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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 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 who 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.
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.
Learning 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.
Are you interested in improving your data literacy? Download our Beginner’s Guide to Data & Analytics to learn how you can leverage the power of data for professional and organizational success.
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- Innovation Process: Formulating Strong Hypotheses
3 Key Characteristics of Well-Formed Hypotheses
Written by Tendayi Viki on December 03, 2019
When you are working on a new business idea, you may be confident that you are on the right path. However, each building block of your business model and value proposition represents an area of risk that needs to be tested: Customer segments, key resources, channels, value propositions, revenues and costs. In order to run good experiments to test our business ideas we need to formulate strong hypotheses that are testable, precise and discrete.
Before you start testing you idea you have to identify the risks within each building block of your business model and value proposition by formulating hypotheses. We define hypotheses as:
- An assumption that your value proposition, business model, or strategy builds on.
- All the things that would have to be true for you business idea to work.
At the core of identifying hypotheses, you have to ask yourself the question: What would have to be true for our idea to work? You capture each hypothesis by writing phrases that begin with “we believe that...”
We believe that college students prefer mobile banking compared to the traditional banking experience.
Once you’ve identified your hypotheses, you then need to rank them in terms of risk and importance. By using this simple process you can identify which hypothesis you may want to test first. After selecting a hypothesis to test, you then choose the right experiment to run.
But wait! Before you start designing your experiments, it’s important to put on your scientific glasses and refine your hypotheses, to make sure that they’re well formulated.
This helps you ensure that you have a clear connection between the hypothesis you are testing and the results you will get from your experiments. It is difficult to design good experiments if you have badly formulated hypotheses.
Well-formed business hypotheses have three key characteristics; they are testable, precise , and discrete . Let’s look at each one of these in turn.
Your hypothesis is testable when it can be shown true (validated) or false (invalidated), based on evidence (and guided by experience).
It would be difficult to design and execute an experiment for the hypothesis on the left side. To build a good experiment, we need to be more clear about what several of the terms mean, which then makes it easy for us to be clear about exactly what we are expecting to happen. Clarity on exactly what we expect to happen is what make a hypothesis testable.
- Generation Z → Adults between 18-24
- Prefers → spend more time… compared to
- pop-up store → temporary pop-up stores that are placed in co-working spaces
- Branches → traditional banking branches
Precise
Your hypothesis is precise when you know what success looks like. Ideally, it describes the precise what, who, and when of your assumptions.
If you design an experiment for the hypothesis on the left side, different members of your team may end up testing different types of ‘planning for the future’. This is why we have to be more precise about exactly what type of planning we want test.
The hypothesis on the right side ensures that the team are all testing the same thing:
- Young adults → majority of young adults between 18-24
- Plan → don’t save more than $100 per month
- Future → retirement
Discrete
Your hypothesis is discrete when it describes only one distinct, testable, and precise thing you want to investigate.
With the hypothesis on the left side, we’re mixing together two different hypotheses, that one experiment probably couldn’t really test. Instead, we should split these into two hypotheses.
The first hypothesis on the right side is about testing your channels, which can be made more precise and testable by saying how much you want to increase the conversion rates.
The second hypothesis is about the viability of your idea: Can it save money?
Technically, we could use one experiment to test both hypotheses. But by separating the hypothesis into two discrete hypotheses we at least now know that these are two very different things we want to test.
While it may seem bothersome to write hypotheses this way, it is important to do it because it will lead you to design stronger experiments.
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- Encyclopedia of Management
- Hypothesis Testing
HYPOTHESIS TESTING
Social science research, and by extension business research, uses a number of different approaches to study a variety of issues. This research may be a very informal, simple process or it may be a formal, somewhat sophisticated process. Regardless of the type of process, all research begins with a generalized idea in the form of a research question or a hypothesis. A research question usually is posed in the beginning of a research effort or in a specific area of study that has had little formal research. A research question may take the form of a basic question about some issue or phenomena or a question about the relationship between two or more variables. For example, a research question might be: "Do flexible work hours improve employee productivity?" Another question might be: "How do flexible hours influence employees' work?"
A hypothesis differs from a research question; it is more specific and makes a prediction. It is a tentative statement about the relationship between two or more variables. The major difference between a research question and a hypothesis is that a hypothesis predicts an experimental outcome. For example, a hypothesis might state: "There is a positive relationship between the availability of flexible work hours and employee productivity."
Hypotheses provide the following benefits:
- They determine the focus and direction for a research effort.
- Their development forces the researcher to clearly state the purpose of the research activity.
- They determine what variables will not be considered in a study, as well as those that will be considered.
- They require the researcher to have an operational definition of the variables of interest.
The worth of a hypothesis often depends on the researcher's skills. Since the hypothesis is the basis of a research study, it is necessary for the hypothesis be developed with a great deal of thought and contemplation. There are basic criteria to consider when developing a hypothesis, in order to ensure that it meets the needs of the study and the researcher. A good hypothesis should:
- Have logical consistency. Based on the current research literature and knowledge base, does this hypothesis make sense?
- Be in step with the current literature and/or provide a good basis for any differences. Though it does not have to support the current body of literature, it is necessary to provide a good rationale for stepping away from the mainstream.
- Be testable. If one cannot design the means to conduct the research, the hypothesis means nothing.
- Be stated in clear and simple terms in order to reduce confusion.
HYPOTHESIS TESTING PROCESS
Hypothesis testing is a systematic method used to evaluate data and aid the decision-making process. Following is a typical series of steps involved in hypothesis testing:
- State the hypotheses of interest
- Determine the appropriate test statistic
- Specify the level of statistical significance
- Determine the decision rule for rejecting or not rejecting the null hypothesis
- Collect the data and perform the needed calculations
- Decide to reject or not reject the null hypothesis
Each step in the process will be discussed in detail, and an example will follow the discussion of the steps.
STATING THE HYPOTHESES.
A research study includes at least two hypotheses—the null hypothesis and the alternative hypothesis. The hypothesis being tested is referred to as the null hypothesis and it is designated as H It also is referred to as the hypothesis of no difference and should include a statement of equality (=, ≥, or £). The alternative hypothesis presents the alternative to the null and includes a statement of inequality (≠). The null hypothesis and the alternative hypothesis are complementary.
The null hypothesis is the statement that is believed to be correct throughout the analysis, and it is the null hypothesis upon which the analysis is based. For example, the null hypothesis might state that the average age of entering college freshmen is 21 years. H 0 The average age of entering college freshman = 21 years
If the data one collects and analyzes indicates that the average age of entering college freshmen is greater than or less than 21 years, the null hypothesis is rejected. In this case the alternative hypothesis could be stated in the following three ways: (1) the average age of entering college freshman is not 21 years (the average age of entering college freshmen ≠ 21); (2) the average age of entering college freshman is less than 21 years (the average age of entering college freshmen < 21); or (3) the average age of entering college freshman is greater than 21 years (the average age of entering college freshmen > 21 years).
The choice of which alternative hypothesis to use is generally determined by the study's objective. The preceding second and third examples of alternative hypotheses involve the use of a "one-tailed" statistical test. This is referred to as "one-tailed" because a direction (greater than [>] or less than [<]) is implied in the statement. The first example represents a "two-tailed" test. There is inequality expressed (age ≠ 21 years), but the inequality does not imply direction. One-tailed tests are used more often in management and marketing research because there usually is a need to imply a specific direction in the outcome. For example, it is more likely that a researcher would want to know if Product A performed better than Product B (Product A performance > Product B performance), or vice versa (Product A performance < Product B performance), rather than whether Product A performed differently than Product B (Product A performance ≠ Product B performance). Additionally, more useful information is gained by knowing that employees who work from 7:00 a.m. to 4:00 p.m. are more productive than those who work from 3:00 p.m. to 12:00 a.m. (early shift employee production > late shift employee production), rather than simply knowing that these employees have different levels of productivity (early shift employee production ≠ late shift employee production).
Both the alternative and the null hypotheses must be determined and stated prior to the collection of data. Before the alternative and null hypotheses can be formulated it is necessary to decide on the desired or expected conclusion of the research. Generally, the desired conclusion of the study is stated in the alternative hypothesis. This is true as long as the null hypothesis can include a statement of equality. For example, suppose that a researcher is interested in exploring the effects of amount of study time on tests scores. The researcher believes that students who study longer perform better on tests. Specifically, the research suggests that students who spend four hours studying for an exam will get a better score than those who study two hours. In this case the hypotheses might be: H 0 The average test scores of students who study 4 hours for the test = the average test scores of those who study 2 hours. H 1 The average test score of students who study 4 hours for the test < the average test scores of those who study 2 hours.
As a result of the statistical analysis, the null hypothesis can be rejected or not rejected. As a principle of rigorous scientific method, this subtle but important point means that the null hypothesis cannot be accepted. If the null is rejected, the alternative hypothesis can be accepted; however, if the null is not rejected, we can't conclude that the null hypothesis is true. The rationale is that evidence that supports a hypothesis is not conclusive, but evidence that negates a hypothesis is ample to discredit a hypothesis. The analysis of study time and test scores provides an example. If the results of one study indicate that the test scores of students who study 4 hours are significantly better than the test scores of students who study two hours, the null hypothesis can be rejected because the researcher has found one case when the null is not true. However, if the results of the study indicate that the test scores of those who study 4 hours are not significantly better than those who study 2 hours, the null hypothesis cannot be rejected. One also cannot conclude that the null hypothesis is accepted because these results are only one set of score comparisons. Just because the null hypothesis is true in one situation does not mean it is always true.
DETERMINING THE APPROPRIATE TEST STATISTIC.
The appropriate test statistic (the statistic to be used in statistical hypothesis testing) is based on various characteristics of the sample population of interest, including sample size and distribution. The test statistic can assume many numerical values. Since the value of the test statistic has a significant effect on the decision, one must use the appropriate statistic in order to obtain meaningful results. Most test statistics follow this general pattern:
For example, the appropriate statistic to use when testing a hypothesis about a population means is:
In this formula Z = test statistic, Χ̅ = mean of the sample, μ = mean of the population, σ = standard deviation of the sample, and η = number in the sample.
SPECIFYING THE STATISTICAL SIGNIFICANCE SEVEL.
As previously noted, one can reject a null hypothesis or fail to reject a null hypothesis. A null hypothesis that is rejected may, in reality, be true or false. Additionally, a null hypothesis that fails to be rejected may, in reality, be true or false. The outcome that a researcher desires is to reject a false null hypothesis or to fail to reject a true null hypothesis. However, there always is the possibility of rejecting a true hypothesis or failing to reject a false hypothesis.
Rejecting a null hypothesis that is true is called a Type I error and failing to reject a false null hypothesis is called a Type II error. The probability of committing a Type I error is termed α and the probability of committing a Type II error is termed β. As the value of α increases, the probability of committing a Type I error increases. As the value of β increases, the probability of committing a Type II error increases. While one would like to decrease the probability of committing of both types of errors, the reduction of α results in the increase of β and vice versa. The best way to reduce the probability of decreasing both types of error is to increase sample size.
The probability of committing a Type I error, α, is called the level of significance. Before data is collected one must specify a level of significance, or the probability of committing a Type I error (rejecting a true null hypothesis). There is an inverse relationship between a researcher's desire to avoid making a Type I error and the selected value of α; if not making the error is particularly important, a low probability of making the error is sought. The greater the desire is to not reject a true null hypothesis, the lower the selected value of α. In theory, the value of α can be any value between 0 and 1. However, the most common values used in social science research are .05, .01, and .001, which respectively correspond to the levels of 95 percent, 99 percent, and 99.9 percent likelihood that a Type I error is not being made. The tradeoff for choosing a higher level of certainty (significance) is that it will take much stronger statistical evidence to ever reject the null hypothesis.
DETERMINING THE DECISION RULE.
Before data are collected and analyzed it is necessary to determine under what circumstances the null hypothesis will be rejected or fail to be rejected. The decision rule can be stated in terms of the computed test statistic, or in probabilistic terms. The same decision will be reached regardless of which method is chosen.
COLLECTING THE DATA AND PERFORMING THE CALCULATIONS.
The method of data collection is determined early in the research process. Once a research question is determined, one must make decisions regarding what type of data is needed and how the data will be collected. This decision establishes the bases for how the data will be analyzed. One should use only approved research methods for collecting and analyzing data.
DECIDING WHETHER TO REJECT THE NULL HYPOTHESIS.
This step involves the application of the decision rule. The decision rule allows one to reject or fail to reject the null hypothesis. If one rejects the null hypothesis, the alternative hypothesis can be accepted. However, as discussed earlier, if one fails to reject the null he or she can only suggest that the null may be true.
XYZ Corporation is a company that is focused on a stable workforce that has very little turnover. XYZ has been in business for 50 years and has more than 10,000 employees. The company has always promoted the idea that its employees stay with them for a very long time, and it has used the following line in its recruitment brochures: "The average tenure of our employees is 20 years." Since XYZ isn't quite sure if that statement is still true, a random sample of 100 employees is taken and the average age turns out to be 19 years with a standard deviation of 2 years. Can XYZ continue to make its claim, or does it need to make a change?
- State the hypotheses. H 0 = 20 years H 1 ≠ 20 years
- Determine the test statistic. Since we are testing a population mean that is normally distributed, the appropriate test statistic is:
- Specify the significance level. Since the firm would like to keep its present message to new recruits, it selects a fairly weak significance level (α = .05). Since this is a two-tailed test, half of the alpha will be assigned to each tail of the distribution. In this situation the critical values of Z = +1.96 and −1.96.
- State the decision rule. If the computed value of Z is greater than or equal to +1.96 or less than or equal to −1.96, the null hypothesis is rejected.
- Calculations.
- Reject or fail to reject the null. Since 2.5 is greater than 1.96, the null is rejected. The mean tenure is not 20 years, therefore XYZ needs to change its statement.
SEE ALSO: Research Methods and Processes ; Statistics
Donna T. Mayo
Revised by Marcia Simmering
FURTHER READING:
Anderson, David R., Dennis J. Sweeney, and Thomas A. Williams. Statistics for Business and Economics. 9th ed. Mason, OH: South-Western College Publishing, 2004.
Kerlinger, Fred N., and Howard B. Lee. Foundations of Behavioral Research. 4th ed. Fort Worth, TX: Harcourt College Publishers, 2000.
Pedhazur, Elazar J., and Liora Pedhazur Schmelkin. Measurement, Design, and Analysis: An Integrated Approach. Hillsdale, NJ: Lawrence Erlbaum Associates, 1991.
Schwab, Donald P. Research Methods for Organizational Studies. Mahwah, NJ: Lawrence Erlbaum Associates, 1999.
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“A fact is a simple statement that everyone believes. It is innocent, unless found guilty. A hypothesis is a novel suggestion that no one wants to believe. It is guilty until found effective.”
– Edward Teller, Nuclear Physicist
During my first brainstorming meeting on my first project at McKinsey, this very serious partner, who had a PhD in Physics, looked at me and said, “So, Joe, what are your main hypotheses.” I looked back at him, perplexed, and said, “Ummm, my what?” I was used to people simply asking, “what are your best ideas, opinions, thoughts, etc.” Over time, I began to understand the importance of hypotheses and how it plays an important role in McKinsey’s problem solving of separating ideas and opinions from facts.
What is a Hypothesis?
“Hypothesis” is probably one of the top 5 words used by McKinsey consultants. And, being hypothesis-driven was required to have any success at McKinsey. A hypothesis is an idea or theory, often based on limited data, which is typically the beginning of a thread of further investigation to prove, disprove or improve the hypothesis through facts and empirical data.
The first step in being hypothesis-driven is to focus on the highest potential ideas and theories of how to solve a problem or realize an opportunity.
Let’s go over an example of being hypothesis-driven.
Let’s say you own a website, and you brainstorm ten ideas to improve web traffic, but you don’t have the budget to execute all ten ideas. The first step in being hypothesis-driven is to prioritize the ten ideas based on how much impact you hypothesize they will create.
The second step in being hypothesis-driven is to apply the scientific method to your hypotheses by creating the fact base to prove or disprove your hypothesis, which then allows you to turn your hypothesis into fact and knowledge. Running with our example, you could prove or disprove your hypothesis on the ideas you think will drive the most impact by executing:
1. An analysis of previous research and the performance of the different ideas 2. A survey where customers rank order the ideas 3. An actual test of the ten ideas to create a fact base on click-through rates and cost
While there are many other ways to validate the hypothesis on your prioritization, I find most people do not take this critical step in validating a hypothesis. Instead, they apply bad logic to many important decisions. An idea pops into their head, and then somehow it just becomes a fact.
One of my favorite lousy logic moments was a CEO who stated,
“I’ve never heard our customers talk about price, so the price doesn’t matter with our products, and I’ve decided we’re going to raise prices.”
Luckily, his management team was able to do a survey to dig deeper into the hypothesis that customers weren’t price-sensitive. Well, of course, they were and through the survey, they built a fantastic fact base that proved and disproved many other important hypotheses.
Why is being hypothesis-driven so important?
Imagine if medicine never actually used the scientific method. We would probably still be living in a world of lobotomies and bleeding people. Many organizations are still stuck in the dark ages, having built a house of cards on opinions disguised as facts, because they don’t prove or disprove their hypotheses. Decisions made on top of decisions, made on top of opinions, steer organizations clear of reality and the facts necessary to objectively evolve their strategic understanding and knowledge. I’ve seen too many leadership teams led solely by gut and opinion. The problem with intuition and gut is if you don’t ever prove or disprove if your gut is right or wrong, you’re never going to improve your intuition. There is a reason why being hypothesis-driven is the cornerstone of problem solving at McKinsey and every other top strategy consulting firm.
How do you become hypothesis-driven?
Most people are idea-driven, and constantly have hypotheses on how the world works and what they or their organization should do to improve. Though, there is often a fatal flaw in that many people turn their hypotheses into false facts, without actually finding or creating the facts to prove or disprove their hypotheses. These people aren’t hypothesis-driven; they are gut-driven.
The conversation typically goes something like “doing this discount promotion will increase our profits” or “our customers need to have this feature” or “morale is in the toilet because we don’t pay well, so we need to increase pay.” These should all be hypotheses that need the appropriate fact base, but instead, they become false facts, often leading to unintended results and consequences. In each of these cases, to become hypothesis-driven necessitates a different framing.
• Instead of “doing this discount promotion will increase our profits,” a hypothesis-driven approach is to ask “what are the best marketing ideas to increase our profits?” and then conduct a marketing experiment to see which ideas increase profits the most.
• Instead of “our customers need to have this feature,” ask the question, “what features would our customers value most?” And, then conduct a simple survey having customers rank order the features based on value to them.
• Instead of “morale is in the toilet because we don’t pay well, so we need to increase pay,” conduct a survey asking, “what is the level of morale?” what are potential issues affecting morale?” and what are the best ideas to improve morale?”
Beyond, watching out for just following your gut, here are some of the other best practices in being hypothesis-driven:
Listen to Your Intuition
Your mind has taken the collision of your experiences and everything you’ve learned over the years to create your intuition, which are those ideas that pop into your head and those hunches that come from your gut. Your intuition is your wellspring of hypotheses. So listen to your intuition, build hypotheses from it, and then prove or disprove those hypotheses, which will, in turn, improve your intuition. Intuition without feedback will over time typically evolve into poor intuition, which leads to poor judgment, thinking, and decisions.
Constantly Be Curious
I’m always curious about cause and effect. At Sports Authority, I had a hypothesis that customers that received service and assistance as they shopped, were worth more than customers who didn’t receive assistance from an associate. We figured out how to prove or disprove this hypothesis by tying surveys to transactional data of customers, and we found the hypothesis was true, which led us to a broad initiative around improving service. The key is you have to be always curious about what you think does or will drive value, create hypotheses and then prove or disprove those hypotheses.
Validate Hypotheses
You need to validate and prove or disprove hypotheses. Don’t just chalk up an idea as fact. In most cases, you’re going to have to create a fact base utilizing logic, observation, testing (see the section on Experimentation), surveys, and analysis.
Be a Learning Organization
The foundation of learning organizations is the testing of and learning from hypotheses. I remember my first strategy internship at Mercer Management Consulting when I spent a good part of the summer combing through the results, findings, and insights of thousands of experiments that a banking client had conducted. It was fascinating to see the vastness and depth of their collective knowledge base. And, in today’s world of knowledge portals, it is so easy to disseminate, learn from, and build upon the knowledge created by companies.
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Statistics Made Easy
4 Examples of Hypothesis Testing in Real Life
In statistics, hypothesis tests are used to test whether or not some hypothesis about a population parameter is true.
To perform a hypothesis test in the real world, researchers will obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:
- Null Hypothesis (H 0 ): The sample data occurs purely from chance.
- Alternative Hypothesis (H A ): The sample data is influenced by some non-random cause.
If the p-value of the hypothesis test is less than some significance level (e.g. α = .05), then we can reject the null hypothesis and conclude that we have sufficient evidence to say that the alternative hypothesis is true.
The following examples provide several situations where hypothesis tests are used in the real world.
Example 1: Biology
Hypothesis tests are often used in biology to determine whether some new treatment, fertilizer, pesticide, chemical, etc. causes increased growth, stamina, immunity, etc. in plants or animals.
For example, suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than they normally do, which is currently 20 inches. To test this, she applies the fertilizer to each of the plants in her laboratory for one month.
She then performs a hypothesis test using the following hypotheses:
- H 0 : μ = 20 inches (the fertilizer will have no effect on the mean plant growth)
- H A : μ > 20 inches (the fertilizer will cause mean plant growth to increase)
If the p-value of the test is less than some significance level (e.g. α = .05), then she can reject the null hypothesis and conclude that the fertilizer leads to increased plant growth.
Example 2: Clinical Trials
Hypothesis tests are often used in clinical trials to determine whether some new treatment, drug, procedure, etc. causes improved outcomes in patients.
For example, suppose a doctor believes that a new drug is able to reduce blood pressure in obese patients. To test this, he may measure the blood pressure of 40 patients before and after using the new drug for one month.
He then performs a hypothesis test using the following hypotheses:
- H 0 : μ after = μ before (the mean blood pressure is the same before and after using the drug)
- H A : μ after < μ before (the mean blood pressure is less after using the drug)
If the p-value of the test is less than some significance level (e.g. α = .05), then he can reject the null hypothesis and conclude that the new drug leads to reduced blood pressure.
Example 3: Advertising Spend
Hypothesis tests are often used in business to determine whether or not some new advertising campaign, marketing technique, etc. causes increased sales.
For example, suppose a company believes that spending more money on digital advertising leads to increased sales. To test this, the company may increase money spent on digital advertising during a two-month period and collect data to see if overall sales have increased.
They may perform a hypothesis test using the following hypotheses:
- H 0 : μ after = μ before (the mean sales is the same before and after spending more on advertising)
- H A : μ after > μ before (the mean sales increased after spending more on advertising)
If the p-value of the test is less than some significance level (e.g. α = .05), then the company can reject the null hypothesis and conclude that increased digital advertising leads to increased sales.
Example 4: Manufacturing
Hypothesis tests are also used often in manufacturing plants to determine if some new process, technique, method, etc. causes a change in the number of defective products produced.
For example, suppose a certain manufacturing plant wants to test whether or not some new method changes the number of defective widgets produced per month, which is currently 250. To test this, they may measure the mean number of defective widgets produced before and after using the new method for one month.
They can then perform a hypothesis test using the following hypotheses:
- H 0 : μ after = μ before (the mean number of defective widgets is the same before and after using the new method)
- H A : μ after ≠ μ before (the mean number of defective widgets produced is different before and after using the new method)
If the p-value of the test is less than some significance level (e.g. α = .05), then the plant can reject the null hypothesis and conclude that the new method leads to a change in the number of defective widgets produced per month.
Additional Resources
Introduction to Hypothesis Testing Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test
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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 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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- How to Write a Strong Hypothesis | Steps & Examples
How to Write a Strong Hypothesis | Steps & Examples
Published on May 6, 2022 by Shona McCombes . Revised on December 2, 2022.
A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .
Example: Hypothesis
Daily apple consumption leads to fewer doctor’s visits.
Table of contents
What is a hypothesis, developing a hypothesis (with example), hypothesis examples, frequently asked questions about writing hypotheses.
A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.
A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).
Variables in hypotheses
Hypotheses propose a relationship between two or more types of variables .
- An independent variable is something the researcher changes or controls.
- A dependent variable is something the researcher observes and measures.
If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias will affect your results.
In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .
Step 1. Ask a question
Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.
Step 2. Do some preliminary research
Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.
At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.
Step 3. Formulate your hypothesis
Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.
4. Refine your hypothesis
You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:
- The relevant variables
- The specific group being studied
- The predicted outcome of the experiment or analysis
5. Phrase your hypothesis in three ways
To identify the variables, you can write a simple prediction in if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.
In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.
If you are comparing two groups, the hypothesis can state what difference you expect to find between them.
6. Write a null hypothesis
If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .
- H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
- H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.
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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).
Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.
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Published on May 6, 2022 by Shona McCombes. Revised on December 2, 2022. A hypothesis is a statement that can be tested by scientific research.
Hypothesis testing can be used in business applications to help validate an assumption being made about data relationships. This lesson looks at...