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Chapter 4: Problem 12

A situation is described for a statistical test. In each case, define the relevant parameter(s) and state the null and alternative hypotheses. Testing to see if average sales are higher in stores where customers are approached by salespeople than in stores where they aren't.

Short Answer

Expert verified
The parameter is average sales. The null hypothesis is \(H_0: \mu_1 - \mu_2 = 0\), and the alternative hypothesis is \(H_A: \mu_1 - \mu_2 > 0\).

Step by step solution

01

Identify the Parameter

The parameter that will be compared in this study is the average sales. We will denote the average sales where customers are approached by salespeople as \(\mu_1\) and where they aren't approached as \(\mu_2\).
02

State the Null Hypothesis

The null hypothesis suggests that there is no difference between the parameters. So we state it as \(H_0: \mu_1 - \mu_2 = 0\), meaning the average sales are the same whether or not customers are approached by salespeople.
03

State the Alternative Hypothesis

The alternative hypothesis represents the scenario we are interested in proving, i.e., the average sales are higher in stores where customers are approached by salespeople than in stores where they aren't. Thus we phrase it as \(H_A: \mu_1 - \mu_2 > 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
The null hypothesis is a foundational concept in statistical hypothesis testing. It's the default assumption or claim we are testing. Think of the null hypothesis as saying "nothing interesting is happening here." In our example about store sales, the null hypothesis (\(H_0\)) would claim that there is no difference in average sales between the stores where salespeople approach customers and those where they don't. This is expressed mathematically as:
  • \(H_0: \mu_1 - \mu_2 = 0\)
The aid of a null hypothesis is to provide a benchmark to test against. By assuming that there is no effect or difference, it allows us to use statistical evidence to challenge this assumption. When the evidence against the null hypothesis is strong enough, researchers can consider rejecting the null hypothesis in favor of the alternative.
Alternative Hypothesis
The alternative hypothesis is what researchers want to prove or support. It is an assertion that proposes an effect or a difference exists. In our sales example, the alternative hypothesis (\(H_A\)) claims that average sales are higher in stores where customers are approached by salespeople, compared to those where they are not. Thus, it is expressed as:
  • \(H_A: \mu_1 - \mu_2 > 0\)
This hypothesis aligns with our interest in finding if approaching customers has a positive effect on sales. Remember, the alternative hypothesis needs evidence to be considered valid. This means collecting and analyzing data to see if it significantly contradicts the null hypothesis. If it does, we gain support for the alternative hypothesis.
Statistical Parameters
Statistical parameters are numerical characteristics that describe a part of a population. Common parameters include means, variances, and proportions. They help us summarize and understand the population鈥檚 characteristics without analyzing the entire group. In our example, the parameter of interest is the average sales (\(\mu\)) of stores in two different scenarios:
  • \(\mu_1\): average sales where customers are approached by salespeople.
  • \(\mu_2\): average sales where they are not approached.
These parameters are critical because they help set up hypotheses and test them statistically. Knowing whether one parameter differs from another or remains the same helps us draw conclusions about the effects of different business strategies. Using the right parameters in hypothesis testing also ensures that the test is aligned with the study's objectives.

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Most popular questions from this chapter

You roll a die 60 times and record the sample proportion of 5 's, and you want to test whether the die is biased to give more 5 's than a fair die would ordinarily give. To find the p-value for your sample data, you create a randomization distribution of proportions of 5 's in many simulated samples of size 60 with a fair die. (a) State the null and alternative hypotheses. (b) Where will the center of the distribution be? Why? (c) Give an example of a sample proportion for which the number of 5 's obtained is less than what you would expect in a fair die. (d) Will your answer to part (c) lie on the left or the right of the center of the randomization distribution? (e) To find the p-value for your answer to part (c), would you look at the left, right, or both tails? (f) For your answer in part (c), can you say anything about the size of the p-value?

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) (a) 95\% confidence interval for \(p: \quad 0.53\) to 0.57 (b) \(95 \%\) confidence interval for \(p: \quad 0.41\) to 0.52 (c) 99\% confidence interval for \(p: \quad 0.35\) to 0.55

Influencing Voters: Is a Phone Call More Effective? Suppose, as in Exercise \(4.38,\) that we wish to compare methods of influencing voters to support a particular candidate, but in this case we are specifically interested in testing whether a phone call is more effective than a flyer. Suppose also that our random sample consists of only 200 voters, with 100 chosen at random to get the flyer and the rest getting a phone call. (a) State the null and alternative hypotheses in this situation. (b) Display in a two-way table possible sample results that would offer clear evidence that the phone call is more effective. (c) Display in a two-way table possible sample results that offer no evidence at all that the phone call is more effective. (d) Display in a two-way table possible sample results for which the outcome is not clear: there is some evidence in the sample that the phone call is more effective but it is possibly only due to random chance and likely not strong enough to generalize to the population.

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Testing to see whether taking a vitamin supplement each day has significant health benefits. There are no (known) harmful side effects of the supplement.

The consumption of caffeine to benefit alertness is a common activity practiced by \(90 \%\) of adults in North America. Often caffeine is used in order to replace the need for sleep. One study \(^{24}\) compares students' ability to recall memorized information after either the consumption of caffeine or a brief sleep. A random sample of 35 adults (between the ages of 18 and 39 ) were randomly divided into three groups and verbally given a list of 24 words to memorize. During a break, one of the groups takes a nap for an hour and a half, another group is kept awake and then given a caffeine pill an hour prior to testing, and the third group is given a placebo. The response variable of interest is the number of words participants are able to recall following the break. The summary statistics for the three groups are in Table 4.9. We are interested in testing whether there is evidence of a difference in average recall ability between any two of the treatments. Thus we have three possible tests between different pairs of groups: Sleep vs Caffeine, Sleep vs Placebo, and Caffeine vs Placebo. (a) In the test comparing the sleep group to the caffeine group, the p-value is \(0.003 .\) What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, do you think sleep is really better than caffeine for recall ability? (b) In the test comparing the sleep group to the placebo group, the p-value is 0.06 . What is the conclusion of the test using a \(5 \%\) significance level? If we use a \(10 \%\) significance level? How strong is the evidence of a difference in mean recall ability between these two treatments? (c) In the test comparing the caffeine group to the placebo group, the p-value is 0.22 . What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, would we be justified in concluding that caffeine impairs recall ability? (d) According to this study, what should you do before an exam that asks you to recall information?
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