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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 relevant parameters are the average sales in stores where customers are approached by salespeople (\( \mu_1 \)) and the average sales in stores where they're not approached (\( \mu_2 \)). The null hypothesis (\( H_0 \)) is \( \mu_1 = \mu_2 \) (i.e., there's no difference in average sales between the two types of stores), while the alternative hypothesis (\( H_a \)) is \( \mu_1 > \mu_2 \) (i.e., average sales are higher in stores where customers are approached by salespeople).

Step by step solution

01

- Identifying the parameters

First, let鈥檚 name the parameters related to the problem. We want to compare the average sales figures in stores where customers are approached by salespeople (\( \mu_1 \)) and stores where they aren't (\( \mu_2 \)).
02

- Formulating the null hypothesis

The null hypothesis, typically denoted as \( H_0 \), is always a statement of no effect or no difference. Hence in this case, the null hypothesis would be: There is no difference in the average sales between the stores with salespeople approaching customers and the ones where they aren't. In terms of parameters, it can be written as: \( H_0: \mu_1 = \mu_2 \). This means the null hypothesis assumes the average sales in both types of stores are equal.
03

- Formulating the alternative hypothesis

The alternative hypothesis, denoted as \( H_a \), is a statement that contradicts the null hypothesis and what we are trying to prove. In this case, the alternative hypothesis is: Average sales are higher in stores where customers are approached by salespeople than where they aren't. In terms of parameters, it can be written as: \( H_a: \mu_1 > \mu_2 \). This alternative hypothesis assumes that the average sales in stores where salespeople approach customers are higher than those where they don't.

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Key Concepts

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

Statistical Parameters
Statistical parameters are fundamental components in hypothesis testing. In simple terms, parameters are values that represent a characteristic of a population. In the context of our exercise, the parameters focus on the average sales in different store environments.

In this example, we identify two parameters:
  • \( \mu_1 \): the average sales in stores where customers are approached by salespeople.
  • \( \mu_2 \): the average sales in stores where customers are not approached by salespeople.
These parameters help frame the test and determine whether there's any statistical difference between two different groups. Identifying these parameters is crucial, as they form the basis on which comparisons are made during hypothesis testing.
Null Hypothesis
The null hypothesis (denoted as \( H_0 \)) is a central concept in hypothesis testing. It represents a default position or a statement of no effect or no difference. It is what you attempt to test against through the analysis.

In practice, the null hypothesis is formulated to reflect the assumption that there is no relationship or difference in the context being studied. In our exercise, the null hypothesis is that there is no difference in average sales between the two types of stores. It is mathematically expressed as:

\[ H_0: \mu_1 = \mu_2 \]

By establishing a null hypothesis, researchers can use statistical tests to determine whether there is enough evidence to reject this assumption.
Alternative Hypothesis
The alternative hypothesis (denoted as \( H_a \)) offers a statement that contradicts the null hypothesis. It's what the researcher wants to prove or confirm, showing the presence of an effect or difference.

For our exercise, the alternative hypothesis suggests that stores where customers are approached by salespeople have higher average sales than those where they aren't. It is mathematically denoted as:

\[ H_a: \mu_1 > \mu_2 \]

The alternative hypothesis is crucial because it guides the direction of the test. If the data provides sufficient evidence, the null hypothesis can be rejected in favor of the alternative hypothesis, further supporting the presence of a difference as believed.

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Most popular questions from this chapter

In Exercises 4.146 to \(4.149,\) 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. 4.148 Hypotheses: \(H_{0}: \rho=0\) vs \(H_{a}: \rho \neq 0 .\) In addition, in each case for which the results are significant, give the sign of the correlation. (a) \(95 \%\) confidence interval for \(\rho: \quad 0.07\) to 0.15 (b) \(90 \%\) confidence interval for \(\rho: \quad-0.39\) to -0.78 (c) \(99 \%\) confidence interval for \(\rho:-0.06\) to 0.03

In Exercises 4.146 to \(4.149,\) 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: \(H_{0}: \mu=15\) vs \(H_{a}: \mu \neq 15\) (a) 95\% confidence interval for \(\mu: \quad 13.9\) to 16.2 (b) 95\% confidence interval for \(\mu: \quad 12.7\) to 14.8 (c) 90\% confidence interval for \(\mu: \quad 13.5\) to 16.5

Definition of a P-value Using the definition of a p-value, explain why the area in the tail of a randomization distribution is used to compute a p-value.

Arsenic in Chicken Data 4.5 on page 228 introduces a situation in which a restaurant chain is measuring the levels of arsenic in chicken from its suppliers. The question is whether there is evidence that the mean level of arsenic is greater than 80 ppb, so we are testing \(H_{0}: \mu=80\) vs \(H_{a}: \mu>80\), where \(\mu\) represents the average level of arsenic in all chicken from a certain supplier. It takes money and time to test for arsenic so samples are often small. Suppose \(n=6\) chickens from one supplier are tested, and the levels of arsenic (in ppb) are: \(68, \quad 75\) 81, \(\quad 93\) 134 (a) What is the sample mean for the data? (b) Translate the original sample data by the appropriate amount to create a new dataset in which the null hypothesis is true. How do the sample size and standard deviation of this new dataset compare to the sample size and standard deviation of the original dataset? (c) Write the six new data values from part (b) on six cards. Sample from these cards with replacement to generate one randomization sample. (Select a card at random, record the value, put it back, select another at random, until you have a sample of size \(6,\) to match the original sample size.) List the values in the sample and give the sample mean. (d) Generate 9 more simulated samples, for a total of 10 samples for a randomization distribution. Give the sample mean in each case and create a small dotplot. Use an arrow to locate the original sample mean on your dotplot.

Radiation from Cell Phones and Brain Activity Does heavy cell phone use affect brain activity? There is some concern about possible negative effects of radiofrequency signals delivered to the brain. In a randomized matched-pairs study, \(^{24}\) 47 healthy participants had cell phones placed on the left and right ears. Brain glucose metabolism (a measure of brain activity) was measured for all participants under two conditions: with one cell phone turned on for 50 minutes (the "on" condition) and with both cell phones off (the "off" condition). The amplitude of radiofrequency waves emitted by the cell phones during the "on" condition was also measured. (a) Is this an experiment or an observational study? Explain what it means to say that this was a "matched-pairs" study. (b) How was randomization likely used in the study? Why did participants have cell phones on their ears during the "off" condition? (c) The investigators were interested in seeing whether average brain glucose metabolism was different based on whether the cell phones were turned on or off. State the null and alternative hypotheses for this test. (d) The p-value for the test in part (c) is 0.004 . State the conclusion of this test in context. (e) The investigators were also interested in seeing if brain glucose metabolism was significantly correlated with the amplitude of the radiofrequency waves. What graph might we use to visualize this relationship? (f) State the null and alternative hypotheses for the test in part (e). (g) The article states that the p-value for the test in part (e) satisfies \(p<0.001\). State the conclusion of this test in context.
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