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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 4.5 to 4.8 , state the null and alternative hvpotheses for the statistical test described. Testing to see if there is evidence that the mean of group \(\mathrm{A}\) is not the same as the mean of group \(\mathrm{B}\).

Exercises 4.117 to 4.122 give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.6\) vs \(H_{a}: p>0.6\) Sample data: \(\hat{p}=52 / 80=0.65\) with \(n=80\)

State the null and alternative hvpotheses for the statistical test described. Testing to see if there is evidence that a proportion is greater than 0.3

Rolling Dice You roll a die 60 times and record the sample proportion of fives, and you want to test whether the die is biased to give more fives than a fair die would ordinarily give. To find the p-value for your sample data, you create a randomization distribution of proportions of fives 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?

Car Window Skin Cancer? A new study suggests that exposure to UV rays through the car window may increase the risk of skin cancer. \(^{43}\) The study reviewed the records of all 1050 skin cancer patients referred to the St. Louis University Cancer Center in 2004 . Of the 42 patients with melanoma, the cancer occurred on the left side of the body in 31 patients and on the right side in the other 11 . (a) Is this an experiment or an observational study? (b) Of the patients with melanoma, what proportion had the cancer on the left side? (c) A bootstrap \(95 \%\) confidence interval for the proportion of melanomas occurring on the left is 0.579 to \(0.861 .\) Clearly interpret the confidence interval in the context of the problem. (d) Suppose the question of interest is whether melanomas are more likely to occur on the left side than on the right. State the null and alternative hypotheses. (e) Is this a one-tailed or two-tailed test? (f) Use the confidence interval given in part (c) to predict the results of the hypothesis test in part (d). Explain your reasoning. (g) A randomization distribution gives the p-value as 0.003 for testing the hypotheses given in part (d). What is the conclusion of the test in the context of this study? (h) The authors hypothesize that skin cancers are more prevalent on the left because of the sunlight coming in through car windows. (Windows protect against UVB rays but not UVA rays.) Do the data in this study support a conclusion that more melanomas occur on the left side because of increased exposure to sunlight on that side for drivers?
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