hypothesis generation in business analytics pdf

Hypothesis Testing in Business Analytics – A Beginner’s Guide

Introduction  

Organizations must understand how their decisions can impact the business in this data-driven age. Hypothesis testing enables organizations to analyze and examine their decisions’ causes and effects before making important management decisions. Based on research by the Harvard Business School Online, prior to making any decision, organizations like to explore the advantages of hypothesis testing and the investigation of decisions in a proper “laboratory” setting. By performing such tests, organizations can be more confident with their decisions. Read on to learn all about hypothesis testing , o ne of the essential concepts in Business Analytics.  

What Is Hypothesis Testing?  

To learn about hypothesis testing, it is crucial that you first understand what the term hypothesis is.   

A hypothesis statement or hypothesis tries to explain why something happened or what may happen under specific conditions. A hypothesis can also help understand how various variables are connected to each other. These are generally compiled as if-then statements; for example, “If something specific were to happen, then a specific condition will come true and vice versa.” Thus, the hypothesis is an arithmetical method of testing a hypothesis or an assumption that has been stated in the hypothesis.  

Turning into a decision-maker who is driven by data can add several advantages to an organization, such as allowing one to recognize new opportunities to follow and reducing the number of threats. In analytics, a hypothesis is nothing but an assumption or a supposition made about a specific population parameter, such as any measurement or quantity about the population that is set and that can be used as a value to the distribution variable. General examples of parameters used in hypothesis testing are variance and mean. In simpler words, hypothesis testing in business analytics is a method that helps researchers, scientists, or anyone for that matter, test the legitimacy or the authenticity of their hypotheses or claims about real-life or real-world events.  

To understand the example of hypothesis testing in business analytics, consider a restaurant owner interested in learning how adding extra house sauce to their chicken burgers can impact customer satisfaction. Or, you could also consider a social media marketing organization. A hypothesis test can be set up to explain how an increase in labor impacts productivity. Thus, hypothesis testing aims to discover the connection between two or more than two variables in the experimental setting.  

How Does Hypothesis Testing Work?  

Generally, each research begins with a hypothesis; the investigator makes a certain claim and experiments to prove that the claim is false or true. For example, if you claim that students drinking milk before class accomplish tasks better than those who do not, then this is a kind of hypothesis that can be refuted or confirmed using an experiment. There are different kinds of hypotheses. They are:  

• Simple Hypothesis : Simple hypothesis, also known as a basic hypothesis, proposes that an independent variable is accountable for the corresponding dependent variable. In simpler words, the occurrence of independent variable results in the existence of the dependent variable. Generally, simple hypotheses are thought of as true and they create a causal relationship between the two variables. One example of a simple hypothesis is smoking cigarettes daily leads to cancer.  
• Complex Hypothesis : This type of hypothesis is also termed a modal. It holds for the relationship between two variables that are independent and result in a dependent variable. This means that the amalgamation of independent variables results in the dependent variables. An example of this kind of hypothesis can be “adults who don’t drink and smoke are less likely to have liver-related problems.  
• Null Hypothesis : A null hypothesis is created when a researcher thinks that there is no connection between the variables that are being observed. An example of this kind of hypothesis can be “A student’s performance is not impacted if they drink tea or coffee before classes.  
• Alternative Hypothesis : If a researcher wants to disapprove of a null hypothesis, then the researcher has to develop an opposite assumption—known as an alternative hypothesis. For example, beginning your day with tea instead of coffee can keep you more alert.  
• Logical Hypothesis: A proposed explanation supported by scant data is called a logical hypothesis. Generally, you wish to test your hypotheses or postulations by converting a logical hypothesis into an empirical hypothesis. For example, waking early helps one to have a productive day.  
• Empirical Hypothesis : This type of hypothesis is based on real evidence, evidence that is verifiable by observation as opposed to something that is correct in theory or by some kind of reckoning or logic. This kind of hypothesis depends on various variables that can result in specific outcomes. For example, individuals eating more fish can run faster than those eating meat.   
• Statistical Hypothesis : This kind of hypothesis is most common in systematic investigations that involve a huge target audience. For example, in Louisiana, 45% of students have middle-income parents.  

Four Steps of Hypothesis Testing  

There are four main steps in hypothesis testing in business analytics :  

Step 1: State the Null and Alternate Hypothesis  

After the initial research hypothesis, it is essential to restate it as a null (Ho) hypothesis and an alternate (Ha) hypothesis so that it can be tested mathematically.  

Step 2: Collate Data  

For a test to be valid, it is essential to do some sampling and collate data in a manner designed to test the hypothesis. If your data are not representative, then statistical inferences cannot be made about the population you are trying to analyze.  

Step 3: Perform a Statistical Test  

Various statistical tests are present, but all of them depend on the contrast of within-group variance (how to spread out the data in a group) against between-group variance (how dissimilar the groups are from one another).  

Step 4: Decide to Reject or Accept Your Null Hypothesis  

Based on the result of your statistical test, you need to decide whether you want to accept or reject your null hypothesis.  

Hypothesis Testing in Business   

When we talk about data-driven decision-making, a specific amount of risk can deceive a professional. This could result from flawed observations or thinking inaccurate or incomplete information , or unknown variables. The threat over here is that if key strategic decisions are made on incorrect insights, it can lead to catastrophic outcomes for an organization. The actual importance of hypothesis testing is that it enables professionals to analyze their assumptions and theories before putting them into action. This enables an organization to confirm the accuracy of its analysis before making key decisions.  

Key Considerations for Hypothesis Testing  

Let us look at the following key considerations of hypothesis testing:  

• Alternative Hypothesis and Null Hypothesis : If a researcher wants to disapprove of a null hypothesis, then the researcher has to develop an opposite assumption—known as an alternative hypothesis. A null hypothesis is created when a researcher thinks that there is no connection between the variables that are being observed.  
• Significance Level and P-Value : The statistical significance level is generally expressed as a p-value that lies between 0 and 1. The lesser the p-value, the more it suggests that you reject the null hypothesis. A p-value of less than 0.05 (generally ≤ 0.05) is significant statistically.  
• One-Sided vs. Two-Sided Testing : One-sided tests suggest the possibility of an effect in a single direction only. Two-sided tests test for the likelihood of the effect in two directions—negative and positive. One-sided tests comprise more statistical power to identify an effect in a single direction than a two-sided test with the same significance level and design.   
• Sampling: For hypothesis testing , you are required to collate a sample of data that has to be examined. In hypothesis testing, an analyst can test a statistical sample with the aim of providing proof of the credibility of the null hypothesis. Statistical analysts can test a hypothesis by examining and measuring a random sample of the population that is being examined.  

Real-World Example of Hypothesis Testing  

The following two examples give a glimpse of the various situations in which hypothesis testing is used in real-world scenarios.  

Example: BioSciences  

Hypothesis tests are frequently used in biological sciences. For example, consider that a biologist is sure that a certain kind of fertilizer will lead to better growth of plants which is at present 10 inches. To test this, the fertilizer is sprayed on the plants in the laboratory for a month. A hypothesis test is then done using the following:  

• H0: μ = 10 inches (the fertilizer has no effect on the plant growth)  
• HA: μ > 10 inches (the fertilizer leads to an increase in plant growth)  

Suppose the p-value is lesser than the significance level (e.g., α = .04). In that case, the null hypothesis can be rejected, and it can be concluded that the fertilizer results in increased plant growth.  

Example: Clinical Trials  

Consider an example where a doctor feels that a new medicine can decrease blood sugar in patients. To confirm this, he can measure the sugar of 20 diabetic patients prior to and after administering the new drug for a month. A hypothesis test is then done using the following:  

• H0: μafter = μbefore (the blood sugar is the same as before and after administering the new drug)  
• HA: μafter < μbefore (the blood sugar is less after the drug)  

If the p-value is less than the significance level (e.g., α = .04), then the null hypothesis can be rejected, and it can be proven that the new drug leads to reduced blood sugar.  

Conclusion  

Now you are aware of the need for hypotheses in Business Analytics . A hypothesis is not just an assumption— it has to be based on prior knowledge and theories. It also needs to be, which means that you can accept or reject it using scientific research methods (such as observations, experiments, and statistical data analysis). Most genuine Hypothesis testing programs teach you how to use hypothesis testing in real-world scenarios. If you are interested in getting a certificate degree in Integrated Program In Business Analytics , UNext Jigsaw is highly recommended.

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Chapter 4. Hypothesis Testing

Hypothesis testing is the other widely used form of inferential statistics. It is different from estimation because you start a hypothesis test with some idea of what the population is like and then test to see if the sample supports your idea. Though the mathematics of hypothesis testing is very much like the mathematics used in interval estimation, the inference being made is quite different. In estimation, you are answering the question, “What is the population like?” While in hypothesis testing you are answering the question, “Is the population like this or not?”

A hypothesis is essentially an idea about the population that you think might be true, but which you cannot prove to be true. While you usually have good reasons to think it is true, and you often hope that it is true, you need to show that the sample data support your idea. Hypothesis testing allows you to find out, in a formal manner, if the sample supports your idea about the population. Because the samples drawn from any population vary, you can never be positive of your finding, but by following generally accepted hypothesis testing procedures, you can limit the uncertainty of your results.

As you will learn in this chapter, you need to choose between two statements about the population. These two statements are the hypotheses. The first, known as the null hypothesis , is basically, “The population is like this.” It states, in formal terms, that the population is no different than usual. The second, known as the alternative hypothesis , is, “The population is like something else.” It states that the population is different than the usual, that something has happened to this population, and as a result it has a different mean, or different shape than the usual case. Between the two hypotheses, all possibilities must be covered. Remember that you are making an inference about a population from a sample. Keeping this inference in mind, you can informally translate the two hypotheses into “I am almost positive that the sample came from a population like this” and “I really doubt that the sample came from a population like this, so it probably came from a population that is like something else”. Notice that you are never entirely sure, even after you have chosen the hypothesis, which is best. Though the formal hypotheses are written as though you will choose with certainty between the one that is true and the one that is false, the informal translations of the hypotheses, with “almost positive” or “probably came”, is a better reflection of what you actually find.

Hypothesis testing has many applications in business, though few managers are aware that that is what they are doing. As you will see, hypothesis testing, though disguised, is used in quality control, marketing, and other business applications. Many decisions are made by thinking as though a hypothesis is being tested, even though the manager is not aware of it. Learning the formal details of hypothesis testing will help you make better decisions and better understand the decisions made by others.

The next section will give an overview of the hypothesis testing method by following along with a young decision-maker as he uses hypothesis testing. Additionally, with the provided interactive Excel template, you will learn how the results of the examples from this chapter can be adjusted for other circumstances. The final section will extend the concept of hypothesis testing to categorical data, where we test to see if two categorical variables are independent of each other. The rest of the chapter will present some specific applications of hypothesis tests as examples of the general method.

The strategy of hypothesis testing

Usually, when you use hypothesis testing, you have an idea that the world is a little bit surprising; that it is not exactly as conventional wisdom says it is. Occasionally, when you use hypothesis testing, you are hoping to confirm that the world is not surprising, that it is like conventional wisdom predicts. Keep in mind that in either case you are asking, “Is the world different from the usual, is it surprising?” Because the world is usually not surprising and because in statistics you are never 100 per cent sure about what a sample tells you about a population, you cannot say that your sample implies that the world is surprising unless you are almost positive that it does. The dull, unsurprising, usual case not only wins if there is a tie, it gets a big lead at the start. You cannot say that the world is surprising, that the population is unusual, unless the evidence is very strong. This means that when you arrange your tests, you have to do it in a manner that makes it difficult for the unusual, surprising world to win support.

The first step in the basic method of hypothesis testing is to decide what value some measure of the population would take if the world was unsurprising. Second, decide what the sampling distribution of some sample statistic would look like if the population measure had that unsurprising value. Third, compute that statistic from your sample and see if it could easily have come from the sampling distribution of that statistic if the population was unsurprising. Fourth, decide if the population your sample came from is surprising because your sample statistic could not easily have come from the sampling distribution generated from the unsurprising population.

That all sounds complicated, but it is really pretty simple. You have a sample and the mean, or some other statistic, from that sample. With conventional wisdom, the null hypothesis that the world is dull, and not surprising, tells you that your sample comes from a certain population. Combining the null hypothesis with what statisticians know tells you what sampling distribution your sample statistic comes from if the null hypothesis is true. If you are almost positive that the sample statistic came from that sampling distribution, the sample supports the null. If the sample statistic “probably came” from a sampling distribution generated by some other population, the sample supports the alternative hypothesis that the population is “like something else”.

Imagine that Thad Stoykov works in the marketing department of Pedal Pushers, a company that makes clothes for bicycle riders. Pedal Pushers has just completed a big advertising campaign in various bicycle and outdoor magazines, and Thad wants to know if the campaign has raised the recognition of the Pedal Pushers brand so that more than 30 per cent of the potential customers recognize it. One way to do this would be to take a sample of prospective customers and see if at least 30 per cent of those in the sample recognize the Pedal Pushers brand. However, what if the sample is small and just barely 30 per cent of the sample recognizes Pedal Pushers? Because there is variance among samples, such a sample could easily have come from a population in which less than 30 per cent recognize the brand. If the population actually had slightly less than 30 per cent recognition, the sampling distribution would include quite a few samples with sample proportions a little above 30 per cent, especially if the samples are small. In order to be comfortable that more than 30 per cent of the population recognizes Pedal Pushers, Thad will want to find that a bit more than 30 per cent of the sample does. How much more depends on the size of the sample, the variance within the sample, and how much chance he wants to take that he’ll conclude that the campaign did not work when it actually did.

Let us follow the formal hypothesis testing strategy along with Thad. First, he must explicitly describe the population his sample could come from in two different cases. The first case is the unsurprising case, the case where there is no difference between the population his sample came from and most other populations. This is the case where the ad campaign did not really make a difference, and it generates the null hypothesis. The second case is the surprising case when his sample comes from a population that is different from most others. This is where the ad campaign worked, and it generates the alternative hypothesis. The descriptions of these cases are written in a formal manner. The null hypothesis is usually called H o . The alternative hypothesis is called either H 1 or H a . For Thad and the Pedal Pushers marketing department, the null hypothesis will be:

H o : proportion of the population recognizing Pedal Pushers brand < .30

and the alternative will be:

H a : proportion of the population recognizing Pedal Pushers brand >.30

Notice that Thad has stacked the deck against the campaign having worked by putting the value of the population proportion that means that the campaign was successful in the alternative hypothesis. Also notice that between H o and H a all possible values of the population proportion (>, =, and < .30) have been covered.

Second, Thad must create a rule for deciding between the two hypotheses. He must decide what statistic to compute from his sample and what sampling distribution that statistic would come from if the null hypothesis,  H o , is true. He also needs to divide the possible values of that statistic into usual and unusual ranges if the null is true. Thad’s decision rule will be that if his sample statistic has a usual value, one that could easily occur if H o is true, then his sample could easily have come from a population like that which described H o . If his sample’s statistic has a value that would be unusual if H o is true, then the sample probably comes from a population like that described in H a . Notice that the hypotheses and the inference are about the original population while the decision rule is about a sample statistic. The link between the population and the sample is the sampling distribution. Knowing the relative frequency of a sample statistic when the original population has a proportion with a known value is what allows Thad to decide what are usual and unusual values for the sample statistic.

The basic idea behind the decision rule is to decide, with the help of what statisticians know about sampling distributions, how far from the null hypothesis’ value for the population the sample value can be before you are uncomfortable deciding that the sample comes from a population like that hypothesized in the null. Though the hypotheses are written in terms of descriptive statistics about the population—means, proportions, or even a distribution of values—the decision rule is usually written in terms of one of the standardized sampling distributions—the t, the normal z, or another of the statistics whose distributions are in the tables at the back of statistics textbooks. It is the sampling distributions in these tables that are the link between the sample statistic and the population in the null hypothesis. If you learn to look at how the sample statistic is computed you will see that all of the different hypothesis tests are simply variations on a theme. If you insist on simply trying to memorize how each of the many different statistics is computed, you will not see that all of the hypothesis tests are conducted in a similar manner, and you will have to learn many different things rather than the variations of one thing.

Thad has taken enough statistics to know that the sampling distribution of sample proportions is normally distributed with a mean equal to the population proportion and a standard deviation that depends on the population proportion and the sample size. Because the distribution of sample proportions is normally distributed, he can look at the bottom line of a t-table and find out that only .05 of all samples will have a proportion more than 1.645 standard deviations above .30 if the null hypothesis is true. Thad decides that he is willing to take a 5 per cent chance that he will conclude that the campaign did not work when it actually did. He therefore decides to conclude that the sample comes from a population with a proportion greater than .30 that has heard of Pedal Pushers, if the sample’s proportion is more than 1.645 standard deviations above .30. After doing a little arithmetic (which you’ll learn how to do later in the chapter), Thad finds that his decision rule is to decide that the campaign was effective if the sample has a proportion greater than .375 that has heard of Pedal Pushers. Otherwise the sample could too easily have come from a population with a proportion equal to or less than .30.

The final step is to compute the sample statistic and apply the decision rule. If the sample statistic falls in the usual range, the data support H o , the world is probably unsurprising, and the campaign did not make any difference. If the sample statistic is outside the usual range, the data support H a , the world is a little surprising, and the campaign affected how many people have heard of Pedal Pushers. When Thad finally looks at the sample data, he finds that .39 of the sample had heard of Pedal Pushers. The ad campaign was successful!

A straightforward example: testing for goodness-of-fit

There are many different types of hypothesis tests, including many that are used more often than the goodness-of-fit test . This test will be used to help introduce hypothesis testing because it gives a clear illustration of how the strategy of hypothesis testing is put to use, not because it is used frequently. Follow this example carefully, concentrating on matching the steps described in previous sections with the steps described in this section. The arithmetic is not that important right now.

We will go back to Chapter 1 , where the Chargers’ equipment manager, Ann, at Camosun College, collected some data on the size of the Chargers players’ sport socks. Recall that she asked both the basketball and volleyball team managers to collect these data, shown in Table 4.2.

David, the marketing manager of the company that produces these socks, contacted Ann to tell her that he is planning to send out some samples to convince the Chargers players that wearing Easy Bounce socks will be more comfortable than wearing other socks. He needs to include an assortment of sizes in those packages and is trying to find out what sizes to include. The Production Department knows what mix of sizes they currently produce, and Ann has collected a sample of 97 basketball and volleyball players’ sock sizes. David needs to test to see if his sample supports the hypothesis that the collected sample from Camosun college players has the same distribution of sock sizes as the company is currently producing. In other words, is the distribution of Chargers players’ sock sizes a good fit to the distribution of sizes now being produced (see Table 4.2)?

From the Production Department, the current relative frequency distribution of Easy Bounce socks in production is shown in Table 4.3.

If the world is unsurprising, the players will wear the socks sized in the same proportions as other athletes, so David writes his hypotheses:

H o : Chargers players’ sock sizes are distributed just like current production.

H a : Chargers players’ sock sizes are distributed differently.

Ann’s sample has n =97. By applying the relative frequencies in the current production mix, David can find out how many players would be expected to wear each size if the sample was perfectly representative of the distribution of sizes in current production. This would give him a description of what a sample from the population in the null hypothesis would be like. It would show what a sample that had a very good fit with the distribution of sizes in the population currently being produced would look like.

Statisticians know the sampling distribution of a statistic that compares the expected  frequency of a sample with the actual, or observed , frequency. For a sample with c different classes (the sizes here), this statistic is distributed like χ 2 with c-1 df. The χ 2 is computed by the formula:

[latex]sample\;chi^2 = \sum{((O-E)^2)/E}[/latex]

O = observed frequency in the sample in this class

E = expected frequency in the sample in this class

The expected frequency, E, is found by multiplying the relative frequency of this class in the H o hypothesized population by the sample size. This gives you the number in that class in the sample if the relative frequency distribution across the classes in the sample exactly matches the distribution in the population.

Notice that χ 2 is always > 0 and equals 0 only if the observed is equal to the expected in each class. Look at the equation and make sure that you see that a larger value of  χ 2 goes with samples with large differences between the observed and expected frequencies.

David now needs to come up with a rule to decide if the data support H o or H a . He looks at the table and sees that for 5 df (there are 6 classes—there is an expected frequency for size 11 socks), only .05 of samples drawn from a given population will have a χ 2 > 11.07 and only .10 will have a χ 2 > 9.24. He decides that it would not be all that surprising if the players had a different distribution of sock sizes than the athletes who are currently buying Easy Bounce, since all of the players are women and many of the current customers are men. As a result, he uses the smaller .10 value of 9.24 for his decision rule. Now David must compute his sample χ 2 . He starts by finding the expected frequency of size 6 socks by multiplying the relative frequency of size 6 in the population being produced by 97, the sample size. He gets E = .06*97=5.82. He then finds O-E = 3-5.82 = -2.82, squares that, and divides by 5.82, eventually getting 1.37. He then realizes that he will have to do the same computation for the other five sizes, and quickly decides that a spreadsheet will make this much easier (see Table 4.4).

David performs his third step, computing his sample statistic, using the spreadsheet. As you can see, his sample χ 2 = 26.46, which is well into the unusual range that starts at 9.24 according to his decision rule. David has found that his sample data support the hypothesis that the distribution of sock sizes of the players is different from the distribution of sock sizes that are currently being manufactured. If David’s employer is going to market Easy Bounce socks to the BC college players, it is going to have to send out packages of samples that contain a different mix of sizes than it is currently making. If Easy Bounce socks are successfully marketed to the BC college players, the mix of sizes manufactured will have to be altered.

Now review what David has done to test to see if the data in his sample support the hypothesis that the world is unsurprising and that the players have the same distribution of sock sizes as the manufacturer is currently producing for other athletes. The essence of David’s test was to see if his sample χ 2 could easily have come from the sampling distribution of χ 2 ’s generated by taking samples from the population of socks currently being produced. Since his sample χ 2 would be way out in the tail of that sampling distribution, he judged that his sample data supported the other hypothesis, that there is a difference between the Chargers players and the athletes who are currently buying Easy Bounce socks.

Formally, David first wrote null and alternative hypotheses, describing the population his sample comes from in two different cases. The first case is the null hypothesis; this occurs if the players wear socks of the same sizes in the same proportions as the company is currently producing. The second case is the alternative hypothesis; this occurs if the players wear different sizes. After he wrote his hypotheses, he found that there was a sampling distribution that statisticians knew about that would help him choose between them. This is the χ 2 distribution. Looking at the formula for computing χ 2 and consulting the tables, David decided that a sample χ 2 value greater than 9.24 would be unusual if his null hypothesis was true. Finally, he computed his sample statistic and found that his χ 2 , at 26.46, was well above his cut-off value. David had found that the data in his sample supported the alternative χ 2 : that the distribution of the players’ sock sizes is different from the distribution that the company is currently manufacturing. Acting on this finding, David will include a different mix of sizes in the sample packages he sends to team coaches.

Testing population proportions

As you learned in Chapter 3 , sample proportions can be used to compute a statistic that has a known sampling distribution. Reviewing, the z-statistic is:

[latex]z = (p-\pi)/\sqrt{\dfrac{(\pi)(1-\pi)}{n}}[/latex]

p = the proportion of the sample with a certain characteristic

π = the proportion of the population with that characteristic

[latex]\sqrt{\dfrac{(\pi)(1-\pi)}{n}}[/latex] = the standard deviation (error) of the proportion of the population with that characteristic

As long as the two technical conditions of   π*n and (1-π)*n are held, these sample z-statistics are distributed normally so that by using the bottom line of the t-table, you can find what portion of all samples from a population with a given population proportion, π , have z-statistics within different ranges. If you look at the z-table, you can see that .95 of all samples from any population have z-statistics between ±1.96, for instance.

If you have a sample that you think is from a population containing a certain proportion, π , of members with some characteristic, you can test to see if the data in your sample support what you think. The basic strategy is the same as that explained earlier in this chapter and followed in the goodness-of-fit example: (a) write two hypotheses, (b) find a sample statistic and sampling distribution that will let you develop a decision rule for choosing between the two hypotheses, and (c) compute your sample statistic and choose the hypothesis supported by the data.

Foothill Hosiery recently received an order for children’s socks decorated with embroidered patches of cartoon characters. Foothill did not have the right machinery to sew on the embroidered patches and contracted out the sewing. While the order was filled and Foothill made a profit on it, the sewing contractor’s price seemed high, and Foothill had to keep pressure on the contractor to deliver the socks by the date agreed upon. Foothill’s CEO, John McGrath, has explored buying the machinery necessary to allow Foothill to sew patches on socks themselves. He has discovered that if more than a quarter of the children’s socks they make are ordered with patches, the machinery will be a sound investment. John asks Kevin to find out if more than 35 per cent of children’s socks are being sold with patches.

Kevin calls the major trade organizations for the hosiery, embroidery, and children’s clothes industries, and no one can answer his question. Kevin decides it must be time to take a sample and test to see if more than 35 per cent of children’s socks are decorated with patches. He calls the sales manager at Foothill, and she agrees to ask her salespeople to look at store displays of children’s socks, counting how many pairs are displayed and how many of those are decorated with patches. Two weeks later, Kevin gets a memo from the sales manager, telling him that of the 2,483 pairs of children’s socks on display at stores where the salespeople counted, 826 pairs had embroidered patches.

Kevin writes his hypotheses, remembering that Foothill will be making a decision about spending a fair amount of money based on what he finds. To be more certain that he is right if he recommends that the money be spent, Kevin writes his hypotheses so that the unusual world would be the one where more than 35 per cent of children’s socks are decorated:

H o : π decorated socks <  .35

H a : π decorated socks > .35

When writing his hypotheses, Kevin knows that if his sample has a proportion of decorated socks well below .35, he will want to recommend against buying the machinery. He only wants to say the data support the alternative if the sample proportion is well above .35. To include the low values in the null hypothesis and only the high values in the alternative, he uses a one-tail test, judging that the data support the alternative only if his z-score is in the upper tail. He will conclude that the machinery should be bought only if his z-statistic is too large to have easily come from the sampling distribution drawn from a population with a proportion of .35. Kevin will accept H a only if his z is large and positive.

Checking the bottom line of the t-table, Kevin sees that .95 of all z-scores associated with the proportion are less than -1.645. His rule is therefore to conclude that his sample data support the null hypothesis that 35 per cent or less of children’s socks are decorated if his sample (calculated) z is less than -1.645. If his sample z is greater than -1.645, he will conclude that more than 35 per cent of children’s socks are decorated and that Foothill Hosiery should invest in the machinery needed to sew embroidered patches on socks.

Using the data the salespeople collected, Kevin finds the proportion of the sample that is decorated:

[latex]\pi = 826/2483 = .333[/latex]

Using this value, he computes his sample z-statistic:

[latex]z = (p-\pi)/(\sqrt{\dfrac{(\pi)(1-\pi)}{n}}) = (.333-.35)/(\sqrt{\dfrac{(.35)(1-.35)}{2483}}) = \dfrac{-.0173}{.0096} = -1.0811[/latex]

All these calculations, along with the plots of both sampling distribution of π and the associated standard normal distributions, are computed by the interactive Excel template in Figure 4.1.

Kevin’s collected numbers, shown in the yellow cells of Figure 4.1., can be changed to other numbers of your choice to see how the business decision may be changed under alternative circumstances.

Because his sample (calculated) z-score is larger than -1.645, it is unlikely that his sample z came from the sampling distribution of z’s drawn from a population where π <  .35, so it is unlikely that his sample comes from a population with π <  .35. Kevin can tell John McGrath that the sample the salespeople collected supports the conclusion that more than 35 per cent of children’s socks are decorated with embroidered patches. John can feel comfortable making the decision to buy the embroidery and sewing machinery.

Testing independence and categorical variables

We also use hypothesis testing when we deal with categorical variables. Categorical variables are associated with categorical data. For instance, gender is a categorical variable as it can be classified into two or more categories. In business, and predominantly in marketing, we want to determine on which factor(s) customers base their preference for one type of product over others. Since customers’ preferences are not the same even in a specific geographical area, marketing strategists and managers are often keen to know the association among those variables that affect shoppers’ choices. In other words, they want to know whether customers’ decisions are statistically independent of a hypothesized factor such as age.

For example, imagine that the owner of a newly established family restaurant in Burnaby, BC, with branches in North Vancouver, Langley, and Kelowna, is interested in determining whether the age of the restaurant’s customers affects which dishes they order. If it does, she will explore the idea of charging different prices for dishes popular with different age groups. The sales manager has collected data on 711 sales of different dishes over the last six months, along with the approximate age of the customers, and divided the customers into three categories. Table 4.5 shows the breakdown of orders and age groups.

The owner writes her hypotheses:

H o : Customers’ preferences for dishes are independent of their ages

H a : Customers’ preferences for dishes depend on their ages

The underlying test for this contingency table is known as the chi-square test . This will determine if customers’ ages and preferences are independent of each other.

We compute both the observed and expected frequencies as we did in the earlier example involving sports socks where  O = observed frequency in the sample in each class, and E = expected frequency in the sample in each class. Then we calculate the expected frequency for the above table with i  rows and j  columns, using the following formula:

This chi-square distribution will have ( i -1)( j -1) degrees of freedom. One technical condition for this test is that the value for each of the cells must not be less than 5. Figure 4.2 provides the hypothesized values for different levels of significance.

The expected frequency, E ij , is found by multiplying the relative frequency of each row and column, and then dividing this amount by the total sample size.  Thus,

For each of the expected frequencies, we select the associated total row from each of the age groups, and multiply it by the total of the same column, then divide it by the total sample size. For the first row and column, we multiply (82 *216)/711=24.95. Table 4.6 summarizes all expected frequencies for this example.

Now we use the calculated expected frequencies and the observed frequencies to compute the chi-square test statistic:

We computed the sample test statistic as 21.13, which is above the 12.592 cut-off value of the chi-square table associated with (3-1)*(4-1) = 6 df at .05 level. To find out the exact cut-off point from the chi-square table, you can enter the alpha level of .05 and the degrees of freedom, 6, directly into the yellow cells in the following interactive Excel template (Figure 4.2). This template contains two sheets; it will plot the chi-square distribution for this example and will automatically show the exact cut-off point.

The result indicates that our sample data supported the alternative hypothesis. In other words, customers’ preferences for different dishes depended on their age groups. Based on this outcome, the owner may differentiate price based on these different age groups.

Using the test of independence, the owner may also go further to find out if such dependency exists among any other pairs of categorical data. This time, she may want to collect data for the selected age groups at different locations of her restaurant in British Columbia. The results of this test will reveal more information about the types of customers these restaurants attract at different locations. Depending on the availability of data, such statistical analysis can also be carried out to help determine an improved pricing policy for different groups in different locations, at different times of day, or on different days of the week. Finally, the owner may also redo this analysis by including other characteristics of these customers, such as education, gender, etc., and their choice of dishes.

This chapter has been an introduction to hypothesis testing. You should be able to see the relationship between the mathematics and strategies of hypothesis testing and the mathematics and strategies of interval estimation. When making an interval estimate, you construct an interval around your sample statistic based on a known sampling distribution. When testing a hypothesis, you construct an interval around a hypothesized population parameter, using a known sampling distribution to determine the width of that interval. You then see if your sample statistic falls within that interval to decide if your sample probably came from a population with that hypothesized population parameter. Hypothesis testing also has implications for decision-making in marketing, as we saw when we extended our discussion to include the test of independence for categorical data.

Hypothesis testing is a widely used statistical technique. It forces you to think ahead about what you might find. By forcing you to think ahead, it often helps with decision-making by forcing you to think about what goes into your decision. All of statistics requires clear thinking, and clear thinking generally makes better decisions. Hypothesis testing requires very clear thinking and often leads to better decision-making.

Introductory Business Statistics with Interactive Spreadsheets - 1st Canadian Edition Copyright © 2015 by Mohammad Mahbobi and Thomas K. Tiemann is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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A Beginner’s Guide to Hypothesis Testing in Business

• 30 Mar 2021

Becoming a more data-driven decision-maker can bring several benefits to your organization, enabling you to identify new opportunities to pursue and threats to abate. Rather than allowing subjective thinking to guide your business strategy, backing your decisions with data can empower your company to become more innovative and, ultimately, profitable.

If you’re new to data-driven decision-making, you might be wondering how data translates into business strategy. The answer lies in generating a hypothesis and verifying or rejecting it based on what various forms of data tell you.

Below is a look at hypothesis testing and the role it plays in helping businesses become more data-driven.

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What Is Hypothesis Testing?

To understand what hypothesis testing is, it’s important first to understand what a hypothesis is.

A hypothesis or hypothesis statement seeks to explain why something has happened, or what might happen, under certain conditions. It can also be used to understand how different variables relate to each other. Hypotheses are often written as if-then statements; for example, “If this happens, then this will happen.”

Hypothesis testing , then, is a statistical means of testing an assumption stated in a hypothesis. While the specific methodology leveraged depends on the nature of the hypothesis and data available, hypothesis testing typically uses sample data to extrapolate insights about a larger population.

Hypothesis Testing in Business

When it comes to data-driven decision-making, there’s a certain amount of risk that can mislead a professional. This could be due to flawed thinking or observations, incomplete or inaccurate data , or the presence of unknown variables. The danger in this is that, if major strategic decisions are made based on flawed insights, it can lead to wasted resources, missed opportunities, and catastrophic outcomes.

The real value of hypothesis testing in business is that it allows professionals to test their theories and assumptions before putting them into action. This essentially allows an organization to verify its analysis is correct before committing resources to implement a broader strategy.

As one example, consider a company that wishes to launch a new marketing campaign to revitalize sales during a slow period. Doing so could be an incredibly expensive endeavor, depending on the campaign’s size and complexity. The company, therefore, may wish to test the campaign on a smaller scale to understand how it will perform.

In this example, the hypothesis that’s being tested would fall along the lines of: “If the company launches a new marketing campaign, then it will translate into an increase in sales.” It may even be possible to quantify how much of a lift in sales the company expects to see from the effort. Pending the results of the pilot campaign, the business would then know whether it makes sense to roll it out more broadly.

Related: 9 Fundamental Data Science Skills for Business Professionals

Key Considerations for Hypothesis Testing

1. alternative hypothesis and null hypothesis.

In hypothesis testing, the hypothesis that’s being tested is known as the alternative hypothesis . Often, it’s expressed as a correlation or statistical relationship between variables. The null hypothesis , on the other hand, is a statement that’s meant to show there’s no statistical relationship between the variables being tested. It’s typically the exact opposite of whatever is stated in the alternative hypothesis.

For example, consider a company’s leadership team that historically and reliably sees $12 million in monthly revenue. They want to understand if reducing the price of their services will attract more customers and, in turn, increase revenue.

In this case, the alternative hypothesis may take the form of a statement such as: “If we reduce the price of our flagship service by five percent, then we’ll see an increase in sales and realize revenues greater than $12 million in the next month.”

The null hypothesis, on the other hand, would indicate that revenues wouldn’t increase from the base of $12 million, or might even decrease.

Check out the video below about the difference between an alternative and a null hypothesis, and subscribe to our YouTube channel for more explainer content.

2. Significance Level and P-Value

Statistically speaking, if you were to run the same scenario 100 times, you’d likely receive somewhat different results each time. If you were to plot these results in a distribution plot, you’d see the most likely outcome is at the tallest point in the graph, with less likely outcomes falling to the right and left of that point.

With this in mind, imagine you’ve completed your hypothesis test and have your results, which indicate there may be a correlation between the variables you were testing. To understand your results' significance, you’ll need to identify a p-value for the test, which helps note how confident you are in the test results.

In statistics, the p-value depicts the probability that, assuming the null hypothesis is correct, you might still observe results that are at least as extreme as the results of your hypothesis test. The smaller the p-value, the more likely the alternative hypothesis is correct, and the greater the significance of your results.

3. One-Sided vs. Two-Sided Testing

When it’s time to test your hypothesis, it’s important to leverage the correct testing method. The two most common hypothesis testing methods are one-sided and two-sided tests , or one-tailed and two-tailed tests, respectively.

Typically, you’d leverage a one-sided test when you have a strong conviction about the direction of change you expect to see due to your hypothesis test. You’d leverage a two-sided test when you’re less confident in the direction of change.

4. Sampling

To perform hypothesis testing in the first place, you need to collect a sample of data to be analyzed. Depending on the question you’re seeking to answer or investigate, you might collect samples through surveys, observational studies, or experiments.

A survey involves asking a series of questions to a random population sample and recording self-reported responses.

Observational studies involve a researcher observing a sample population and collecting data as it occurs naturally, without intervention.

Finally, an experiment involves dividing a sample into multiple groups, one of which acts as the control group. For each non-control group, the variable being studied is manipulated to determine how the data collected differs from that of the control group.

Learn How to Perform Hypothesis Testing

Hypothesis testing is a complex process involving different moving pieces that can allow an organization to effectively leverage its data and inform strategic decisions.

If you’re interested in better understanding hypothesis testing and the role it can play within your organization, one option is to complete a course that focuses on the process. Doing so can lay the statistical and analytical foundation you need to succeed.

Do you want to learn more about hypothesis testing? Explore Business Analytics —one of our online business essentials courses —and download our Beginner’s Guide to Data & Analytics .

About the Author

Hypothesis Generation and Interpretation

Design Principles and Patterns for Big Data Applications

• © 2024
• Hiroshi Ishikawa 0

Department of Systems Design, Tokyo Metropolitan University, Hino, Japan

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• Provides an integrated perspective on why decisions are made and how the process is modeled
• Presentation of design patterns enables use in a wide variety of big-data applications
• Multiple practical use cases indicate the broad real-world significance of the methods presented

Part of the book series: Studies in Big Data (SBD, volume 139)

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The novel methods and technologies proposed in  Hypothesis Generation and Interpretation are supported by the incorporation of historical perspectives on science and an emphasis on the origin and development of the ideas behind their design principles and patterns.

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Analysis, Visualization and Exploration Scenarios: Formal Methods for Systematic Meta Studies of Big Data Applications

The Nexus Between Big Data and Decision-Making: A Study of Big Data Techniques and Technologies

• Hypothesis Generation
• Hypothesis Interpretation
• Data Engineering
• Data Science
• Data Management
• Machine Learning
• Data Mining
• Design Patterns
• Design Principles

Table of contents (8 chapters)

Front matter, basic concept.

Hiroshi Ishikawa

Science and Hypothesis

Machine learning and integrated approach, hypothesis generation by difference, methods for integrated hypothesis generation, interpretation, back matter, authors and affiliations, about the author.

He has published actively in international, refereed journals and conferences, such as ACM Transactions on Database Systems , IEEE Transactions on Knowledge and Data Engineering , The VLDB Journal , IEEE International Conference on Data Engineering, and ACM SIGSPATIAL and Management of Emergent Digital EcoSystems (MEDES). He has authored and co-authored a dozen books, including Social Big Data Mining (CRC, 2015) and Object-Oriented Database System (Springer-Verlag, 1993).

Bibliographic Information

Book Title : Hypothesis Generation and Interpretation

Book Subtitle : Design Principles and Patterns for Big Data Applications

Authors : Hiroshi Ishikawa

Series Title : Studies in Big Data

DOI : https://doi.org/10.1007/978-3-031-43540-9

Publisher : Springer Cham

eBook Packages : Computer Science , Computer Science (R0)

Copyright Information : The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024

Hardcover ISBN : 978-3-031-43539-3 Published: 02 February 2024

Softcover ISBN : 978-3-031-43542-3 Due: 15 February 2025

eBook ISBN : 978-3-031-43540-9 Published: 01 January 2024

Series ISSN : 2197-6503

Series E-ISSN : 2197-6511

Edition Number : 1

Number of Pages : XII, 372

Number of Illustrations : 52 b/w illustrations, 125 illustrations in colour

Topics : Theory of Computation , Database Management , Data Mining and Knowledge Discovery , Machine Learning , Big Data , Complex Systems

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January 13, 2024

Demystifying Hypothesis Generation: A Guide to AI-Driven Insights

Hypothesis generation involves making informed guesses about various aspects of a business, market, or problem that need further exploration and testing. This article discusses the process you need to follow while generating hypothesis and how an AI tool, like Akaike's BYOB can help you achieve the process quicker and better.

What is Hypothesis Generation?

Hypothesis generation involves making informed guesses about various aspects of a business, market, or problem that need further exploration and testing. It's a crucial step while applying the scientific method to business analysis and decision-making. 

Here is an example from a popular B-school marketing case study: 

A bicycle manufacturer noticed that their sales had dropped significantly in 2002 compared to the previous year. The team investigating the reasons for this had many hypotheses. One of them was: “many cycling enthusiasts have switched to walking with their iPods plugged in.” The Apple iPod was launched in late 2001 and was an immediate hit among young consumers. Data collected manually by the team seemed to show that the geographies around Apple stores had indeed shown a sales decline.

Traditionally, hypothesis generation is time-consuming and labour-intensive. However, the advent of Large Language Models (LLMs) and Generative AI (GenAI) tools has transformed the practice altogether. These AI tools can rapidly process extensive datasets, quickly identifying patterns, correlations, and insights that might have even slipped human eyes, thus streamlining the stages of hypothesis generation.

These tools have also revolutionised experimentation by optimising test designs, reducing resource-intensive processes, and delivering faster results. LLMs' role in hypothesis generation goes beyond mere assistance, bringing innovation and easy, data-driven decision-making to businesses.

Hypotheses come in various types, such as simple, complex, null, alternative, logical, statistical, or empirical. These categories are defined based on the relationships between the variables involved and the type of evidence required for testing them. In this article, we aim to demystify hypothesis generation. We will explore the role of LLMs in this process and outline the general steps involved, highlighting why it is a valuable tool in your arsenal.

Understanding Hypothesis Generation

A hypothesis is born from a set of underlying assumptions and a prediction of how those assumptions are anticipated to unfold in a given context. Essentially, it's an educated, articulated guess that forms the basis for action and outcome assessment.

A hypothesis is a declarative statement that has not yet been proven true. Based on past scholarship , we could sum it up as the following: 

• A definite statement, not a question
• Based on observations and knowledge
• Testable and can be proven wrong
• Predicts the anticipated results clearly
• Contains a dependent and an independent variable where the dependent variable is the phenomenon being explained and the independent variable does the explaining

In a business setting, hypothesis generation becomes essential when people are made to explain their assumptions. This clarity from hypothesis to expected outcome is crucial, as it allows people to acknowledge a failed hypothesis if it does not provide the intended result. Promoting such a culture of effective hypothesising can lead to more thoughtful actions and a deeper understanding of outcomes. Failures become just another step on the way to success, and success brings more success.

Hypothesis generation is a continuous process where you start with an educated guess and refine it as you gather more information. You form a hypothesis based on what you know or observe.

Say you're a pen maker whose sales are down. You look at what you know:

• I can see that pen sales for my brand are down in May and June.
• I also know that schools are closed in May and June and that schoolchildren use a lot of pens.
• I hypothesise that my sales are down because school children are not using pens in May and June, and thus not buying newer ones.

The next step is to collect and analyse data to test this hypothesis, like tracking sales before and after school vacations. As you gather more data and insights, your hypothesis may evolve. You might discover that your hypothesis only holds in certain markets but not others, leading to a more refined hypothesis.

Once your hypothesis is proven correct, there are many actions you may take - (a) reduce supply in these months (b) reduce the price so that sales pick up (c) release a limited supply of novelty pens, and so on.

Once you decide on your action, you will further monitor the data to see if your actions are working. This iterative cycle of formulating, testing, and refining hypotheses - and using insights in decision-making - is vital in making impactful decisions and solving complex problems in various fields, from business to scientific research.

How do Analysts generate Hypotheses? Why is it iterative?

A typical human working towards a hypothesis would start with:

    1. Picking the Default Action

    2. Determining the Alternative Action

    3. Figuring out the Null Hypothesis (H0)

    4. Inverting the Null Hypothesis to get the Alternate Hypothesis (H1)

    5. Hypothesis Testing

The default action is what you would naturally do, regardless of any hypothesis or in a case where you get no further information. The alternative action is the opposite of your default action.

The null hypothesis, or H0, is what brings about your default action. The alternative hypothesis (H1) is essentially the negation of H0.

For example, suppose you are tasked with analysing a highway tollgate data (timestamp, vehicle number, toll amount) to see if a raise in tollgate rates will increase revenue or cause a volume drop. Following the above steps, we can determine:

Now, we can start looking at past data of tollgate traffic in and around rate increases for different tollgates. Some data might be irrelevant. For example, some tollgates might be much cheaper so customers might not have cared about an increase. Or, some tollgates are next to a large city, and customers have no choice but to pay. 

Ultimately, you are looking for the level of significance between traffic and rates for comparable tollgates. Significance is often noted as its P-value or probability value . P-value is a way to measure how surprising your test results are, assuming that your H0 holds true.

The lower the p-value, the more convincing your data is to change your default action.

Usually, a p-value that is less than 0.05 is considered to be statistically significant, meaning there is a need to change your null hypothesis and reject your default action. In our example, a low p-value would suggest that a 10% increase in the toll rate causes a significant dip in traffic (>3%). Thus, it is better if we keep our rates as is if we want to maintain revenue. 

In other examples, where one has to explore the significance of different variables, we might find that some variables are not correlated at all. In general, hypothesis generation is an iterative process - you keep looking for data and keep considering whether that data convinces you to change your default action.

Internal and External Data 

Hypothesis generation feeds on data. Data can be internal or external. In businesses, internal data is produced by company owned systems (areas such as operations, maintenance, personnel, finance, etc). External data comes from outside the company (customer data, competitor data, and so on).

Let’s consider a real-life hypothesis generated from internal data: 

Multinational company Johnson & Johnson was looking to enhance employee performance and retention.  Initially, they favoured experienced industry candidates for recruitment, assuming they'd stay longer and contribute faster. However, HR and the people analytics team at J&J hypothesised that recent college graduates outlast experienced hires and perform equally well.  They compiled data on 47,000 employees to test the hypothesis and, based on it, Johnson & Johnson increased hires of new graduates by 20% , leading to reduced turnover with consistent performance. 

For an analyst (or an AI assistant), external data is often hard to source - it may not be available as organised datasets (or reports), or it may be expensive to acquire. Teams might have to collect new data from surveys, questionnaires, customer feedback and more. 

Further, there is the problem of context. Suppose an analyst is looking at the dynamic pricing of hotels offered on his company’s platform in a particular geography. Suppose further that the analyst has no context of the geography, the reasons people visit the locality, or of local alternatives; then the analyst will have to learn additional context to start making hypotheses to test. 

Internal data, of course, is internal, meaning access is already guaranteed. However, this probably adds up to staggering volumes of data. 

Looking Back, and Looking Forward

Data analysts often have to generate hypotheses retrospectively, where they formulate and evaluate H0 and H1 based on past data. For the sake of this article, let's call it retrospective hypothesis generation.

Alternatively, a prospective approach to hypothesis generation could be one where hypotheses are formulated before data collection or before a particular event or change is implemented. 

For example: 

A pen seller has a hypothesis that during the lean periods of summer, when schools are closed, a Buy One Get One (BOGO) campaign will lead to a 100% sales recovery because customers will buy pens in advance.  He then collects feedback from customers in the form of a survey and also implements a BOGO campaign in a single territory to see whether his hypothesis is correct, or not.

The HR head of a multi-office employer realises that some of the company’s offices have been providing snacks at 4:30 PM in the common area, and the rest have not. He has a hunch that these offices have higher productivity. The leader asks the company’s data science team to look at employee productivity data and the employee location data. “Am I correct, and to what extent?”, he asks. 

These examples also reflect another nuance, in which the data is collected differently: 

• Observational: Observational testing happens when researchers observe a sample population and collect data as it occurs without intervention. The data for the snacks vs productivity hypothesis was observational. 
• Experimental: In experimental testing, the sample is divided into multiple groups, with one control group. The test for the non-control groups will be varied to determine how the data collected differs from that of the control group. The data collected by the pen seller in the single territory experiment was experimental.

Such data-backed insights are a valuable resource for businesses because they allow for more informed decision-making, leading to the company's overall growth. Taking a data-driven decision, from forming a hypothesis to updating and validating it across iterations, to taking action based on your insights reduces guesswork, minimises risks, and guides businesses towards strategies that are more likely to succeed.

How can GenAI help in Hypothesis Generation?

Of course, hypothesis generation is not always straightforward. Understanding the earlier examples is easy for us because we're already inundated with context. But, in a situation where an analyst has no domain knowledge, suddenly, hypothesis generation becomes a tedious and challenging process.

AI, particularly high-capacity, robust tools such as LLMs, have radically changed how we process and analyse large volumes of data. With its help, we can sift through massive datasets with precision and speed, regardless of context, whether it's customer behaviour, financial trends, medical records, or more. Generative AI, including LLMs, are trained on diverse text data, enabling them to comprehend and process various topics.

Now, imagine an AI assistant helping you with hypothesis generation. LLMs are not born with context. Instead, they are trained upon vast amounts of data, enabling them to develop context in a completely unfamiliar environment. This skill is instrumental when adopting a more exploratory approach to hypothesis generation. For example, the HR leader from earlier could simply ask an LLM tool: “Can you look at this employee productivity data and find cohorts of high-productivity and see if they correlate to any other employee data like location, pedigree, years of service, marital status, etc?” 

For an LLM-based tool to be useful, it requires a few things:

• Domain Knowledge: A human could take months to years to acclimatise to a particular field fully, but LLMs, when fed extensive information and utilising Natural Language Processing (NLP), can familiarise themselves in a very short time.
• Explainability:   Explainability is its ability to explain its thought process and output to cease being a "black box".
• Customisation: For consistent improvement, contextual AI must allow tweaks, allowing users to change its behaviour to meet their expectations. Human intervention and validation is a necessary step in adoptingAI tools. NLP allows these tools to discern context within textual data, meaning it can read, categorise, and analyse data with unimaginable speed. LLMs, thus, can quickly develop contextual understanding and generate human-like text while processing vast amounts of unstructured data, making it easier for businesses and researchers to organise and utilise data effectively.LLMs have the potential to become indispensable tools for businesses. The future rests on AI tools that harness the powers of LLMs and NLP to deliver actionable insights, mitigate risks, inform decision-making, predict future trends, and drive business transformation across various sectors.

Together, these technologies empower data analysts to unravel hidden insights within their data. For our pen maker, for example, an AI tool could aid data analytics. It can look through historical data to track when sales peaked or go through sales data to identify the pens that sold the most. It can refine a hypothesis across iterations, just as a human analyst would. It can even be used to brainstorm other hypotheses. Consider the situation where you ask the LLM, " Where do I sell the most pens? ". It will go through all of the data you have made available - places where you sell pens, the number of pens you sold - to return the answer. Now, if we were to do this on our own, even if we were particularly meticulous about keeping records, it would take us at least five to ten minutes, that too, IF we know how to query a database and extract the needed information. If we don't, there's the added effort required to find and train such a person. An AI assistant, on the other hand, could share the answer with us in mere seconds. Its finely-honed talents in sorting through data, identifying patterns, refining hypotheses iteratively, and generating data-backed insights enhance problem-solving and decision-making, supercharging our business model.

Top-Down and Bottom-Up Hypothesis Generation

As we discussed earlier, every hypothesis begins with a default action that determines your initial hypotheses and all your subsequent data collection. You look at data and a LOT of data. The significance of your data is dependent on the effect and the relevance it has to your default action. This would be a top-down approach to hypothesis generation.

There is also the bottom-up method , where you start by going through your data and figuring out if there are any interesting correlations that you could leverage better. This method is usually not as focused as the earlier approach and, as a result, involves even more data collection, processing, and analysis. AI is a stellar tool for Exploratory Data Analysis (EDA). Wading through swathes of data to highlight trends, patterns, gaps, opportunities, errors, and concerns is hardly a challenge for an AI tool equipped with NLP and powered by LLMs.

EDA can help with: 

• Cleaning your data
• Understanding your variables
• Analysing relationships between variables

An AI assistant performing EDA can help you review your data, remove redundant data points, identify errors, note relationships, and more. All of this ensures ease, efficiency, and, best of all, speed for your data analysts.

Good hypotheses are extremely difficult to generate. They are nuanced and, without necessary context, almost impossible to ascertain in a top-down approach. On the other hand, an AI tool adopting an exploratory approach is swift, easily running through available data - internal and external. 

If you want to rearrange how your LLM looks at your data, you can also do that. Changing the weight you assign to the various events and categories in your data is a simple process. That’s why LLMs are a great tool in hypothesis generation - analysts can tailor them to their specific use cases. 

Ethical Considerations and Challenges

There are numerous reasons why you should adopt AI tools into your hypothesis generation process. But why are they still not as popular as they should be?

Some worry that AI tools can inadvertently pick up human biases through the data it is fed. Others fear AI and raise privacy and trust concerns. Data quality and ability are also often questioned. Since LLMs and Generative AI are developing technologies, such issues are bound to be, but these are all obstacles researchers are earnestly tackling.

One oft-raised complaint against LLM tools (like OpenAI's ChatGPT) is that they 'fill in' gaps in knowledge, providing information where there is none, thus giving inaccurate, embellished, or outright wrong answers; this tendency to "hallucinate" was a major cause for concern. But, to combat this phenomenon, newer AI tools have started providing citations with the insights they offer so that their answers become verifiable. Human validation is an essential step in interpreting AI-generated hypotheses and queries in general. This is why we need a collaboration between the intelligent and artificially intelligent mind to ensure optimised performance.

Clearly, hypothesis generation is an immensely time-consuming activity. But AI can take care of all these steps for you. From helping you figure out your default action, determining all the major research questions, initial hypotheses and alternative actions, and exhaustively weeding through your data to collect all relevant points, AI can help make your analysts' jobs easier. It can take any approach - prospective, retrospective, exploratory, top-down, bottom-up, etc. Furthermore, with LLMs, your structured and unstructured data are taken care of, meaning no more worries about messy data! With the wonders of human intuition and the ease and reliability of Generative AI and Large Language Models, you can speed up and refine your process of hypothesis generation based on feedback and new data to provide the best assistance to your business.

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Statistics for Business Analytics

12.2 hypothesis testing.

Another way to look at the uncertainty of parameters is to test a statistical hypothesis. As it was discussed in Section 7 , I personally think that hypothesis testing is a less useful instrument for these purposes than the confidence interval and that it might be misleading in some circumstances. Nonetheless, it has its merits and can be helpful if an analyst knows what they are doing. In order to test the hypothesis, we need to follow the procedure, described in Section 7 .

12.2.1 Regression parameters

The classical hypotheses for the parameters are formulated in the following way: \[\begin{equation} \begin{aligned} \mathrm{H}_0: \beta_i = 0 \\ \mathrm{H}_1: \beta_i \neq 0 \end{aligned} . \tag{12.3} \end{equation}\] This formulation of hypotheses comes from the idea that we want to check if the effect estimated by the regression is indeed there (i.e. statistically significantly different from zero). Note however, that as in any other hypothesis testing, if you fail to reject the null hypothesis, this only means that you do not know, we do not have enough evidence to conclude anything. This does not mean that there is no effect and that the respective variable can be removed from the model. In case of simple linear regression, the null and alternative hypothesis can be represented graphically as shown in Figure 12.4 .

Figure 12.4: Graphical presentation of null and alternative hypothesis in regression context

The graph on the left in Figure 12.4 demonstrates how the true model could look if the null hypothesis was true - it would be just a straight line, parallel to x-axis. The graph on the right demonstrates the alternative situation, when the parameter is not equal to zero. We do not know the true model, and hypothesis testing does not tell us, whether the hypothesis is true or false, but if we have enough evidence to reject H \(_0\) , then we might conclude that we see an effect of one variable on another in the data. Note, as discussed in Section 7 , the null hypothesis is always wrong, and it will inevitably be rejected with the increase of sample size.

Given the discussion in the previous subsection, we know that the parameters of regression model will follow normal distribution, as long as all assumptions are satisfied (including those for CLT ). We also know that because the standard errors of parameters are estimated, we need to use Student’s distribution, which takes the uncertainty about the variance into account. Based on this, we can say that the following statistics will follow t with \(n-k\) degrees of freedom: \[\begin{equation} \frac{b_i - 0}{s_{b_i}} \sim t(n-k) . \tag{12.4} \end{equation}\] After calculating the value and comparing it with the critical t-value on the selected significance level or directly comparing p-value based on (12.4) with the significance level, we can make conclusions about the hypothesis.

The context of regression provides a great example, why we never accept hypothesis and why in the case of “Fail to reject H \(_0\) ”, we should not remove a variable (unless we have more fundamental reasons for doing that). Consider an example, where the estimated parameter \(b_1=0.5\) , and its standard error is \(s_{b_1}=1\) , we estimated a simple linear regression on a sample of 30 observations, and we want to test, whether the parameter in the population is zero (i.e. hypothesis (12.3) ) on 1% significance level. Inserting the values in formula (12.4) , we get: \[\begin{equation*} \frac{|0.5 - 0|}{1} = 0.5, \end{equation*}\] with the critical value for two-tailed test of \(t_{0.01}(30-2)\approx 2.76\) . Comparing t-value with the critical one, we would conclude that we fail to reject H \(_0\) and thus the parameter is not statistically different from zero. But what would happen if we check another hypothesis: \[\begin{equation*} \begin{aligned} \mathrm{H}_0: \beta_1 = 1 \\ \mathrm{H}_1: \beta_1 \neq 1 \end{aligned} . \end{equation*}\] The procedure is the same, the calculated t-value is: \[\begin{equation*} \frac{|0.5 - 1|}{1} = 0.5, \end{equation*}\] which leads to exactly the same conclusion as before: on 1% significance level, we fail to reject the new H \(_0\) , so the value is not distinguishable from 1. So, which of the two is correct? The correct answer is “we do not know”. The non-rejection region just tells us that uncertainty about the parameter is so high that it also include the value of interest (0 in case of the classical regression analysis). If we constructed the confidence interval for this problem, we would not have such confusion, as we would conclude that on 1% significance level the true parameter lies in the region \((-2.26, 3.26)\) and can be any of these numbers.

In R, if you want to test the hypothesis for parameters, I would recommend using lm() function for regression:

This output tells us that when we consider the parameter for the variable speed, we reject the standard H \(_0\) on the pre-selected 1% significance level (comparing the level with p-value in the last column of the output). Note that we should first select the significance level and only then conduct the test, otherwise we would be bending reality for our needs.

12.2.2 Regression line

Finally, in regression context, we can test another hypothesis, which becomes useful, when a lot of parameters of the model are very close to zero and seem to be insignificant on the selected level: \[\begin{equation} \begin{aligned} \mathrm{H}_0: \beta_1 = \beta_2 = \dots = \beta_{k-1} = 0 \\ \mathrm{H}_1: \beta_1 \neq 0 \vee \beta_2 \neq 0 \vee \dots \vee \beta_{k-1} \neq 0 \end{aligned} , \tag{12.5} \end{equation}\] which translates into normal language as “H \(_0\) : all parameters (except for intercept) are equal to zero; H \(_1\) : at least one parameter is not equal to zero”. This hypothesis is only needed, when you have a model with many statistically insignificant variables and want to see if the model explains anything. This is done using F-test, which can be calculated based on sums of squares: \[\begin{equation*} F = \frac{ SSR / (k-1)}{SSE / (n-k)} \sim F(k-1, n-k) , \end{equation*}\] where the sums of squares are divided by their degrees of freedom. The test is conducted in the similar manner as any other test (see Section 7 ): after choosing the significance level, we can either calculate the critical value of F for the specified degrees of freedom, or compare it with the p-value from the test to make a conclusion about the null hypothesis.

This hypothesis is not very useful, when the parameter are significant and coefficient of determination is high. It only becomes useful in difficult situations of poor fit. The test on its own does not tell if the model is adequate or not. And the F value and related p-value is not comparable with respective values of other models. Graphically, this test checks, whether in the true model the slope of the straight line on the plot of actuals vs fitted is different from zero. An example with the same stopping distance model is provided in Figure 12.5 .

Figure 12.5: Graphical presentation of F test for regression model.

What the test is tries to get insight about, is whether in the true model the blue line coincides with the red line (i.e. the slope is equal to zero, which is only possible, when all parameters are zero). If we have enough evidence to reject the null hypothesis, then this means that the slopes are different on the selected significance level.

Here is an example with the speed model discussed above with the significance level of 1%:

In the output above, the critical value is lower than the calculated, so we can reject the H \(_0\) , which means that there is something in the model that explains the variability in the variable dist . Alternatively, we could focus on p-value. We see that the it is lower than the significance level of 1%, so we reject the H \(_0\) and come to the same conclusion as above.

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Understanding Hypothesis Testing

Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

What is Hypothesis Testing?

A hypothesis is an assumption or idea, specifically a statistical claim about an unknown population parameter. For example, a judge assumes a person is innocent and verifies this by reviewing evidence and hearing testimony before reaching a verdict.

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. 

To test the validity of the claim or assumption about the population parameter:

• A sample is drawn from the population and analyzed.
• The results of the analysis are used to decide whether the claim is true or not.

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

Defining Hypotheses

• Null hypothesis (H 0 ): In statistics, the null hypothesis is a general statement or default position that there is no relationship between two measured cases or no relationship among groups. In other words, it is a basic assumption or made based on the problem knowledge. Example : A company’s mean production is 50 units/per da H 0 : [Tex]\mu [/Tex] = 50.
• Alternative hypothesis (H 1 ): The alternative hypothesis is the hypothesis used in hypothesis testing that is contrary to the null hypothesis.  Example: A company’s production is not equal to 50 units/per day i.e. H 1 : [Tex]\mu [/Tex] [Tex]\ne [/Tex] 50.

Key Terms of Hypothesis Testing

• Level of significance : It refers to the degree of significance in which we accept or reject the null hypothesis. 100% accuracy is not possible for accepting a hypothesis, so we, therefore, select a level of significance that is usually 5%. This is normally denoted with  [Tex]\alpha[/Tex] and generally, it is 0.05 or 5%, which means your output should be 95% confident to give a similar kind of result in each sample.
• P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
• Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
• Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
• Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

Why do we use Hypothesis Testing?

Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing. 

One-Tailed and Two-Tailed Test

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Tailed Test

There are two types of one-tailed test:

• Left-Tailed (Left-Sided) Test: The alternative hypothesis asserts that the true parameter value is less than the null hypothesis. Example: H 0 ​: [Tex]\mu \geq 50 [/Tex] and H 1 : [Tex]\mu < 50 [/Tex]
• Right-Tailed (Right-Sided) Test : The alternative hypothesis asserts that the true parameter value is greater than the null hypothesis. Example: H 0 : [Tex]\mu \leq50 [/Tex] and H 1 : [Tex]\mu > 50 [/Tex]

Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

Example: H 0 : [Tex]\mu = [/Tex] 50 and H 1 : [Tex]\mu \neq 50 [/Tex]

To delve deeper into differences into both types of test: Refer to link

What are Type 1 and Type 2 errors in Hypothesis Testing?

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

• Type I error: When we reject the null hypothesis, although that hypothesis was true. Type I error is denoted by alpha( [Tex]\alpha [/Tex] ).
• Type II errors : When we accept the null hypothesis, but it is false. Type II errors are denoted by beta( [Tex]\beta [/Tex] ).

How does Hypothesis Testing work?

Step 1: define null and alternative hypothesis.

State the null hypothesis ( [Tex]H_0 [/Tex] ), representing no effect, and the alternative hypothesis ( [Tex]H_1 [/Tex] ​), suggesting an effect or difference.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

Step 2 – Choose significance level

Select a significance level ( [Tex]\alpha [/Tex] ), typically 0.05, to determine the threshold for rejecting the null hypothesis. It provides validity to our hypothesis test, ensuring that we have sufficient data to back up our claims. Usually, we determine our significance level beforehand of the test. The p-value is the criterion used to calculate our significance value.

Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

• Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
• t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
• Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
• F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

• If Test Statistic>Critical Value: Reject the null hypothesis.
• If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Method B: Using P-values

We can also come to an conclusion using the p-value,

• If the p-value is less than or equal to the significance level i.e. ( [Tex]p\leq\alpha [/Tex] ), you reject the null hypothesis. This indicates that the observed results are unlikely to have occurred by chance alone, providing evidence in favor of the alternative hypothesis.
• If the p-value is greater than the significance level i.e. ( [Tex]p\geq \alpha[/Tex] ), you fail to reject the null hypothesis. This suggests that the observed results are consistent with what would be expected under the null hypothesis.

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

1. Z-statistics:

When population means and standard deviations are known.

[Tex]z = \frac{\bar{x} – \mu}{\frac{\sigma}{\sqrt{n}}}[/Tex]

• [Tex]\bar{x} [/Tex] is the sample mean,
• μ represents the population mean, 
• σ is the standard deviation
• and n is the size of the sample.

2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

[Tex]t=\frac{x̄-μ}{s/\sqrt{n}} [/Tex]

• t = t-score,
• x̄ = sample mean
• μ = population mean,
• s = standard deviation of the sample,
• n = sample size

3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

[Tex]\chi^2 = \sum \frac{(O_{ij} – E_{ij})^2}{E_{ij}}[/Tex]

• [Tex]O_{ij}[/Tex] is the observed frequency in cell [Tex]{ij} [/Tex]
• i,j are the rows and columns index respectively.
• [Tex]E_{ij}[/Tex] is the expected frequency in cell [Tex]{ij}[/Tex] , calculated as : [Tex]\frac{{\text{{Row total}} \times \text{{Column total}}}}{{\text{{Total observations}}}}[/Tex]

Real life Examples of Hypothesis Testing

Let’s examine hypothesis testing using two real life situations,

Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

• Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
• After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

Step 1 : Define the Hypothesis

• Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
• Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

• m  = mean of the difference i.e X after, X before
• s  = standard deviation of the difference (d) i.e d i ​= X after, i ​− X before,
• n  = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

• If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
• If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

Python Implementation of Case A

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

import numpy as np from scipy import stats # Data before_treatment = np . array ([ 120 , 122 , 118 , 130 , 125 , 128 , 115 , 121 , 123 , 119 ]) after_treatment = np . array ([ 115 , 120 , 112 , 128 , 122 , 125 , 110 , 117 , 119 , 114 ]) # Step 1: Null and Alternate Hypotheses # Null Hypothesis: The new drug has no effect on blood pressure. # Alternate Hypothesis: The new drug has an effect on blood pressure. null_hypothesis = "The new drug has no effect on blood pressure." alternate_hypothesis = "The new drug has an effect on blood pressure." # Step 2: Significance Level alpha = 0.05 # Step 3: Paired T-test t_statistic , p_value = stats . ttest_rel ( after_treatment , before_treatment ) # Step 4: Calculate T-statistic manually m = np . mean ( after_treatment - before_treatment ) s = np . std ( after_treatment - before_treatment , ddof = 1 ) # using ddof=1 for sample standard deviation n = len ( before_treatment ) t_statistic_manual = m / ( s / np . sqrt ( n )) # Step 5: Decision if p_value <= alpha : decision = "Reject" else : decision = "Fail to reject" # Conclusion if decision == "Reject" : conclusion = "There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different." else : conclusion = "There is insufficient evidence to claim a significant difference in average blood pressure before and after treatment with the new drug." # Display results print ( "T-statistic (from scipy):" , t_statistic ) print ( "P-value (from scipy):" , p_value ) print ( "T-statistic (calculated manually):" , t_statistic_manual ) print ( f "Decision: { decision } the null hypothesis at alpha= { alpha } ." ) print ( "Conclusion:" , conclusion )

T-statistic (from scipy): -9.0 P-value (from scipy): 8.538051223166285e-06 T-statistic (calculated manually): -9.0 Decision: Reject the null hypothesis at alpha=0.05. Conclusion: There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05. 

• The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
• The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

Step 1: Define the Hypothesis

• Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
• Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

The test statistic is calculated by using the z formula Z = [Tex](203.8 – 200) / (5 \div \sqrt{25}) [/Tex] ​ and we get accordingly , Z =2.039999999999992.

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

Python Implementation of Case B

import scipy.stats as stats import math import numpy as np # Given data sample_data = np . array ( [ 205 , 198 , 210 , 190 , 215 , 205 , 200 , 192 , 198 , 205 , 198 , 202 , 208 , 200 , 205 , 198 , 205 , 210 , 192 , 205 , 198 , 205 , 210 , 192 , 205 ]) population_std_dev = 5 population_mean = 200 sample_size = len ( sample_data ) # Step 1: Define the Hypotheses # Null Hypothesis (H0): The average cholesterol level in a population is 200 mg/dL. # Alternate Hypothesis (H1): The average cholesterol level in a population is different from 200 mg/dL. # Step 2: Define the Significance Level alpha = 0.05 # Two-tailed test # Critical values for a significance level of 0.05 (two-tailed) critical_value_left = stats . norm . ppf ( alpha / 2 ) critical_value_right = - critical_value_left # Step 3: Compute the test statistic sample_mean = sample_data . mean () z_score = ( sample_mean - population_mean ) / \ ( population_std_dev / math . sqrt ( sample_size )) # Step 4: Result # Check if the absolute value of the test statistic is greater than the critical values if abs ( z_score ) > max ( abs ( critical_value_left ), abs ( critical_value_right )): print ( "Reject the null hypothesis." ) print ( "There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL." ) else : print ( "Fail to reject the null hypothesis." ) print ( "There is not enough evidence to conclude that the average cholesterol level in the population is different from 200 mg/dL." )

Reject the null hypothesis. There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL.

Limitations of Hypothesis Testing

• Although a useful technique, hypothesis testing does not offer a comprehensive grasp of the topic being studied. Without fully reflecting the intricacy or whole context of the phenomena, it concentrates on certain hypotheses and statistical significance.
• The accuracy of hypothesis testing results is contingent on the quality of available data and the appropriateness of statistical methods used. Inaccurate data or poorly formulated hypotheses can lead to incorrect conclusions.
• Relying solely on hypothesis testing may cause analysts to overlook significant patterns or relationships in the data that are not captured by the specific hypotheses being tested. This limitation underscores the importance of complimenting hypothesis testing with other analytical approaches.

Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.

Frequently Asked Questions (FAQs)

1. what are the 3 types of hypothesis test.

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.

2.What are the 4 components of hypothesis testing?

Null Hypothesis ( [Tex]H_o [/Tex] ): No effect or difference exists. Alternative Hypothesis ( [Tex]H_1 [/Tex] ): An effect or difference exists. Significance Level ( [Tex]\alpha [/Tex] ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.

3.What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

4.What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

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