{36}}}\) = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
• Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) < 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = \(\frac{82-90}{\frac{18}{\sqrt{6}}}\) t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

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FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.

Hypothesis test

A significance test, also referred to as a statistical hypothesis test, is a method of statistical inference in which observed data is compared to a claim (referred to as a hypothesis) in order to assess the truth of the claim. For example, one might wonder whether age affects the number of apples a person can eat, and may use a significance test to determine whether there is any evidence to suggest that it does.

Generally, the process of statistical hypothesis testing involves the following steps:

• State the null hypothesis.
• State the alternative hypothesis.
• Select the appropriate test statistic and select a significance level.
• Compute the observed value of the test statistic and its corresponding p-value.
• Reject the null hypothesis in favor of the alternative hypothesis, or do not reject the null hypothesis.

The null hypothesis

The null hypothesis, H 0 , is the claim that is being tested in a statistical hypothesis test. It typically is a statement that there is no difference between the populations being studied, or that there is no evidence to support a claim being made. For example, "age has no effect on the number of apples a person can eat."

A significance test is designed to test the evidence against the null hypothesis. This is because it is easier to prove that a claim is false than to prove that it is true; demonstrating that the claim is false in one case is sufficient, while proving that it is true requires that the claim be true in all cases.

The alternative hypothesis

The alternative hypothesis is the opposite of the null hypothesis in that it is a statement that there is some difference between the populations being studied. For example, "younger people can eat more apples than older people."

The alternative hypothesis is typically the hypothesis that researchers are trying to prove. A significance test is meant to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. Note that the results of a significance test should either be to reject the null hypothesis in favor of the alternative hypothesis, or to not reject the null hypothesis. The result should not be to reject the alternative hypothesis or to accept the alternative hypothesis.

Test statistics and significance level

A test statistic is a statistic that is calculated as part of hypothesis testing that compares the distribution of observed data to the expected distribution, based on the null hypothesis. Examples of test statistics include the Z-score, T-statistic, F-statistic, and the Chi-square statistic. The test statistic used is dependent on the significance test used, which is dependent on the type of data collected and the type of relationship to be tested.

In many cases, the chosen significance level is 0.05, though 0.01 is also used. A significance level of 0.05 indicates that there is a 5% chance of rejecting the null hypothesis when the null hypothesis is actually true. Thus, a smaller selected significance level will require more evidence if the null hypothesis is to be rejected in favor of the alternative hypothesis.

After the test statistic is computed, the p-value can be determined based on the result of the test statistic. The p-value indicates the probability of obtaining test results that are at least as extreme as the observed results, under the assumption that the null hypothesis is correct. It tells us how likely it is to obtain a result based solely on chance. The smaller the p-value, the less likely a result can occur purely by chance, while a larger p-value makes it more likely. For example, a p-value of 0.01 means that there is a 1% chance that a result occurred solely by chance, given that the null hypothesis is true; a p-value of 0.90 means that there is a 90% chance.

A p-value is significantly affected by sample size. The larger the sample size, the smaller the p-value, even if the difference between populations may not be meaningful. On the other hand, if a sample size is too small, a meaningful difference may not be detected.

The last step in a significance test is to determine whether the p-value provides evidence that the null hypothesis should be rejected in favor of the alternative hypothesis. This is based on the selected significance level. If the p-value is less than or equal to the selected significance level, the null hypothesis is rejected in favor of the alternative hypothesis, and the result is deemed statistically significant. If the p-value is greater than the selected significance level, the null hypothesis is not rejected, and the result is deemed not statistically significant.

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Hypothesis is a testable statement that explains what is happening or observed. It proposes the relation between the various participating variables. Hypothesis is also called Theory, Thesis, Guess, Assumption, or Suggestion. Hypothesis creates a structure that guides the search for knowledge.

In this article, we will learn what is hypothesis, its characteristics, types, and examples. We will also learn how hypothesis helps in scientific research.

What is Hypothesis?

A hypothesis is a suggested idea or plan that has little proof, meant to lead to more study. It’s mainly a smart guess or suggested answer to a problem that can be checked through study and trial. In science work, we make guesses called hypotheses to try and figure out what will happen in tests or watching. These are not sure things but rather ideas that can be proved or disproved based on real-life proofs. A good theory is clear and can be tested and found wrong if the proof doesn’t support it.

Hypothesis Meaning

A hypothesis is a proposed statement that is testable and is given for something that happens or observed.

• It is made using what we already know and have seen, and it’s the basis for scientific research.
• A clear guess tells us what we think will happen in an experiment or study.
• It’s a testable clue that can be proven true or wrong with real-life facts and checking it out carefully.
• It usually looks like a “if-then” rule, showing the expected cause and effect relationship between what’s being studied.

Characteristics of Hypothesis

Here are some key characteristics of a hypothesis:

• Testable: An idea (hypothesis) should be made so it can be tested and proven true through doing experiments or watching. It should show a clear connection between things.
• Specific: It needs to be easy and on target, talking about a certain part or connection between things in a study.
• Falsifiable: A good guess should be able to show it’s wrong. This means there must be a chance for proof or seeing something that goes against the guess.
• Logical and Rational: It should be based on things we know now or have seen, giving a reasonable reason that fits with what we already know.
• Predictive: A guess often tells what to expect from an experiment or observation. It gives a guide for what someone might see if the guess is right.
• Concise: It should be short and clear, showing the suggested link or explanation simply without extra confusion.
• Grounded in Research: A guess is usually made from before studies, ideas or watching things. It comes from a deep understanding of what is already known in that area.
• Flexible: A guess helps in the research but it needs to change or fix when new information comes up.
• Relevant: It should be related to the question or problem being studied, helping to direct what the research is about.
• Empirical: Hypotheses come from observations and can be tested using methods based on real-world experiences.

Sources of Hypothesis

Hypotheses can come from different places based on what you’re studying and the kind of research. Here are some common sources from which hypotheses may originate:

• Existing Theories: Often, guesses come from well-known science ideas. These ideas may show connections between things or occurrences that scientists can look into more.
• Observation and Experience: Watching something happen or having personal experiences can lead to guesses. We notice odd things or repeat events in everyday life and experiments. This can make us think of guesses called hypotheses.
• Previous Research: Using old studies or discoveries can help come up with new ideas. Scientists might try to expand or question current findings, making guesses that further study old results.
• Literature Review: Looking at books and research in a subject can help make guesses. Noticing missing parts or mismatches in previous studies might make researchers think up guesses to deal with these spots.
• Problem Statement or Research Question: Often, ideas come from questions or problems in the study. Making clear what needs to be looked into can help create ideas that tackle certain parts of the issue.
• Analogies or Comparisons: Making comparisons between similar things or finding connections from related areas can lead to theories. Understanding from other fields could create new guesses in a different situation.
• Hunches and Speculation: Sometimes, scientists might get a gut feeling or make guesses that help create ideas to test. Though these may not have proof at first, they can be a beginning for looking deeper.
• Technology and Innovations: New technology or tools might make guesses by letting us look at things that were hard to study before.
• Personal Interest and Curiosity: People’s curiosity and personal interests in a topic can help create guesses. Scientists could make guesses based on their own likes or love for a subject.

Types of Hypothesis

Here are some common types of hypotheses:

Simple Hypothesis

Complex hypothesis, directional hypothesis.

• Non-directional Hypothesis

Null Hypothesis (H0)

Alternative hypothesis (h1 or ha), statistical hypothesis, research hypothesis, associative hypothesis, causal hypothesis.

Simple Hypothesis guesses a connection between two things. It says that there is a connection or difference between variables, but it doesn’t tell us which way the relationship goes.

Complex Hypothesis tells us what will happen when more than two things are connected. It looks at how different things interact and may be linked together.

Directional Hypothesis says how one thing is related to another. For example, it guesses that one thing will help or hurt another thing.

Non-Directional Hypothesis

Non-Directional Hypothesis are the one that don’t say how the relationship between things will be. They just say that there is a connection, without telling which way it goes.

Null hypothesis is a statement that says there’s no connection or difference between different things. It implies that any seen impacts are because of luck or random changes in the information.

Alternative Hypothesis is different from the null hypothesis and shows that there’s a big connection or gap between variables. Scientists want to say no to the null hypothesis and choose the alternative one.

Statistical Hypotheis are used in math testing and include making ideas about what groups or bits of them look like. You aim to get information or test certain things using these top-level, common words only.

Research Hypothesis comes from the research question and tells what link is expected between things or factors. It leads the study and chooses where to look more closely.

Associative Hypotheis guesses that there is a link or connection between things without really saying it caused them. It means that when one thing changes, it is connected to another thing changing.

Causal Hypothesis are different from other ideas because they say that one thing causes another. This means there’s a cause and effect relationship between variables involved in the situation. They say that when one thing changes, it directly makes another thing change.

Hypothesis Examples

Following are the examples of hypotheses based on their types:

Simple Hypothesis Example

• Studying more can help you do better on tests.
• Getting more sun makes people have higher amounts of vitamin D.

Complex Hypothesis Example

• How rich you are, how easy it is to get education and healthcare greatly affects the number of years people live.
• A new medicine’s success relies on the amount used, how old a person is who takes it and their genes.

Directional Hypothesis Example

• Drinking more sweet drinks is linked to a higher body weight score.
• Too much stress makes people less productive at work.

Non-directional Hypothesis Example

• Drinking caffeine can affect how well you sleep.
• People often like different kinds of music based on their gender.
• The average test scores of Group A and Group B are not much different.
• There is no connection between using a certain fertilizer and how much it helps crops grow.

Alternative Hypothesis (Ha)

• Patients on Diet A have much different cholesterol levels than those following Diet B.
• Exposure to a certain type of light can change how plants grow compared to normal sunlight.
• The average smarts score of kids in a certain school area is 100.
• The usual time it takes to finish a job using Method A is the same as with Method B.
• Having more kids go to early learning classes helps them do better in school when they get older.
• Using specific ways of talking affects how much customers get involved in marketing activities.
• Regular exercise helps to lower the chances of heart disease.
• Going to school more can help people make more money.
• Playing violent video games makes teens more likely to act aggressively.
• Less clean air directly impacts breathing health in city populations.

Functions of Hypothesis

Hypotheses have many important jobs in the process of scientific research. Here are the key functions of hypotheses:

• Guiding Research: Hypotheses give a clear and exact way for research. They act like guides, showing the predicted connections or results that scientists want to study.
• Formulating Research Questions: Research questions often create guesses. They assist in changing big questions into particular, checkable things. They guide what the study should be focused on.
• Setting Clear Objectives: Hypotheses set the goals of a study by saying what connections between variables should be found. They set the targets that scientists try to reach with their studies.
• Testing Predictions: Theories guess what will happen in experiments or observations. By doing tests in a planned way, scientists can check if what they see matches the guesses made by their ideas.
• Providing Structure: Theories give structure to the study process by arranging thoughts and ideas. They aid scientists in thinking about connections between things and plan experiments to match.
• Focusing Investigations: Hypotheses help scientists focus on certain parts of their study question by clearly saying what they expect links or results to be. This focus makes the study work better.
• Facilitating Communication: Theories help scientists talk to each other effectively. Clearly made guesses help scientists to tell others what they plan, how they will do it and the results expected. This explains things well with colleagues in a wide range of audiences.
• Generating Testable Statements: A good guess can be checked, which means it can be looked at carefully or tested by doing experiments. This feature makes sure that guesses add to the real information used in science knowledge.
• Promoting Objectivity: Guesses give a clear reason for study that helps guide the process while reducing personal bias. They motivate scientists to use facts and data as proofs or disprovals for their proposed answers.
• Driving Scientific Progress: Making, trying out and adjusting ideas is a cycle. Even if a guess is proven right or wrong, the information learned helps to grow knowledge in one specific area.

How Hypothesis help in Scientific Research?

Researchers use hypotheses to put down their thoughts directing how the experiment would take place. Following are the steps that are involved in the scientific method:

• Initiating Investigations: Hypotheses are the beginning of science research. They come from watching, knowing what’s already known or asking questions. This makes scientists make certain explanations that need to be checked with tests.
• Formulating Research Questions: Ideas usually come from bigger questions in study. They help scientists make these questions more exact and testable, guiding the study’s main point.
• Setting Clear Objectives: Hypotheses set the goals of a study by stating what we think will happen between different things. They set the goals that scientists want to reach by doing their studies.
• Designing Experiments and Studies: Assumptions help plan experiments and watchful studies. They assist scientists in knowing what factors to measure, the techniques they will use and gather data for a proposed reason.
• Testing Predictions: Ideas guess what will happen in experiments or observations. By checking these guesses carefully, scientists can see if the seen results match up with what was predicted in each hypothesis.
• Analysis and Interpretation of Data: Hypotheses give us a way to study and make sense of information. Researchers look at what they found and see if it matches the guesses made in their theories. They decide if the proof backs up or disagrees with these suggested reasons why things are happening as expected.
• Encouraging Objectivity: Hypotheses help make things fair by making sure scientists use facts and information to either agree or disagree with their suggested reasons. They lessen personal preferences by needing proof from experience.
• Iterative Process: People either agree or disagree with guesses, but they still help the ongoing process of science. Findings from testing ideas make us ask new questions, improve those ideas and do more tests. It keeps going on in the work of science to keep learning things.

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Summary – Hypothesis

A hypothesis is a testable statement serving as an initial explanation for phenomena, based on observations, theories, or existing knowledge. It acts as a guiding light for scientific research, proposing potential relationships between variables that can be empirically tested through experiments and observations. The hypothesis must be specific, testable, falsifiable, and grounded in prior research or observation, laying out a predictive, if-then scenario that details a cause-and-effect relationship. It originates from various sources including existing theories, observations, previous research, and even personal curiosity, leading to different types, such as simple, complex, directional, non-directional, null, and alternative hypotheses, each serving distinct roles in research methodology. The hypothesis not only guides the research process by shaping objectives and designing experiments but also facilitates objective analysis and interpretation of data, ultimately driving scientific progress through a cycle of testing, validation, and refinement.

FAQs on Hypothesis

What is a hypothesis.

A guess is a possible explanation or forecast that can be checked by doing research and experiments.

What are Components of a Hypothesis?

The components of a Hypothesis are Independent Variable, Dependent Variable, Relationship between Variables, Directionality etc.

What makes a Good Hypothesis?

Testability, Falsifiability, Clarity and Precision, Relevance are some parameters that makes a Good Hypothesis

Can a Hypothesis be Proven True?

You cannot prove conclusively that most hypotheses are true because it’s generally impossible to examine all possible cases for exceptions that would disprove them.

How are Hypotheses Tested?

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data

Can Hypotheses change during Research?

Yes, you can change or improve your ideas based on new information discovered during the research process.

What is the Role of a Hypothesis in Scientific Research?

Hypotheses are used to support scientific research and bring about advancements in knowledge.

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3.6: Mathematical Induction - An Introduction

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• Save as PDF
• Page ID 24565

• Harris Kwong
• State University of New York at Fredonia via OpenSUNY

Mathematical induction can be used to prove that an identity is valid for all integers \(n\geq1\). Here is a typical example of such an identity: \[1+2+3+\cdots+n = \frac{n(n+1)}{2}.\] More generally, we can use mathematical induction to prove that a propositional function \(P(n)\) is true for all integers \(n\geq a\).

Principal of Mathematical Induction (PMI)

Given a propositional function \(P(n)\) defined for integers \(n\), and a fixed integer \(a.\)

Then, if these two conditions are true

• \(P(a)\) is true.
• if \(P(k)\) is true for some integer \(k\geq a\), then \(P(k+1)\) is also true.

then the  \(P(n)\) is true for all integers \(n\geq a\).

Outline for Mathematical Induction

To show that a propositional function \(P(n)\) is true for all integers \(n\geq a\), follow these steps:

• Base Step : Verify that \(P(a)\) is true.
• Assume \(P(n)\) is true for an arbitrary integer, \(k\) with  \(k\geq a\).  This is the inductive hypothesis .
• With this assumption (the inductive hypothesis) , show \(P(k+1)\) is true.
• Conclude, by the Principle of Mathematical Induction (PMI) that \(P(n)\) is true for all integers \(n\geq a\).

The base step is also called the basis step or the  anchor step or the initial step . 

The base step and the inductive step, together, prove that \[P(a) \Rightarrow P(a+1) \Rightarrow P(a+2) \Rightarrow \cdots\,.\] Therefore, \(P(n)\) is true for all integers \(n\geq a\). Compare induction to falling dominoes. When the first domino falls, it knocks down the next domino. The second domino in turn knocks down the third domino. Eventually, all the dominoes will be knocked down. But it will not happen unless these conditions are met:

• The first domino must fall to start the motion. If it does not fall, no chain reaction will occur. This is the base step.
• The distance between adjacent dominoes must be set up correctly. Otherwise, a certain domino may fall down without knocking over the next. Then the chain reaction will stop, and will never be completed. Maintaining the right inter-domino distance ensures that \(P(k)\Rightarrow P(k+1)\) for each integer \(k\geq a\).

To prove the implication \[P(k) \Rightarrow P(k+1)\] in the inductive step, we need to carry out two steps: assuming that \(P(k)\) is true, then using it to prove \(P(k+1)\) is also true. So we can refine an induction proof into a 3-step procedure:

• Verify that \(P(a)\) is true.
• Assume that \(P(k)\) is true for some integer \(k\geq a\).
• Show that \(P(k+1)\) is also true.

The second step, the assumption that \(P(k)\) is true, is referred to as the inductive hypothesis.  This is how a mathematical induction proof may look:

The idea behind mathematical induction is rather simple. However, it must be delivered with precision.

• Be sure to say “Assume \(P(n)\) holds for some integer \(k\geq a\).” Do not say “Assume it holds for all integers \(k\geq a\).” If we already know the result holds for all \(k\geq a\), then there is no need to prove anything at all.
• Be sure to specify the requirement \(k\geq a\). This ensures that the chain reaction of the falling dominoes starts with the first one.
• Do not say “let \(n=k\)” or “let \(n=k+1\).” The point is, you are not assigning the value of \(k\) and \(k+1\) to \(n\). Rather, you are assuming that the statement is true when \(n\) equals \(k\), and using it to show that the statement also holds when \(n\) equals \(k+1\).

Some proofs by induction 

\(1+2+3+\cdots+n\).

Use mathematical induction to show proposition \(P(n)\) :   \[1+2+3+\cdots+n = \frac{n(n+1)}{2}\] for all integers \(n\geq1\).

Base Step:  consider n = 1

On the Left-Hand Side (LHS) we get 1.  On the Right-Hand Side ( RHS) we get \(\frac{1(1+1)}{2}=\frac{2}{2}=1.\)  Thus \(P(n)\) is true for \(n =1.\)

Inductive step: Assume  \(P(n)\) is true for \(n =k, k \geq 1.\)  In other words, \(P(k)\) is true so our  inductive hypothesis is  \[1+2+3+\cdots+k = \frac{k(k+1)}{2}.\]   

Consider the left-hand side of \(P(k+1)\).   \[1+2+3+\cdots+(k+1) = 1+2+\cdots+k+(k+1),\]

we can regroup this as

\[1+2+3+\cdots+(k+1) = [1+2+\cdots+k]+(k+1),\]

so that \(1+2+\cdots+k\) can be replaced by \(\frac{k(k+1)}{2}\), by the inductive hypothesis.

Using the inductive hypothesis, we find

\[\begin{aligned} 1+2+3+\cdots+(k+1) &=& 1+2+3+\cdots+k+(k+1) \\ &=& \frac{k(k+1)}{2}+(k+1) \\ &=& (k+1)\left(\frac{k}{2}+1\right) \\ &=& (k+1)\cdot\frac{k+2}{2}\\ &=& \frac{(k+1)(k+2)}{2}. \end{aligned}\]

Therefore, the identity also holds when \(n=k+1\).

Thus, by the Principle of Mathematical Induction (PMI),  \[1+2+3+\cdots+n = \frac{n(n+1)}{2}\] for all integers \(n\geq1\).

We can use the summation notation (also called the sigma notation ) to abbreviate a sum. For example, the sum in the last example can be written as

\[\sum_{i=1}^n i.\]

The letter \(i\) is the index of summation . By putting \(i=1\) under \(\sum\) and \(n\) above, we declare that the sum starts with \(i=1\), and ranges through \(i=2\), \(i=3\), and so on, until \(i=n\). The quantity that follows \(\sum\) describes the pattern of the terms that we are adding in the summation. Accordingly,

\[\sum_{i=1}^{10} i^2 = 1^2+2^2+3^2+\cdots+10^2.\]

In general, the sum of the first \(n\) terms in a sequence \(\{a_1,a_2,a_3,\ldots\,\}\) is denoted \(\sum_{i=1}^n a_i\). Observe that

\[\sum_{i=1}^{k+1} a_i = \left(\sum_{i=1}^k a_i\right) + a_{k+1},\]

which provides the link between \(P(k+1)\) and \(P(k)\) in an induction proof.

\(\sum_{i=1}^n i^2\)

Example \(\PageIndex{2}\)

Use mathematical induction to show that, for all integers \(n\geq1\), \[\sum_{i=1}^n i^2 = 1^2+2^2+3^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}.\]

Base Step: When \(n=1\), the left-hand side reduces to \(1^2=1\), and the right-hand side becomes \(\frac{1\cdot2\cdot3}{6}=1\); hence, the identity holds when \(n=1\). Inductive Step: Assume it holds when \(n=k\) for some integer \(k\geq1\); that is, assume for some integer \(k\geq1\) that \[\sum_{i=1}^k i^2 = \frac{k(k+1)(2k+1)}{6}\] . Consider \(n=k+1\).   \[\sum_{i=1}^{k+1} i^2 =1^2+2^2+3^2+\cdots+k^2+(k+1)^2. \] From the inductive hypothesis, we find \[\sum_{i=1}^{k+1} i^2 = \sum_{i=1}^k i^2 + (k+1)^2\] \[=\frac{k(k+1)(2k+1)}{6}+(k+1)^2\]  \[=\frac{k(k+1)(2k+1)+6(k+1)^2}{6}\] \[\frac{(k+1)[k(2k+1)+6(k+1)]}{6}\] \[\frac{(k+1)(2k^2+7k+6)}{6}\] \[\frac{(k+1)(k+2)(2k+3)}{6}\] \[\frac{(k+1)(k+2)(2(k+1)+1)}{6}.\] Therefore, the identity also holds when \(n=k+1\).  Thus, by PMI for all integers \(n\geq1\), \[\sum_{i=1}^n i^2 = 1^2+2^2+3^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}.\]

hands-on exercise \(\PageIndex{1}\label{he:induct1-01}\)

It is time for you to write your own induction proof. Prove that \[1\cdot2 + 2\cdot3 + 3\cdot4 + \cdots + n(n+1) = \frac{n(n+1)(n+2)}{3}\] for all integers \(n\geq1\).

hands-on exercise \(\PageIndex{2}\label{he:induct1-02}\)

Use induction to prove that, for all positive integers \(n\), \[1\cdot2\cdot3 + 2\cdot3\cdot4 + \cdots + n(n+1)(n+2) = \frac{n(n+1)(n+2)(n+3)}{4}.\]

hands-on exercise \(\PageIndex{3}\label{he:sumfourn}\)

Use induction to prove that, for all positive integers \(n\), \[1+4^1+4^2+\cdots+4^n = \frac{4^{n+1}-1}{3}.\]

All three steps in an induction proof must be completed; otherwise, the proof may not be correct.

Example \(\PageIndex{3}\label{eg:induct1-03}\)

Can we just use examples?

Never attempt to prove \(P(k)\Rightarrow P(k+1)\) by examples alone . Consider \[P(n): \qquad n^2+n+11 \mbox{ is prime}.\] In the inductive step, we want to prove that \[P(k) \Rightarrow P(k+1) \qquad\mbox{ for ANY } k\geq1.\] The following table verifies that it is true for \(1\leq k\leq 9\): \[\begin{array}{|*{10}{c|}} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline n^2+n+11 & 13 & 17 & 23 & 31 & 41 & 53 & 67 & 83 & 101 \\ \hline \end{array}\] Nonetheless, when \(n=10\), \(n^2+n+11=121\) is composite. So \(P(9) \Rightarrow P(10)\) is false. The inductive step breaks down when \(k=9\).

Example \(\PageIndex{4}\label{eg:induct1-04}\)

The base step is equally important . Consider proving \[P(n): \qquad 3n+2 = 3q \mbox{ for some integer $q$}\] for all \(n\in\mathbb{N}\). Assume \(P(k)\) is true for some integer \(k\geq1\); that is, assume \(3k+2=3q\) for some integer \(q\). Then \[3(k+1)+2 = 3k+3+2 = 3+3q = 3(1+q).\] Therefore, \(3(k+1)+2\) can be written in the same form. This proves that \(P(k+1)\) is also true. Does it follow that \(P(n)\) is true for all integers \(n\geq1\)? We know that \(3n+2\) cannot be written as a multiple of 3. What is the problem?

The problem is: we need \(P(k)\) to be true for at least one value of \(k\) so as to start the sequence of implications \[P(1) \Rightarrow P(2), \qquad P(2) \Rightarrow P(3), \qquad P(3) \Rightarrow P(4), \qquad\ldots\] The induction fails because we have not established the basis step. In fact, \(P(1)\) is false. Since the first domino does not fall, we cannot even start the chain reaction.

Thus far, we have learned how to use mathematical induction to prove identities. In general, we can use mathematical induction to prove a statement about \(n\). This statement can take the form of an identity, an inequality, or simply a verbal statement about \(n\). We shall learn more about mathematical induction in the next few sections.

Summary and Review

• Mathematical induction can be used to prove that a statement about \(n\) is true for all integers \(n\geq a\).
• We have to complete three steps.
• In the base step, verify the statement for \(n=a\).
• In the inductive hypothesis, assume that the statement holds when \(n=k\) for some integer \(k\geq a\).
• In the inductive step, use the information gathered from the inductive hypothesis to prove that the statement also holds when \(n=k+1\).
• Be sure to complete all three steps.
• Pay attention to the wording. At the beginning, follow the template closely. When you feel comfortable with the whole process, you can start venturing out on your own.

Exercises 

Exercise \(\PageIndex{1}\label{ex:induct1-01}\)

Use induction to prove that \[1^3+2^3+3^3+\cdots+n^3 = \frac{n^2(n+1)^2}{4}\] for all integers \(n\geq1\).

Exercise \(\PageIndex{2}\)

Use induction to prove that the following identity holds for all integers \(n\geq1\): \[1+3+5+\cdots+(2n-1) = n^2.\]

Base Case: consider \(n=1\).  \(2(1)-1=1\) and \(1^2=1\) so the LHS & RHS are both 1. This works for  \(n=1\).

Inductive Step: Assume this works for some integer, \(k \geq 1.\) In other words,  \(1+3+5+\cdots+(2k-1) = k^2.\)  ( Inductive Hypothesis )

Consider the case of  \(n=k+1.\)   \(1+3+5+\cdots +(2k-1)+(2(k+1)-1)\)

 \[=k^2+(2(k+1)-1) \text{   by inductive hypothesis}\] \[=k^2+2k+2-1=k^2+2k+1=(k+1)^2 \text{   by algebra} \]

\(1+3+5+\cdots+(2(k+1)-1)=(k+1)^2\); assuming our proposition works for \(k\) it will also work for \(k+1.\)

By PMI, \(1+3+5+\cdots+(2n-1) = n^2\)  for all integers,  \(n\geq1\).

Exercise \(\PageIndex{3}\label{ex:induct1-03}\)

Use induction to show that \[1+\frac{1}{3}+\frac{1}{3^2}+\cdots+\frac{1}{3^n} = \frac{3}{2}\left(1-\frac{1}{3^{n+1}}\right)\] for all positive integers \(n\).

Exercise \(\PageIndex{4}\label{ex:induct1-04}\)

Use induction to establish the following identity for any integer \(n\geq1\): \[1-3+9-\cdots+(-3)^n = \frac{1-(-3)^{n+1}}{4}.\]

Exercise \(\PageIndex{5}\label{ex:induct1-05}\)

Use induction to show that, for any integer \(n\geq1\): \[\sum_{i=1}^n i\cdot i! = (n+1)!-1.\]

Exercise \(\PageIndex{6}\label{ex:induct1-06}\)

Use induction to prove the following identity for integers \(n\geq1\): \[\sum_{i=1}^n \frac{1}{(2i-1)(2i+1)} = \frac{n}{2n+1}.\]

Exercise \(\PageIndex{7}\)

Prove \(2^{2n}-1\) is divisible by 3, for all integers \(n\geq0.\)

Base Case: consider \(n=0\).  \(2^{2(0)}-1=1-1=0.\)  \(0\) is divisible by 3 because 0 = 0(3).

Inductive Step: Assume this works for some integer, \(k \geq 0.\) In other words, \(2^{2k}-1\) is divisible by 3. ( Inductive Hypothesis )

Since \(2^{2k}-1\) is divisible by 3, there exists some integer, m such that \(2^{2k}-1=3m,\)  by definition of divides.

Consider the case of  \(n=k+1.\)  By algebra: \[2^{2(k+1)}-1=2^{2k+2}-1=2^{2k}\cdot 2^2-1=2^{2k}\cdot 4 -1=2^{2k}\cdot (3+1)-1=3 \cdot 2^{2k}+2^{2k}-1\] \[=3 \cdot 2^{2k}+3m \text{   by inductive hypothesis}\]

\[=3(2^{2k}+m) \text{   by algebra}\]

\(2^{2(k+1)}-1=3(2^{2k}+m)\) and  \((2^{2k}+m)\in \mathbb{Z}\) since the integers are closed under addition and multiplication.  

So, \(2^{2(k+1)}-1\) is divisible by 3 by the definition of divisible.

Thus assuming our proposition works for \(k\) it will also work for \(k+1.\)

By PMI,  \(2^{2n}-1\) is divisible by 3, for all integers \(n\geq0.\)

Exercise \(\PageIndex{8}\label{ex:induct1-08}\)

Evaluate \(\sum_{i=1}^n \frac{1}{i(i+1)}\) for a few values of \(n\). What do you think the result should be? Use induction to prove your conjecture.

Exercise \(\PageIndex{9}\label{ex:induct1-09}\)

Use induction to prove that \[\sum_{i=1}^n (2i-1)^3 = n^2(2n^2-1)\] whenever \(n\) is a positive integer.

Exercise \(\PageIndex{10}\label{ex:induct1-10}\)

Use induction to show that, for any integer \(n\geq1\): \[1^2-2^2+3^2-\cdots+(-1)^{n-1}n^2 = (-1)^{n-1}\,\frac{n(n+1)}{2}.\]

Exercise \(\PageIndex{11}\label{ex:induct1-11}\)

Use mathematical induction to show that \[\sum_{i=1}^n \frac{i+4}{i(i+1)(i+2)} = \frac{n(3n+7)}{2(n+1)(n+2)}\] for all integers \(n\geq1\).

Exercise \(\PageIndex{12}\)

Use mathematical induction to show that \[3+\sum_{i=1}^n (3+5i) = \frac{(n+1)(5n+6)}{2}\] for all integers \(n\geq1\).

No answer here at this time.

Has the U.S. really conserved a third of its waters? Here’s the math.

An early biden administration analysis suggests the country is meeting a key marine conservation goal, but ocean advocates warn against including areas where fishing is allowed..

Almost everyone loves the ocean. But not everyone agrees on what it means to protect it.

The United States is conserving approximately one-third of the country’s ocean areas, according to an early analysis released Friday by the Biden administration — suggesting the president is meeting a key environmental goal laid out at the beginning of his term.

But others say that’s not the case.

Some of those areas still allow for commercial fishing, advocates say, and fall short of protections needed to save marine ecosystems facing dire threats.

“It’s padding the numbers,” said Brad Sewell, oceans director at the Natural Resources Defense Council.

The disagreement comes as the White House on Friday outlined how much progress the country has made in achieving President Biden’s ambitious goal of conserving at least 30 percent of U.S. lands and waters by 2030.

The White House’s Council on Environmental Quality said its preliminary count — outlined in a newly released atlas — shows that approximately “one-third of U.S. marine areas are currently conserved.”

“We are making bold progress to conserve our ocean,” Rick Spinrad, head of the National Oceanic and Atmospheric Administration, said in a statement.

Yet precisely what areas on the map should count as protected has been a subject of considerable debate .

The White House said a majority of that ocean expanse — 26 percent of U.S. waters — is officially designated as “marine protected areas,” where human activity is typically restricted to protect wildlife.

But that one-third tally also includes parts of the ocean where only a type of fishing called bottom trawling is banned to protect coral and other bottom-dwelling creatures from nets that scrape the seafloor. Other types of commercial fishing in those areas, which include swaths of ocean off New England and the Mid-Atlantic, are still allowed.

“These are partial measures,” he added. “We’re at a point in time where the biodiversity crisis is crying out for not partial measures, but full, complete, adequate measures.”

Matt Lee-Ashley, chief of staff for the Council on Environmental Quality, said the Biden administration is using a “more inclusive definition of conservation than strictly looking at protected areas” but added the work conserving ocean areas is not yet complete and the administration is open to feedback on how to measure progress. The White House said the current atlas is in “a beta version.”

“We have a lot of work yet to do to develop the data and information needed” for more precise estimates, Lee-Ashley said.

Sewell said he was grateful the administration is “open to listening to our view that they need a more accurate protective number.”

John Hocevar, oceans campaign director at Greenpeace USA, warned that claiming that a third of U.S. waters are already protected is “a huge mistake because we are definitely not there.” He worries about the example the United States may set for other countries as they try to meet their own conservation goals.

Biden’s 30 percent conservation goal, dubbed “30x30,” comes in tandem with international efforts to protect nearly a third of land and oceans globally as a refuge for the planet’s wild plants and animals in the face of a growing extinction crisis.

“It’s bad enough to mislead the United States public with creative accounting,” Hocevar said. “But this also risks opening a door for other countries to try to get away with similar shenanigans.”

Marine ecosystems upon which people depend on for food face a multitude of threats, including rising temperatures, acidifying waters and growing waves of plastic debris.

When it comes to terrestrial ecosystems, the White House said it is on track but has not yet met its 30 percent conservation goal for land, noting that approximately 13 percent of U.S. lands have permanent protections.

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Guest Essay

Why Biden Has a Narrower Path to the Presidency Than Trump, in 11 Maps

Illustration by Akshita Chandra/The New York Times; Images by PhotoObjects.net, Yuji Sakai, and THEPALMER/Getty Images

By Doug Sosnik Graphics by Quoctrung Bui

Mr. Sosnik was a senior adviser to President Bill Clinton from 1994 to 2000 and has advised over 50 governors and U.S. senators.

While polls show the race for president is tightening, Joe Biden still has a narrower and more challenging path to winning the election than Donald Trump. The reason is the Electoral College: My analysis of voter history and polling shows a map that currently favors Mr. Trump, even though recent developments in Arizona improve Mr. Biden’s chances. The Biden campaign will need to decide this summer which states to contest hardest. Our Electoral College maps below lay out the best scenarios for him and Mr. Trump.

Seven states with close results determined who won both the 2020 and the 2016 presidential elections, and those same seven states will most likely play the same battleground role this fall: three industrial states – Michigan, Pennsylvania and Wisconsin – and four Sun Belt states – Arizona, Georgia, Nevada and North Carolina.

The seven states that will most likely decide the 2024 presidential election

Mr. Biden’s declining popularity in the Sun Belt states is the main reason Mr. Trump has an edge right now. He is especially struggling with young and nonwhite voters there. Let’s take a closer look:

According to 2020 exit polls , Mr. Biden won 65 percent of Latino voters, who comprised roughly a fifth of voters in Arizona and Nevada. And Mr. Biden won 87 percent of Black voters, who made up 29 percent of the Georgia vote and 23 percent of the North Carolina vote. He also won 60 percent of voters aged 18 to 29. Now look at this year: A New York Times/Siena College poll released last weekend showed support for Mr. Biden had dropped 18 points with Black voters, 15 points with Latinos and 14 points with younger voters nationally.

Abortion could be a decisive issue in Mr. Biden stemming this erosion of support in Arizona and Nevada. The Arizona Supreme Court’s ruling last week that largely bans abortions raises the stakes of a likely ballot initiative on the issue there in November. It also appears likely that there will be a similar ballot measure in Nevada.

Nevertheless, the key to Mr. Biden’s victory is to perform well in the three industrial states. If Mr. Trump is able to win one or more of Pennsylvania, Michigan and Wisconsin, Mr. Biden’s path to 270 electoral votes becomes even narrower.

If Mr. Biden and Mr. Trump remain ahead in the states where they are currently running strongest, the outcome of the election could come down to who wins Michigan and the two Sun Belt states where abortion will very likely be on the ballot, Arizona and Nevada.

Based on past voting, Mr. Trump will start out the general election with 219 electoral votes, compared to 226 votes for Mr. Biden, with 93 votes up for grabs.

Voter history and recent polling suggest that Mr. Trump is in a strong position to win North Carolina . Republicans have carried the state in every presidential election since 1976 except in 2008. In a Wall Street Journal battleground poll taken in March, Mr. Biden had only 37 percent job approval in the state. By winning North Carolina , Mr. Trump would have 235 electoral votes and two strong paths to 270.

The first path involves carrying Georgia , a state he lost by fewer than 12,000 votes in 2020. Before then, Republicans won Georgia in every election since 1992. If Mr. Trump carries North Carolina and Georgia , he would have a base of 251 electoral votes with four scenarios that get him to 270.

Scenario 1 Then all Mr. Trump needs is Pennsylvania …

Scenario 2 … or Michigan and Nevada …

Scenario 3 … or Michigan and Arizona …

Scenario 4 … or Arizona and Wisconsin.

The second and harder path for Mr. Trump would be if he carried only one Southern swing state – most likely North Carolina . He would have only 235 electoral votes and would need to win three of the six remaining battleground states.

Scenario 5 Then he would need to win Arizona , Michigan and Wisconsin …

Scenario 6 … or Arizona , Nevada and Pennsylvania .

How Biden Can Win

It is difficult to see how Mr. Biden gets re-elected without doing well in the industrial battleground states – the so-called “ Blue Wall ” for Democrats. This is particularly true of Pennsylvania, given the state’s 19 electoral votes and Mr. Biden’s ties there and appeal to middle-class and blue-collar voters. That’s why he’s spending three days in Pennsylvania this week.

Mr. Biden will most likely need to win at least one other industrial battleground – with Wisconsin the most probable, since his polling numbers there are stronger than in the other battleground states.

A combination of factors have made winning Michigan much more challenging for Mr. Biden. Hamas’s attack on Israel and the war in Gaza have ripped apart the coalitions that enabled Democrats to do so well in the state since 2018. There are over 300,000 Arab Americans there, as well as a large Jewish population. Both groups were crucial to Mr. Biden’s success there in 2020.

In addition, Michigan voters’ perception of the economy is more negative compared with the other battleground states. In the Journal battleground poll , two-thirds of Michigan voters described the national economy negatively; more than half had a negative opinion of the state’s economy.

Now let’s look at Mr. Biden’s map.

Mr. Biden’s best strategy is based on winning Pennsylvania and Wisconsin, which would give him 255 electoral votes (assuming that he carries the 2nd Congressional District in Nebraska). By carrying these states, Mr. Biden has several paths to 270, but the first three scenarios are his most viable.

Scenario 1 He just needs to win Michigan …

Scenario 2 … or Arizona and Nevada …

Scenario 3 ... or Georgia .

There are two other scenarios where Mr. Biden loses Wisconsin and keeps Pennsylvania . But that would mean winning states where Mr. Biden is polling much worse.

Scenario 4 They involve Mr. Biden winning Georgia and Arizona …

Scenario 5 … or Michigan and Georgia .

A Look Ahead

With over six months to go until Election Day, given the volatility in the world and the weaknesses of Mr. Biden and Mr. Trump, it would be foolish to make firm predictions about specific results. And other electoral map scenarios are possible: Recent polling shows Mr. Biden with a narrow lead in Minnesota, a state that usually votes for Democrats for president. While it is mathematically possible for Mr. Biden to win without carrying Minnesota, it is unlikely he will be elected if he cannot carry this traditionally Democratic state.

For the third election cycle in a row, a small number of voters in a handful of states could determine the next president of the United States.

If the election remains close but Mr. Biden is unable to regain support from the core group of voters who propelled him to victory in 2020 — young and nonwhite voters — then we could be headed to a repeat of the 2016 election. The outcome of that election was decided by fewer than 80,000 votes in Michigan, Pennsylvania and Wisconsin.

Last week’s abortion ruling in Arizona, and the likely abortion ballot initiatives in that state and Nevada, give Mr. Biden the possibility of being re-elected even if he loses Michigan. That’s why, if we have another close presidential election, I think Arizona, Michigan and Nevada will likely determine the outcome for Mr. Biden or Mr. Trump.

Based on my experience as Bill Clinton’s White House political director in his 1996 re-election campaign, I would take immediate advantage of Mr. Biden’s significant fund-raising advantage over Mr. Trump to focus on shoring up the president’s chances in Michigan and the must-win states of Pennsylvania and Wisconsin, while at the same time trying to keep Georgia and North Carolina in play. Mr. Biden does not need to win either of those Sun Belt states to get re-elected, but draining Mr. Trump’s resources there could help him in other battleground states.

More on the 2024 presidential election

Democrats Need to Stop Playing Nice

Too often, Democrats litigate; Republicans fight.

By Joe Klein

One Purple State Is ‘Testing the Outer Limits of MAGAism’

North Carolina Republicans are “in the running for the most MAGA party in the nation.”

By Thomas B. Edsall

2024, Meet 1892, Your Doppelgänger

Great political change can unfold when the political system seems woefully stalled.

By Jon Grinspan

Doug Sosnik was a senior adviser to President Bill Clinton from 1994 to 2000 and has advised over 50 governors and U.S. senators.

The Times is committed to publishing a diversity of letters to the editor. We’d like to hear what you think about this or any of our articles. Here are some tips . And here’s our email: [email protected] .

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