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How much customers buy is a direct result of how much time they spend in a store. A study of average shopping times in a large national housewares 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, lefttailed, 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 housewares 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

Understanding Part (a)

For part (a), we focus on women with female companions. We are tasked with challenging the claim that a woman with a female friend spends an average of 8.3 minutes shopping. Since we want to test if the average time is less than this, we set up hypotheses as follows:- Null Hypothesis (H_0): \( \mu = 8.3 \) minutes- Alternate Hypothesis (H_a): \( \mu < 8.3 \) minutesThis is a left-tailed test because we are testing for a value less than the hypothesized mean.

02

Understanding Part (b)

For part (b), we still focus on women with female companions, but now we are interested if the average shopping time is simply different from 8.3 minutes. The hypothesis setup is as follows:- Null Hypothesis (H_0): \( \mu = 8.3 \) minutes- Alternate Hypothesis (H_a): \( \mu eq 8.3 \) minutesThis is a two-tailed test because we are looking for any difference from the hypothesized mean (either less or more).

03

Understanding Part (c)

For part (c), we switch focus to women with male companions. Here, we challenge the claim that the average shopping time is only 4.5 minutes, hypothesizing that it's actually more. The hypotheses are:- Null Hypothesis (H_0): \( \mu = 4.5 \) minutes- Alternate Hypothesis (H_a): \( \mu > 4.5 \) minutesThis is a right-tailed test because we are testing for a mean greater than the hypothesized mean.

04

Understanding Part (d)

For part (d), again focusing on women with male companions, we want to see if the shopping time is different from 4.5 minutes. The hypotheses should be:- Null Hypothesis (H_0): \( \mu = 4.5 \) minutes- Alternate Hypothesis (H_a): \( \mu eq 4.5 \) minutesThis is a two-tailed test, since we're testing for any deviation from the hypothesized mean (either more or less).

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Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis, denoted as \( H_0 \), is a statement of no effect, no difference, or the status quo. It's the claim that researchers aim to test against. In the context of shopping times in a store, for example, if the store claims that women with female companions spend an average of 8.3 minutes there, the null hypothesis would be \( \mu = 8.3 \) minutes. This means we start with the assumption that there is no difference from this value. The idea is that any deviation from this assumed value is due to random chance unless significant evidence suggests otherwise.

Alternate Hypothesis

The alternate hypothesis, denoted as \( H_a \), challenges the null hypothesis by suggesting that there is an effect, a difference, or a change. It reflects what you want to prove. For instance, if you believe that women with female companions spend a different amount of time than the claimed average of 8.3 minutes, the alternate hypothesis would be \( \mu eq 8.3 \) minutes. This contrasts with the null hypothesis. Depending on what you are testing (more, less, or different), the form of the alternate hypothesis can vary. The direction of the test, whether left-tailed, right-tailed, or two-tailed, is determined by how you frame the alternate hypothesis.

Left-Tailed Test

A left-tailed test is used when the alternate hypothesis suggests that the true parameter is less than the null hypothesis value. In our shopping time example, if one hypothesizes that women with female companions spend less than the stated 8.3 minutes, the alternate hypothesis would be \( \mu < 8.3 \) minutes. This type of test focuses on the lower tail of the distribution. Practically, it attempts to find evidence that the true mean is smaller than what is claimed, testing whether the data significantly sits in that left tail.

Right-Tailed Test

A right-tailed test is appropriate when the alternate hypothesis posits that the true parameter is greater than the value claimed by the null hypothesis. Using our housewares store example, if there's a belief that women with male companions actually spend more than the average 4.5 minutes stated, then the alternate hypothesis is \( \mu > 4.5 \) minutes. This involves examining whether the sample data is significantly far out in the right tail of the distribution, indicating a mean larger than expected under the null hypothesis.

Two-Tailed Test

A two-tailed test comes into play when you are interested in any significant difference from the hypothesized value, either less or more. For example, if you're testing whether the shopping time for women with male companions is simply different from 4.5 minutes, the alternate hypothesis becomes \( \mu eq 4.5 \) minutes. This test splits the significance level equally into two tails of the distribution. Its purpose is to detect deviations on either side of the hypothesized mean, making it suitable for identifying any unexpected results that are not specified in one particular direction.