91Ó°ÊÓ

Steve Solomon, the owner of Leonardo's Italian restaurant, wonders whether a redesigned menu will increase, on the average, the amount that customers spend in the restaurant. For the following scenarios, pick a statistical method from this chapter that would be appropriate for analyzing the data, indicating whether the samples are independent or dependent, which parameter is relevant, and what inference method you would use: a. Solomon records the mean sales the week before the change and the week after the change and then wonders whether the difference is "statistically significant." b. Solomon randomly samples 100 people and shows them each both menus, asking them to give a rating between 0 and 10 for each menu. c. Solomon randomly samples 100 people and shows them each both menus, asking them to give an overall rating of positive or negative to each menu. d. Solomon randomly samples 100 people and randomly separates them into two groups of 50 each. He asks those in Group 1 to give a rating to the old menu and those in Group 2 to give a rating to the new menu, using a 0 to 10 rating scale.

Step by step solution

01

Analyzing Scenario (a)

In scenario (a), we need to compare the mean sales from the week before the menu change to the week after the change. Since both sets of sales data are related (same restaurant), we use a paired samples t-test to determine if the change in mean sales is statistically significant. Here, the samples are dependent, and the parameter of interest is the mean sales difference.

02

Analyzing Scenario (b)

In scenario (b), each person rates both menus, making the samples dependent. A paired samples t-test is appropriate for analyzing the mean difference in ratings. The relevant parameter is the mean rating difference for the two menus.

03

Analyzing Scenario (c)

In scenario (c), each person provides a positive or negative rating for both menus, resulting in dependent samples again. The McNemar test, a non-parametric test, is suitable for comparing proportions since the data is categorical (positive or negative ratings). The relevant parameter is the difference in proportions of positive and negative ratings between the two menus.

04

Analyzing Scenario (d)

In scenario (d), we're dealing with two independent groups each rating one menu. An independent samples t-test is best for this scenario as the ratings are continuous, and the samples are independent. The parameter of interest is the difference in mean ratings between the two groups, each representing a different menu.

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Key Concepts

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

Paired Samples t-Test

When you have two related sets of data, the paired samples t-test is an ideal method to determine if there's a significant difference between them. Imagine you want to compare test scores of students before and after a specific teaching method. Because each student's performance before and after the method change is related, a paired samples t-test is suitable. In statistical terms:

• Data sets are related, like before-and-after scenarios.
• You're interested in the mean difference between these two sets.
• The test checks if this difference is statistically significant.

For Steve Solomon's scenario (a), comparing the mean sales of a restaurant before and after a menu change, sales data from the same restaurant is used before and after, making it a perfect fit for a paired samples t-test.

Independent Samples t-Test

The independent samples t-test is applied when comparing two different groups that are not related to each other. Suppose you run a study where group one uses one diet plan and group two uses another. Since participants in one group have no relation to those in the other, this test helps determine disparities in outcomes between these groups. Key factors of this test include:

• Two groups are statistically independent.
• Comparing their means can reveal potential differences.
• Ideal for data that fits a normal distribution.

In scenario (d) from the exercise, 100 people are split into two groups of 50. One group rates the old menu, and the other the new menu. As these groups are distinct and independent, using an independent samples t-test will tell if their rating means differ significantly.

Dependent Samples

Dependent samples often appear when measurements take place on the same individuals under different conditions. This type of sample is frequently involved in experiments measuring effects over time or under varying circumstances. Here’s how dependent samples work:

• Involves the same subjects in two different conditions.
• Any difference in outcomes is within the same subject's experience.
• Commonly analyzed using paired samples t-tests or similar methods.

For instance, in scenario (b) from the exercise, individuals rate both versions of a menu. Since the ratings come from the same persons, the analysis requires recognizing this dependency, just like the paired samples t-test, to properly evaluate differences.

McNemar Test

The McNemar test is a statistical test used for paired nominal data, often involving categorical responses, typically in a 2x2 table. It measures change over time or differences in subjects' attributes that naturally pair together. Consider the important features of the McNemar test:

• Deals with binary, categorical outcomes (e.g., positive/negative).
• Useful for dependent samples where each subject is measured twice.
• Focuses on changes or shifts in categorical ratings between two time points or conditions.

In Steve Solomon's scenario (c), where individuals give positive or negative menu ratings, the data is categorical and dependent (one person rates both menus). The McNemar test helps to evaluate if shifts between positive and negative ratings are statistically significant, making it the right choice in this case.