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Once you have developed a clear and focused research question or set of research questions, you’ll be ready to conduct further research, a literature review, on the topic to help you make an educated guess about the answer to your question(s). This educated guess is called a hypothesis.

In research, there are two types of hypotheses: null and alternative. They work as a complementary pair, each stating that the other is wrong.

• Null Hypothesis (H 0 ) – This can be thought of as the implied hypothesis. “Null” meaning “nothing.”  This hypothesis states that there is no difference between groups or no relationship between variables. The null hypothesis is a presumption of status quo or no change.
• Alternative Hypothesis (H a ) – This is also known as the claim. This hypothesis should state what you expect the data to show, based on your research on the topic. This is your answer to your research question.

Null Hypothesis:   H 0 : There is no difference in the salary of factory workers based on gender. Alternative Hypothesis :  H a : Male factory workers have a higher salary than female factory workers.

Null Hypothesis :  H 0 : There is no relationship between height and shoe size. Alternative Hypothesis :  H a : There is a positive relationship between height and shoe size.

Null Hypothesis :  H 0 : Experience on the job has no impact on the quality of a brick mason’s work. Alternative Hypothesis :  H a : The quality of a brick mason’s work is influenced by on-the-job experience.

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Hypothesis Testing: Null Hypothesis and Alternative Hypothesis

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Figuring out exactly what the null hypothesis and the alternative hypotheses are is not a walk in the park. Hypothesis testing is based on the knowledge that you can acquire by going over what we have previously covered about statistics in our blog.

So, if you don’t want to have a hard time keeping up, make sure you have read all the tutorials about confidence intervals , distributions , z-tables and t-tables .

We've also made a video on null hypothesis vs alternative hypothesis - you can watch it below or just scroll down if you prefer reading.

Confidence intervals provide us with an estimation of where the parameters are located. You can obtain them with our confidence interval calculator and learn more about them in the related article.

However, when we are making a decision, we need a yes or no answer. The correct approach, in this case, is to use a test .

Here we will start learning about one of the fundamental tasks in statistics - hypothesis testing !

The Hypothesis Testing Process

  First off, let’s talk about data-driven decision-making. It consists of the following steps:

• First, we must formulate a hypothesis .
• After doing that, we have to find the right test for our hypothesis .
• Then, we execute the test.
• Finally, we make a decision based on the result.

Let’s start from the beginning.

What is a Hypothesis?

Though there are many ways to define it, the most intuitive must be:

“A hypothesis is an idea that can be tested.”

This is not the formal definition, but it explains the point very well.

So, if we say that apples in New York are expensive, this is an idea or a statement. However, it is not testable, until we have something to compare it with.

For instance, if we define expensive as: any price higher than $1.75 dollars per pound, then it immediately becomes a hypothesis .

What Cannot Be a Hypothesis?

An example may be: would the USA do better or worse under a Clinton administration, compared to a Trump administration? Statistically speaking, this is an idea , but there is no data to test it. Therefore, it cannot be a hypothesis of a statistical test.

Actually, it is more likely to be a topic of another discipline.

Conversely, in statistics, we may compare different US presidencies that have already been completed. For example, the Obama administration and the Bush administration, as we have data on both.

A Two-Sided Test

Alright, let’s get out of politics and get into hypotheses . Here’s a simple topic that CAN be tested.

According to Glassdoor (the popular salary information website), the mean data scientist salary in the US is 113,000 dollars.

So, we want to test if their estimate is correct.

The Null and Alternative Hypotheses

There are two hypotheses that are made: the null hypothesis , denoted H 0 , and the alternative hypothesis , denoted H 1 or H A .

The null hypothesis is the one to be tested and the alternative is everything else. In our example:

The null hypothesis would be: The mean data scientist salary is 113,000 dollars.

While the alternative : The mean data scientist salary is not 113,000 dollars.

Author's note: If you're interested in a data scientist career, check out our articles Data Scientist Career Path , 5 Business Basics for Data Scientists , Data Science Interview Questions , and 15 Data Science Consulting Companies Hiring Now .

An Example of a One-Sided Test

You can also form one-sided or one-tailed tests.

Say your friend, Paul, told you that he thinks data scientists earn more than 125,000 dollars per year. You doubt him, so you design a test to see who’s right.

The null hypothesis of this test would be: The mean data scientist salary is more than 125,000 dollars.

The alternative will cover everything else, thus: The mean data scientist salary is less than or equal to 125,000 dollars.

Important: The outcomes of tests refer to the population parameter rather than the sample statistic! So, the result that we get is for the population.

Important: Another crucial consideration is that, generally, the researcher is trying to reject the null hypothesis . Think about the null hypothesis as the status quo and the alternative as the change or innovation that challenges that status quo. In our example, Paul was representing the status quo, which we were challenging.

Let’s go over it once more. In statistics, the null hypothesis is the statement we are trying to reject. Therefore, the null hypothesis is the present state of affairs, while the alternative is our personal opinion.

Why Hypothesis Testing Works

Right now, you may be feeling a little puzzled. This is normal because this whole concept is counter-intuitive at the beginning. However, there is an extremely easy way to continue your journey of exploring it. By diving into the linked tutorial, you will find out why hypothesis testing actually works.

Interested in learning more? You can take your skills from good to great with our statistics course!

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Next Tutorial:  Hypothesis Testing: Significance Level and Rejection Region

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Module 9: Hypothesis Testing With One Sample

Summary: null and alternative hypotheses, key concepts.

• The null hypothesis is a statement of no change or status quo.
• Symbols used for the null hypothesis are =, ≤ and ≥.
• The alternative hypothesis is sometimes referred to as the research hypothesis; it is what the researcher believes to be true based on the data to be collected.
• Symbols used for the alternative hypothesis are ≠, > and <.

Hypothesis:  a statement about the value of a population parameter. In the case of two hypotheses, the statement assumed to be true is called the null hypothesis (notation [latex]H_0[/latex]) and the contradictory statement is called the alternative hypothesis (notation [latex]H_a[/latex]).

Hypothesis Testing:  Based on sample evidence, a procedure for determining whether the hypothesis stated is a reasonable statement and should not be rejected or is unreasonable and should be rejected.

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8.1 The Elements of Hypothesis Testing

Learning objectives.

• To understand the logical framework of tests of hypotheses.
• To learn basic terminology connected with hypothesis testing.
• To learn fundamental facts about hypothesis testing.

Types of Hypotheses

A hypothesis about the value of a population parameter is an assertion about its value. As in the introductory example we will be concerned with testing the truth of two competing hypotheses, only one of which can be true.

The null hypothesis The statement that is assumed to be true unless there is convincing evidence to the contrary. , denoted H 0 , is the statement about the population parameter that is assumed to be true unless there is convincing evidence to the contrary .

The alternative hypothesis A statement that is accepted as true only if there is convincing evidence in favor of it. , denoted H a , is a statement about the population parameter that is contradictory to the null hypothesis, and is accepted as true only if there is convincing evidence in favor of it.

Hypothesis testing A statistical procedure in which a choice is made between a null hypothesis and a specific alternative hypothesis based on information in a sample. is a statistical procedure in which a choice is made between a null hypothesis and an alternative hypothesis based on information in a sample.

The end result of a hypotheses testing procedure is a choice of one of the following two possible conclusions:

• Reject H 0 (and therefore accept H a ), or
• Fail to reject H 0 (and therefore fail to accept H a ).

The null hypothesis typically represents the status quo, or what has historically been true. In the example of the respirators, we would believe the claim of the manufacturer unless there is reason not to do so, so the null hypotheses is H 0 : μ = 75 . The alternative hypothesis in the example is the contradictory statement H a : μ < 75 . The null hypothesis will always be an assertion containing an equals sign, but depending on the situation the alternative hypothesis can have any one of three forms: with the symbol “<,” as in the example just discussed, with the symbol “>,” or with the symbol “≠” The following two examples illustrate the latter two cases.

A publisher of college textbooks claims that the average price of all hardbound college textbooks is $127.50. A student group believes that the actual mean is higher and wishes to test their belief. State the relevant null and alternative hypotheses.

The default option is to accept the publisher’s claim unless there is compelling evidence to the contrary. Thus the null hypothesis is H 0 : μ = 127.50 . Since the student group thinks that the average textbook price is greater than the publisher’s figure, the alternative hypothesis in this situation is H a : μ > 127.50 .

The recipe for a bakery item is designed to result in a product that contains 8 grams of fat per serving. The quality control department samples the product periodically to insure that the production process is working as designed. State the relevant null and alternative hypotheses.

The default option is to assume that the product contains the amount of fat it was formulated to contain unless there is compelling evidence to the contrary. Thus the null hypothesis is H 0 : μ = 8.0 . Since to contain either more fat than desired or to contain less fat than desired are both an indication of a faulty production process, the alternative hypothesis in this situation is that the mean is different from 8.0, so H a : μ ≠ 8.0 .

In Note 8.8 "Example 1" , the textbook example, it might seem more natural that the publisher’s claim be that the average price is at most $127.50, not exactly $127.50. If the claim were made this way, then the null hypothesis would be H 0 : μ ≤ 127.50 , and the value $127.50 given in the example would be the one that is least favorable to the publisher’s claim, the null hypothesis. It is always true that if the null hypothesis is retained for its least favorable value, then it is retained for every other value.

Thus in order to make the null and alternative hypotheses easy for the student to distinguish, in every example and problem in this text we will always present one of the two competing claims about the value of a parameter with an equality. The claim expressed with an equality is the null hypothesis. This is the same as always stating the null hypothesis in the least favorable light. So in the introductory example about the respirators, we stated the manufacturer’s claim as “the average is 75 minutes” instead of the perhaps more natural “the average is at least 75 minutes,” essentially reducing the presentation of the null hypothesis to its worst case.

The first step in hypothesis testing is to identify the null and alternative hypotheses.

The Logic of Hypothesis Testing

Although we will study hypothesis testing in situations other than for a single population mean (for example, for a population proportion instead of a mean or in comparing the means of two different populations), in this section the discussion will always be given in terms of a single population mean μ .

The null hypothesis always has the form H 0 : μ = μ 0 for a specific number μ 0 (in the respirator example μ 0 = 75 , in the textbook example μ 0 = 127.50 , and in the baked goods example μ 0 = 8.0 ). Since the null hypothesis is accepted unless there is strong evidence to the contrary, the test procedure is based on the initial assumption that H 0 is true. This point is so important that we will repeat it in a display:

The test procedure is based on the initial assumption that H 0 is true.

The criterion for judging between H 0 and H a based on the sample data is: if the value of X - would be highly unlikely to occur if H 0 were true, but favors the truth of H a , then we reject H 0 in favor of H a . Otherwise we do not reject H 0 .

Supposing for now that X - follows a normal distribution, when the null hypothesis is true the density function for the sample mean X - must be as in Figure 8.1 "The Density Curve for " : a bell curve centered at μ 0 . Thus if H 0 is true then X - is likely to take a value near μ 0 and is unlikely to take values far away. Our decision procedure therefore reduces simply to:

• if H a has the form H a : μ < μ 0 then reject H 0 if x - is far to the left of μ 0 ;
• if H a has the form H a : μ > μ 0 then reject H 0 if x - is far to the right of μ 0 ;
• if H a has the form H a : μ ≠ μ 0 then reject H 0 if x - is far away from μ 0 in either direction.

Figure 8.1 The Density Curve for X - if H 0 Is True

Think of the respirator example, for which the null hypothesis is H 0 : μ = 75 , the claim that the average time air is delivered for all respirators is 75 minutes. If the sample mean is 75 or greater then we certainly would not reject H 0 (since there is no issue with an emergency respirator delivering air even longer than claimed).

If the sample mean is slightly less than 75 then we would logically attribute the difference to sampling error and also not reject H 0 either.

Values of the sample mean that are smaller and smaller are less and less likely to come from a population for which the population mean is 75. Thus if the sample mean is far less than 75, say around 60 minutes or less, then we would certainly reject H 0 , because we know that it is highly unlikely that the average of a sample would be so low if the population mean were 75. This is the rare event criterion for rejection: what we actually observed ( X - < 60 ) would be so rare an event if μ = 75 were true that we regard it as much more likely that the alternative hypothesis μ < 75 holds.

In summary, to decide between H 0 and H a in this example we would select a “ rejection region An interval or union of intervals such that the null hypothesis is rejected if and only if the statistic of interest lies in this region. ” of values sufficiently far to the left of 75, based on the rare event criterion, and reject H 0 if the sample mean X - lies in the rejection region, but not reject H 0 if it does not.

The Rejection Region

Each different form of the alternative hypothesis H a has its own kind of rejection region:

• if (as in the respirator example) H a has the form H a : μ < μ 0 , we reject H 0 if x - is far to the left of μ 0 , that is, to the left of some number C , so the rejection region has the form of an interval (−∞, C ];
• if (as in the textbook example) H a has the form H a : μ > μ 0 , we reject H 0 if x - is far to the right of μ 0 , that is, to the right of some number C , so the rejection region has the form of an interval [ C ,∞);
• if (as in the baked good example) H a has the form H a : μ ≠ μ 0 , we reject H 0 if x - is far away from μ 0 in either direction, that is, either to the left of some number C or to the right of some other number C ′, so the rejection region has the form of the union of two intervals (−∞, C ]∪[ C ′,∞).

The key issue in our line of reasoning is the question of how to determine the number C or numbers C and C ′, called the critical value or critical values of the statistic, that determine the rejection region.

The critical value The number or one of a pair of numbers that determines the rejection region. or critical values of a test of hypotheses are the number or numbers that determine the rejection region.

Suppose the rejection region is a single interval, so we need to select a single number C . Here is the procedure for doing so. We select a small probability, denoted α , say 1%, which we take as our definition of “rare event:” an event is “rare” if its probability of occurrence is less than α . (In all the examples and problems in this text the value of α will be given already.) The probability that X - takes a value in an interval is the area under its density curve and above that interval, so as shown in Figure 8.2 (drawn under the assumption that H 0 is true, so that the curve centers at μ 0 ) the critical value C is the value of X - that cuts off a tail area α in the probability density curve of X - . When the rejection region is in two pieces, that is, composed of two intervals, the total area above both of them must be α , so the area above each one is α ∕ 2 , as also shown in Figure 8.2 .

The number α is the total area of a tail or a pair of tails.

In the context of Note 8.9 "Example 2" , suppose that it is known that the population is normally distributed with standard deviation σ = 0.15 gram, and suppose that the test of hypotheses H 0 : μ = 8.0 versus H a : μ ≠ 8.0 will be performed with a sample of size 5. Construct the rejection region for the test for the choice α = 0.10 . Explain the decision procedure and interpret it.

If H 0 is true then the sample mean X - is normally distributed with mean and standard deviation

Since H a contains the ≠ symbol the rejection region will be in two pieces, each one corresponding to a tail of area α ∕ 2 = 0.10 ∕ 2 = 0.05 . From Figure 12.3 "Critical Values of " , z 0.05 = 1.645 , so C and C ′ are 1.645 standard deviations of X - to the right and left of its mean 8.0:

The result is shown in Figure 8.3 "Rejection Region for the Choice " .

Figure 8.3 Rejection Region for the Choice α = 0.10

The decision procedure is: take a sample of size 5 and compute the sample mean x - . If x - is either 7.89 grams or less or 8.11 grams or more then reject the hypothesis that the average amount of fat in all servings of the product is 8.0 grams in favor of the alternative that it is different from 8.0 grams. Otherwise do not reject the hypothesis that the average amount is 8.0 grams.

The reasoning is that if the true average amount of fat per serving were 8.0 grams then there would be less than a 10% chance that a sample of size 5 would produce a mean of either 7.89 grams or less or 8.11 grams or more. Hence if that happened it would be more likely that the value 8.0 is incorrect (always assuming that the population standard deviation is 0.15 gram).

Because the rejection regions are computed based on areas in tails of distributions, as shown in Figure 8.2 , hypothesis tests are classified according to the form of the alternative hypothesis in the following way.

If H a has the form μ ≠ μ 0 the test is called a two-tailed test .

If H a has the form μ < μ 0 the test is called a left-tailed test .

If H a has the form μ > μ 0 the test is called a right-tailed test .

Each of the last two forms is also called a one-tailed test .

Two Types of Errors

The format of the testing procedure in general terms is to take a sample and use the information it contains to come to a decision about the two hypotheses. As stated before our decision will always be either

• reject the null hypothesis H 0 in favor of the alternative H a presented, or
• do not reject the null hypothesis H 0 in favor of the alternative H a presented.

There are four possible outcomes of hypothesis testing procedure, as shown in the following table:

As the table shows, there are two ways to be right and two ways to be wrong. Typically to reject H 0 when it is actually true is a more serious error than to fail to reject it when it is false, so the former error is labeled “Type I” and the latter error “Type II.”

In a test of hypotheses, a Type I error Rejection of a true null hypothesis. is the decision to reject H 0 when it is in fact true. A Type II error Failure to reject a false null hypothesis. is the decision not to reject H 0 when it is in fact not true.

Unless we perform a census we do not have certain knowledge, so we do not know whether our decision matches the true state of nature or if we have made an error. We reject H 0 if what we observe would be a “rare” event if H 0 were true. But rare events are not impossible: they occur with probability α . Thus when H 0 is true, a rare event will be observed in the proportion α of repeated similar tests, and H 0 will be erroneously rejected in those tests. Thus α is the probability that in following the testing procedure to decide between H 0 and H a we will make a Type I error.

The number α that is used to determine the rejection region is called the level of significance of the test The probability α that defines an event as “rare;” the probability that the test procedure will lead to a Type I error. . It is the probability that the test procedure will result in a Type I error.

The probability of making a Type II error is too complicated to discuss in a beginning text, so we will say no more about it than this: for a fixed sample size, choosing α smaller in order to reduce the chance of making a Type I error has the effect of increasing the chance of making a Type II error. The only way to simultaneously reduce the chances of making either kind of error is to increase the sample size.

Standardizing the Test Statistic

Hypotheses testing will be considered in a number of contexts, and great unification as well as simplification results when the relevant sample statistic is standardized by subtracting its mean from it and then dividing by its standard deviation. The resulting statistic is called a standardized test statistic . In every situation treated in this and the following two chapters the standardized test statistic will have either the standard normal distribution or Student’s t -distribution.

A standardized test statistic The standardized statistic used in performing the test. for a hypothesis test is the statistic that is formed by subtracting from the statistic of interest its mean and dividing by its standard deviation.

For example, reviewing Note 8.14 "Example 3" , if instead of working with the sample mean X - we instead work with the test statistic

then the distribution involved is standard normal and the critical values are just ± z 0.05 . The extra work that was done to find that C = 7.89 and C ′ = 8.11 is eliminated. In every hypothesis test in this book the standardized test statistic will be governed by either the standard normal distribution or Student’s t -distribution. Information about rejection regions is summarized in the following tables:

Every instance of hypothesis testing discussed in this and the following two chapters will have a rejection region like one of the six forms tabulated in the tables above.

No matter what the context a test of hypotheses can always be performed by applying the following systematic procedure, which will be illustrated in the examples in the succeeding sections.

Systematic Hypothesis Testing Procedure: Critical Value Approach

• Identify the null and alternative hypotheses.
• Identify the relevant test statistic and its distribution.
• Compute from the data the value of the test statistic.
• Construct the rejection region.
• Compare the value computed in Step 3 to the rejection region constructed in Step 4 and make a decision. Formulate the decision in the context of the problem, if applicable.

The procedure that we have outlined in this section is called the “Critical Value Approach” to hypothesis testing to distinguish it from an alternative but equivalent approach that will be introduced at the end of Section 8.3 "The Observed Significance of a Test" .

Key Takeaways

• A test of hypotheses is a statistical process for deciding between two competing assertions about a population parameter.
• The testing procedure is formalized in a five-step procedure.

State the null and alternative hypotheses for each of the following situations. (That is, identify the correct number μ 0 and write H 0 : μ = μ 0 and the appropriate analogous expression for H a .)

• The average July temperature in a region historically has been 74.5°F. Perhaps it is higher now.
• The average weight of a female airline passenger with luggage was 145 pounds ten years ago. The FAA believes it to be higher now.
• The average stipend for doctoral students in a particular discipline at a state university is $14,756. The department chairman believes that the national average is higher.
• The average room rate in hotels in a certain region is $82.53. A travel agent believes that the average in a particular resort area is different.
• The average farm size in a predominately rural state was 69.4 acres. The secretary of agriculture of that state asserts that it is less today.
• The average time workers spent commuting to work in Verona five years ago was 38.2 minutes. The Verona Chamber of Commerce asserts that the average is less now.
• The mean salary for all men in a certain profession is $58,291. A special interest group thinks that the mean salary for women in the same profession is different.
• The accepted figure for the caffeine content of an 8-ounce cup of coffee is 133 mg. A dietitian believes that the average for coffee served in a local restaurants is higher.
• The average yield per acre for all types of corn in a recent year was 161.9 bushels. An economist believes that the average yield per acre is different this year.
• An industry association asserts that the average age of all self-described fly fishermen is 42.8 years. A sociologist suspects that it is higher.

Describe the two types of errors that can be made in a test of hypotheses.

Under what circumstance is a test of hypotheses certain to yield a correct decision?

• H 0 : μ = 74.5 vs. H a : μ > 74.5
• H 0 : μ = 145 vs. H a : μ > 145
• H 0 : μ = 14756 vs. H a : μ > 14756
• H 0 : μ = 82.53 vs. H a : μ ≠ 82.53
• H 0 : μ = 69.4 vs. H a : μ < 69.4

A Type I error is made when a true H 0 is rejected. A Type II error is made when a false H 0 is not rejected.

6a.1 - Introduction to Hypothesis Testing

Basic terms.

The first step in hypothesis testing is to set up two competing hypotheses. The hypotheses are the most important aspect. If the hypotheses are incorrect, your conclusion will also be incorrect.

The two hypotheses are named the null hypothesis and the alternative hypothesis.

The goal of hypothesis testing is to see if there is enough evidence against the null hypothesis. In other words, to see if there is enough evidence to reject the null hypothesis. If there is not enough evidence, then we fail to reject the null hypothesis.

Consider the following example where we set up these hypotheses.

Example 6-1

A man, Mr. Orangejuice, goes to trial and is tried for the murder of his ex-wife. He is either guilty or innocent. Set up the null and alternative hypotheses for this example.

Putting this in a hypothesis testing framework, the hypotheses being tested are:

• The man is guilty
• The man is innocent

Let's set up the null and alternative hypotheses.

\(H_0\colon \) Mr. Orangejuice is innocent

\(H_a\colon \) Mr. Orangejuice is guilty

Remember that we assume the null hypothesis is true and try to see if we have evidence against the null. Therefore, it makes sense in this example to assume the man is innocent and test to see if there is evidence that he is guilty.

The Logic of Hypothesis Testing

We want to know the answer to a research question. We determine our null and alternative hypotheses. Now it is time to make a decision.

The decision is either going to be...

• reject the null hypothesis or...
• fail to reject the null hypothesis.

Consider the following table. The table shows the decision/conclusion of the hypothesis test and the unknown "reality", or truth. We do not know if the null is true or if it is false. If the null is false and we reject it, then we made the correct decision. If the null hypothesis is true and we fail to reject it, then we made the correct decision.

So what happens when we do not make the correct decision?

When doing hypothesis testing, two types of mistakes may be made and we call them Type I error and Type II error. If we reject the null hypothesis when it is true, then we made a type I error. If the null hypothesis is false and we failed to reject it, we made another error called a Type II error.

Types of errors

The “reality”, or truth, about the null hypothesis is unknown and therefore we do not know if we have made the correct decision or if we committed an error. We can, however, define the likelihood of these events.

\(\alpha\) and \(\beta\) are probabilities of committing an error so we want these values to be low. However, we cannot decrease both. As \(\alpha\) decreases, \(\beta\) increases.

Example 6-1 Cont'd...

A man, Mr. Orangejuice, goes to trial and is tried for the murder of his ex-wife. He is either guilty or not guilty. We found before that...

• \( H_0\colon \) Mr. Orangejuice is innocent
• \( H_a\colon \) Mr. Orangejuice is guilty

Interpret Type I error, \(\alpha \), Type II error, \(\beta \).

As you can see here, the Type I error (putting an innocent man in jail) is the more serious error. Ethically, it is more serious to put an innocent man in jail than to let a guilty man go free. So to minimize the probability of a type I error we would choose a smaller significance level.

An inspector has to choose between certifying a building as safe or saying that the building is not safe. There are two hypotheses:

• Building is safe
• Building is not safe

Set up the null and alternative hypotheses. Interpret Type I and Type II error.

\( H_0\colon\) Building is not safe vs \(H_a\colon \) Building is safe

Power and \(\beta \) are complements of each other. Therefore, they have an inverse relationship, i.e. as one increases, the other decreases.

9.2 Outcomes and the Type I and Type II Errors

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H 0 and the decision to reject or not. The outcomes are summarized in the following table:

The four possible outcomes in the table are:

• The decision is cannot reject H 0 when H 0 is true (correct decision).
• The decision is cannot accept H 0 when H 0 is true (incorrect decision known as a Type I error ). This case is described as "rejecting a good null". As we will see later, it is this type of error that we will guard against by setting the probability of making such an error. The goal is to NOT take an action that is an error.
• The decision is cannot reject H 0 when, in fact, H 0 is false (incorrect decision known as a Type II error ). This is called "accepting a false null". In this situation you have allowed the status quo to remain in force when it should be overturned. As we will see, the null hypothesis has the advantage in competition with the alternative.
• The decision is cannot accept H 0 when H 0 is false ( correct decision ).

Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities.

α = probability of a Type I error = P (Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true: rejecting a good null.

β = probability of a Type II error = P (Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false. (1 − β ) is called the Power of the Test .

α and β should be as small as possible because they are probabilities of errors.

Statistics allows us to set the probability that we are making a Type I error. The probability of making a Type I error is α. Recall that the confidence intervals in the last unit were set by choosing a value called Z α (or t α ) and the alpha value determined the confidence level of the estimate because it was the probability of the interval failing to capture the true mean (or proportion parameter p). This alpha and that one are the same.

The easiest way to see the relationship between the alpha error and the level of confidence is with the following figure.

In the center of Figure 9.2 is a normally distributed sampling distribution marked H 0 . This is a sampling distribution of X ¯ X ¯ and by the Central Limit Theorem it is normally distributed. The distribution in the center is marked H 0 and represents the distribution for the null hypotheses H 0 : µ = 100. This is the value that is being tested. The formal statements of the null and alternative hypotheses are listed below the figure.

The distributions on either side of the H 0 distribution represent distributions that would be true if H 0 is false, under the alternative hypothesis listed as H a . We do not know which is true, and will never know. There are, in fact, an infinite number of distributions from which the data could have been drawn if H a is true, but only two of them are on Figure 9.2 representing all of the others.

To test a hypothesis we take a sample from the population and determine if it could have come from the hypothesized distribution with an acceptable level of significance. This level of significance is the alpha error and is marked on Figure 9.2 as the shaded areas in each tail of the H 0 distribution. (Each area is actually α/2 because the distribution is symmetrical and the alternative hypothesis allows for the possibility for the value to be either greater than or less than the hypothesized value--called a two-tailed test).

If the sample mean marked as X ¯ 1 X ¯ 1 is in the tail of the distribution of H 0 , we conclude that the probability that it could have come from the H 0 distribution is less than alpha. We consequently state, "the null hypothesis cannot be accepted with (α) level of significance". The truth may be that this X ¯ 1 X ¯ 1 did come from the H 0 distribution, but from out in the tail. If this is so then we have falsely rejected a true null hypothesis and have made a Type I error. What statistics has done is provide an estimate about what we know, and what we control, and that is the probability of us being wrong, α.

We can also see in Figure 9.2 that the sample mean could be really from an H a distribution, but within the boundary set by the alpha level. Such a case is marked as X ¯ 2 X ¯ 2 . There is a probability that X ¯ 2 X ¯ 2 actually came from H a but shows up in the range of H 0 between the two tails. This probability is the beta error, the probability of accepting a false null.

Our problem is that we can only set the alpha error because there are an infinite number of alternative distributions from which the mean could have come that are not equal to H 0 . As a result, the statistician places the burden of proof on the alternative hypothesis. That is, we will not reject a null hypothesis unless there is a greater than 90, or 95, or even 99 percent probability that the null is false: the burden of proof lies with the alternative hypothesis. This is why we called this the tyranny of the status quo earlier.

By way of example, the American judicial system begins with the concept that a defendant is "presumed innocent". This is the status quo and is the null hypothesis. The judge will tell the jury that they cannot find the defendant guilty unless the evidence indicates guilt beyond a "reasonable doubt" which is usually defined in criminal cases as 95% certainty of guilt. If the jury cannot accept the null, innocent, then action will be taken, jail time. The burden of proof always lies with the alternative hypothesis. (In civil cases, the jury needs only to be more than 50% certain of wrongdoing to find culpability, called "a preponderance of the evidence").

The example above was for a test of a mean, but the same logic applies to tests of hypotheses for all statistical parameters one may wish to test.

The following are examples of Type I and Type II errors.

Example 9.4

Suppose the null hypothesis, H 0 , is: Navah's rock climbing equipment is safe.

Type I error : Navah thinks that his rock climbing equipment may not be safe when, in fact, it really is safe.

Type II error : Navah thinks that her rock climbing equipment may be safe when, in fact, it is not safe.

α = probability that Navah thinks her rock climbing equipment may not be safe when, in fact, it really is safe. β = probability that Navah thinks her rock climbing equipment may be safe when, in fact, it is not safe.

Notice that, in this case, the error with the greater consequence is the Type II error. (If Navah thinks her rock climbing equipment is safe, she will go ahead and use it.)

This is a situation described as "accepting a false null".

Suppose the null hypothesis, H 0 , is: the blood cultures contain no traces of pathogen X . State the Type I and Type II errors.

Example 9.5

Suppose the null hypothesis, H 0 , is: The victim of an automobile accident is alive when he arrives at the emergency room of a hospital. This is the status quo and requires no action if it is true. If the null hypothesis cannot be accepted then action is required and the hospital will begin appropriate procedures.

Type I error : The emergency crew thinks that the victim is dead when, in fact, the victim is alive. Type II error : The emergency crew does not know if the victim is alive when, in fact, the victim is dead.

α = probability that the emergency crew thinks the victim is dead when, in fact, he is really alive = P (Type I error). β = probability that the emergency crew does not know if the victim is alive when, in fact, the victim is dead = P (Type II error).

The error with the greater consequence is the Type I error. (If the emergency crew thinks the victim is dead, they will not treat him.)

Suppose the null hypothesis, H 0 , is: a patient is not sick. Which type of error has the greater consequence, Type I or Type II?

Example 9.6

A company called Genetic Labs claims to be able to increase the likelihood that a pregnancy will result in a male being born. Statisticians want to test the claim. Suppose that the null hypothesis, H 0 , is: Genetic Labs has no effect on sex outcome. The status quo is that the claim is false. The burden of proof always falls to the person making the claim, in this case the Genetics Lab.

Type I error : This results when a true null hypothesis is rejected. In the context of this scenario, we would state that we believe that Genetic Labs influences the sex outcome, when in fact it has no effect. The probability of this error occurring is denoted by the Greek letter alpha, α .

Type II error : This results when we fail to reject a false null hypothesis. In context, we would state that Genetic Labs does not influence the sex outcome of a pregnancy when, in fact, it does. The probability of this error occurring is denoted by the Greek letter beta, β .

The error of greater consequence would be the Type I error since people would use the Genetic Labs product in hopes of increasing the chances of having a male.

“Red tide” is a bloom of poison-producing algae–a few different species of a class of plankton called dinoflagellates. When the weather and water conditions cause these blooms, shellfish such as clams living in the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of Marine Fisheries (DMF) monitors levels of the toxin in shellfish by regular sampling of shellfish along the coastline. If the mean level of toxin in clams exceeds 800 μg (micrograms) of toxin per kg of clam meat in any area, clam harvesting is banned there until the bloom is over and levels of toxin in clams subside. Describe both a Type I and a Type II error in this context, and state which error has the greater consequence.

Example 9.7

A certain experimental drug claims a cure rate of at least 75% for males with prostate cancer. Describe both the Type I and Type II errors in context. Which error is the more serious?

Type I : A cancer patient believes the cure rate for the drug is less than 75% when it actually is at least 75%.

Type II : A cancer patient believes the experimental drug has at least a 75% cure rate when it has a cure rate that is less than 75%.

In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75% of the time, this most likely will influence the patient’s (and doctor’s) choice about whether to use the drug as a treatment option.

Determine both Type I and Type II errors for the following scenario:

Assume a null hypothesis, H 0 , that states the percentage of adults with jobs is at least 88%.

Identify the Type I and Type II errors from these four statements.

• Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%
• Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.
• Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.
• Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%.

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• Authors: Alexander Holmes, Barbara Illowsky, Susan Dean
• Publisher/website: OpenStax
• Book title: Introductory Business Statistics 2e
• Publication date: Dec 13, 2023
• Location: Houston, Texas
• Book URL: https://openstax.org/books/introductory-business-statistics-2e/pages/1-introduction
• Section URL: https://openstax.org/books/introductory-business-statistics-2e/pages/9-2-outcomes-and-the-type-i-and-type-ii-errors

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• Why Study Statistics?
• Descriptive & Inferential Statistics
• Fundamental Elements of Statistics
• Quantitative and Qualitative Data
• Measurement Data Levels
• Collecting Data
• Ethics in Statistics
• Describing Qualitative Data
• Describing Quantitative Data
• Stem-and-Leaf Plots
• Measures of Central Tendency
• Measures of Variability
• Describing Data using the Mean and Standard Deviation
• Measures of Position
• Counting Techniques
• Simple & Compound Events
• Independent and Dependent Events
• Mutually Exclusive and Non-Mutually Exclusive Events
• Permutations and Combinations
• Normal Distribution
• Central Limit Theorem
• Confidence Intervals
• Determining the Sample Size

Hypothesis Testing

• Hypothesis Testing Process

What is hypothesis testing?

Hypothesis testing in statistics is a way to test sample(s) with a population parameter. You can use hypothesis testing to test the data whether the data you gathered is a meaningful result. More specifically, how probable is your sample data compared to population parameters or to other samples. 

The null hypothesis,  (\(H_0\)), represents the status quo. Meaning the evidence collected is not strong enough to warrant a change.

The  alternative hypothesis,  (\(H_1\) or \(H_A\)),   represents what is required to make a change to the status quo. That means there is statistical evidence using probability that it is different from the status quo. 

For example, you believe your internet is slower than the average. You decide to collect the data on your connection speed every day for a month. The null hypothesis would be that there is no significant difference between your connection speed and the average. And the alternative hypothesis would be that your internet is slower.

So what is considered convincing evidence?  To consider what evidence is convincing we use a  test statistic  to decide whether to reject the null hypothesis or not. The distribution is divided up into two regions, one to reject the null hypothesis and one that remains the same. These regions are determined by the  significance level  or  critical value . 

The  test statistic  depends on the distribution. For normal or approximately normal distributions, we can use the z- or t-statistics.

The  significance level  or  critical value , splits the rejection region from the non-rejection region. For example, a significance level \(\alpha =0.05) in a two-tailed test for a normal approximation results in a critical value of 1.96 (found using z-table).

Two-tailed test:  The diagrams above represents a two-tailed hypothesis test. This means that there is a rejection region at the two extreme ends of the distribution. The null and alternative hypothesis is stated as the following when comparing it to the population mean:

\begin{align} H_0 &= \mu \\ H_1 &\neq \mu \end{align}

One-tailed test:

There are two different  one-tailed  tests. One where the rejection region is in the upper extreme of the distribution and the other the lower extreme:

The upper one-tailed test is usually stated: 

\begin{align} H_0 &\leq \mu \\ H_1 &> \mu \end{align}

The lower one-tailed test:

\begin{align} H_0 &\geq \mu \\ H_1 &< \mu \end{align}

In some text, the \(\leq\) and \(\geq\) signs are replaced with the \(=\) sign.

Making a decision . The decision during hypothesis testing is whether to keep the status quo or there is evidence to consider the alternative hypothesis. If the test statistic falls inside the rejection region, then there is evidence to reject the status quo. This does not mean that the alternative hypothesis is true population parameter, it just means there is evidence that this sample is different from the population parameter.

Setting up the null and alternative hypothesis

Example 1: A formal hypothesis test is to be conducted using the claim that the mean height of men is equal to 174.1 cm. What is the null and alternative hypothesis?

Solution:  Since the test is to see if the population mean is equal to 174.1 cm. Any mean that is significantly  higher  or  lower  will reject the claim. This is a  two-tailed  test  because there are rejection regions both in the upper and lower extremes.

The null hypothesis or status quo is that the mean height of men is equal to 174.1 cm.

\[H_0=174.1\,cm\]

The alternative hypothesis is that there is evidence that there is a mean height statistically higher or lower than 174.1 cm.

\[H_1\neq 174.1\,cm\]

Example 2: Fewer than 95% of adults have a cell phone. In a poll of 1128 adults, 87% said they have a cell phone. 

Solution:  In this example, the status quo is that fewer than 95% of adults have a cell phone. The sample of 1128 adults shows a mean that may is lower. You want to test if this sample mean is significantly lower to warrant a change to the population mean of 95%.

You are performing a lower one-tailed test because you want to show this sample is significantly lower. A value significantly higher does not challenge this claim for this test. Therefore, the null and alternative hypothesis are (note that the population mean, 95%, is used and not the sample mean):

\begin{align} H_0 &\geq 95\% \\  H_1 &< 95\% \end{align}

Example 3: A survey of online reports that 68% of college administration believe that their online education courses are as good as or superior that utilize traditional face-to-face instruction. You collect a survey that shows that 75% of college administration makes such a claim.

Solution:  You are testing that your sample is showing a significant increase to the original claim. This is an upper one-tailed test since the rejection region of the claim will be values that are statistically higher.

\begin{align} H_0 &\leq 68\% \\  H_1 &> 68\% \end{align}

Significance level and Type I and II errors

The  significance level  \(\alpha\) for a hypothesis test is the probability used as the cutoff for what constitutes significant evidence against the null hypothesis. 

For example, a significance level of \(\alpha=0.05\) in an upper one-tailed test in a normal distribution represents the following rejection region

The critical value of \(z=1.645\) was found using the z-table looking up the top 5% of the normal distribution. 

However, a significance level of \(\alpha=0.05\) during a  two-tailed test means that the upper and lower rejections regions total 5%. That means on each side there is a rejection region of 2.5% or 0.025.

The critical value \(z=\pm 1.96\) was found by looking for the upper and lower 2.5% of the z-table.

Note that determining if it is a  one-tailed  or  two-tailed  test when setting up the alternative hypothesis is important to setting up the rejection region and finding the critical value.   

You may think "Why don't I just the rejection region as large as possible, so I can verify that it is different?" Or "Let's make the rejection region as small as possible, so the status quo remains." This is not the best method to claim something is statistically different because of the error that is created. There are two types of errors:  Type I  and  Type II . 

Type I error:  The mistake of rejecting the null hypothesis when it is actually true. \(\alpha\), same as the significance level, is the type I error. So \(\alpha=0.05\) means that there is a 5% chance of rejecting the status quo, when in fact the status quo was the right decision. 

Type II error:  The mistake of failing to reject the null hypothesis when it is actually false. Type II error is represented by the symbol \(\beta\) and can be calculated using the Power of a Test .

 As a result, there are four outcomes possible. Two of those outcomes are correct, and two are errors. 

\(\alpha\) and \(\beta\) are inversely related (as one increases, the other decreases). Therefore, it is impossible to reduce both errors to 0. In an actual study, you will be choosing your own significance level, so keep type I and I errors in mind. 

Example 1: There is a medicine that claims to cure a disease. Describe the type I and II errors.

Example 2: You decide to attend an online lecture because it will discuss a topic you are not familiar with.

Each possibility contains a probability value depending on the information provided.

• Practice Questions - Hypothesis Testing Concepts

• << Previous: Determining the Sample Size
• Next: Hypothesis Testing Process >>
• Last Updated: Apr 20, 2023 12:47 PM
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