91Ó°ÊÓ

How much customers buy is a direct result of how much time they spend in the store. A study of average shopping times in a large national houseware store gave the following information (Source: Why We Buy: The Science of Shopping by P. Underhill): Women with female companion: \(8.3 \mathrm{~min}\). Women with male companion: \(4.5 \mathrm{~min}\). Suppose you want to set up a statistical test to challenge the claim that a woman with a female friend spends an average of \(8.3\) minutes shopping in such a store. (a) What would you use for the null and alternate hypotheses if you believe the average shopping time is less than \(8.3\) minutes? Is this a right-tailed, left-tailed, or two-tailed test? (b) What would you use for the null and alternate hypotheses if you believe the average shopping time is different from \(8.3\) minutes? Is this a right- tailed, left-tailed, or two-tailed test? Stores that sell mainly to women should figure out a way to engage the interest of men! Perhaps comfortable seats and a big TV with sports programs. Suppose such an entertainment center was installed and you now wish to challenge the claim that a woman with a male friend spends only \(4.5\) minutes shopping in a houseware store. (c) What would you use for the null and alternate hypotheses if you believe the average shopping time is more than \(4.5\) minutes? Is this a right-tailed, lefttailed, or two-tailed test? (d) What would you use for the null and alternate hypotheses if you believe the average shopping time is different from \(4.5\) minutes? Is this a right- tailed, left-tailed, or two-tailed test?

Step by step solution

01

Define Hypotheses for Less Than 8.3 Minutes

To set up a statistical test, start by identifying the null and alternate hypotheses. For part (a), we suspect that women with a female companion spend less than 8.3 minutes shopping. - Null Hypothesis: \( H_0: \mu = 8.3 \) minutes- Alternate Hypothesis: \( H_a: \mu < 8.3 \) minutesSince we are checking for less than the average, this is a left-tailed test.

02

Define Hypotheses for Different from 8.3 Minutes

For part (b), we want to test if the average time differs from 8.3 minutes. This requires examining both directions (less than and greater than).- Null Hypothesis: \( H_0: \mu = 8.3 \) minutes- Alternate Hypothesis: \( H_a: \mu eq 8.3 \) minutesThis is a two-tailed test as we are interested in deviations on both sides.

03

Define Hypotheses for More Than 4.5 Minutes

In part (c), we challenge the claim that the time spent is more than 4.5 minutes when a woman shops with a male companion.- Null Hypothesis: \( H_0: \mu = 4.5 \) minutes- Alternate Hypothesis: \( H_a: \mu > 4.5 \) minutesThis indicates a right-tailed test as we are checking for more than the average.

04

Define Hypotheses for Different from 4.5 Minutes

For part (d), we suspect that the average shopping time differs from 4.5 minutes.- Null Hypothesis: \( H_0: \mu = 4.5 \) minutes- Alternate Hypothesis: \( H_a: \mu eq 4.5 \) minutesThis is a two-tailed test as it involves checking deviations on both ends of the mean.

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 91Ó°ÊÓ!

Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement that suggests no effect or no difference. It's a starting point for any hypothesis test. By default, the null hypothesis assumes that any kind of difference or change is due to random chance.

For example, in problems (a) and (c) regarding shopping times, we have these null hypotheses:

• For women with female companions:
  • Null Hypothesis: \( H_0: \mu = 8.3 \) minutes
• For women with male companions:
  • Null Hypothesis: \( H_0: \mu = 4.5 \) minutes

In both cases, the null hypothesis claims that average shopping times are exactly \(8.3\) minutes and \(4.5\) minutes, respectively. The null hypothesis is often believed false only if there is enough statistical evidence.

Alternate Hypothesis

The alternate hypothesis proposes a condition that is different from the null hypothesis. It suggests that there is an actual, observed effect or difference. In hypothesis testing, scientists or researchers try to find evidence against the null hypothesis, often with the hope that they can support the alternate hypothesis.

Consider these problems once again:

• For part (a), with women and female companions, where we suspect less time is spent:
  • Alternate Hypothesis: \( H_a: \mu < 8.3 \) minutes
• For part (c), with women and male companions, where we suspect more time is spent:
  • Alternate Hypothesis: \( H_a: \mu > 4.5 \) minutes

These hypotheses indicate that average times are respectively less or greater than the mentioned averages.

Tail Test Types

In statistical hypothesis testing, the concept of 'tail' relates to how we look at the probability of observing a sample statistic as extreme as, or more extreme than, the statistic actually observed, under the assumption that the null hypothesis is true.

There are three types of tail tests:

• **Left-Tailed Test**: This is used when the alternate hypothesis suggests that the actual parameter is less than the null hypothesis value. In part (a), because we are testing if shopping time is less than 8.3 minutes, it is a left-tailed test.
• **Right-Tailed Test**: This test applies when the alternate hypothesis indicates that the parameter is more than the null hypothesis value. In part (c), since we are suspecting more than 4.5 minutes, it's a right-tailed test.
• **Two-Tailed Test**: If the alternate hypothesis implies that the parameter differs from the null hypothesis value irrespective of direction—meaning it could be either more or less—then a two-tailed test is appropriate. As in parts (b) and (d), where the claim is just different, not specifically higher or lower, making it a two-tailed test.

Choosing the correct tail test type is crucial as it directly influences the critical regions and how we will make decisions about the null hypothesis.