/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 A situation is described for a s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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.

Beer and Mosquitoes Does consuming beer attract mosquitoes? A study done in Burkino Faso, Africa, about the spread of malaria investigated the connection between beer consumption and mosquito attraction. \(^{9}\) In the experiment, 25 volunteers consumed a liter of beer while 18 volunteers consumed a liter of water. The volunteers \({ }^{8}\) Bouchard, M., Bellinger, D., Wright, \(\mathrm{R}\), and Weisskopf, M. "Attention-Deficit/Hyperactivity Disorder and Urinary Metabolites of Organophosphate Pesticides," Pediatrics, \(2010 ; 125:\) e1270-e1277. \({ }^{9}\) Lefvre, T., et al., "Beer Consumption Increases Human Attractiveness to Malaria Mosquitoes," PLoS ONE, 2010; 5(3): e9546.were assigned to the two groups randomly. The attractiveness to mosquitoes of each volunteer was tested twice: before the beer or water and after. Mosquitoes were released and caught in traps as they approached the volunteers. For the beer group, the total number of mosquitoes caught in the traps before consumption was 434 and the total was 590 after consumption. For the water group, the total was 337 before and 345 after. (a) Define the relevant parameter(s) and state the null and alternative hypotheses for a test to see if, after consumption, the average number of mosquitoes is higher for the volunteers who drank beer. (b) Compute the average number of mosquitoes per volunteer before consumption for each group and compare the results. Are the two sample means different? Do you expect that this difference is just the result of random chance? (c) Compute the average number of mosquitoes per volunteer after consumption for each group and compare the results. Are the two sample means different? Do you expect that this difference is just the result of random chance? (d) If the difference in part (c) is unlikely to happen by random chance, what can we conclude about beer consumption and mosquitoes? (e) If the difference in part (c) is statistically significant, do we have evidence that beer consumption increases mosquito attraction? Why or why not?

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 \(\mathrm{B}\).

Determine whether the sets of hypotheses given are valid hypotheses. State whether each set of hypotheses is valid for a statistical test. If not valid, explain why not. (a) \(H_{0}: \rho=0 \quad\) vs \(\quad H_{a}: \rho<0\) (b) \(H_{0}: \hat{p}=0.3 \quad\) vs \(\quad H_{a}: \hat{p} \neq 0.3\) (c) \(H_{0}: \mu_{1} \neq \mu_{2} \quad\) vs \(\quad H_{a}: \mu_{1}=\mu_{2}\) (d) \(H_{0}: p=25 \quad\) vs \(\quad H_{a}: p \neq 25\)

Mercury Levels in Fish Figure 4.26 shows a scatterplot of the acidity (pH) for a sample of \(n=53\) Florida lakes vs the average mercury level (ppm) found in fish taken from each lake. The full dataset is introduced in Data 2.4 on page 68 and is available in FloridaLakes. There appears to be a negative trend in the scatterplot, and we wish to test whether there is significant evidence of a negative association between \(\mathrm{pH}\) and mercury levels. (a) What are the null and alternative hypotheses? (b) For these data, a statistical software package produces the following output: $$ r=-0.575 \quad p \text { -value }=0.000017 $$ Use the p-value to give the conclusion of the test. Include an assessment of the strength of the evidence and state your result in terms of rejecting or failing to reject \(H_{0}\) and in terms of \(\mathrm{pH}\) and mercury. (c) Is this convincing evidence that low pH causes the average mercury level in fish to increase? Why or why not?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.