91Ó°ÊÓ

State which inference procedure from Chapter \(8,9,\) or 10 you would use. Be specific. For example, you might say, "Two-sample z test for the difference between two proportions." You do not need to carry out any procedures. Which inference method? (a) How do young adults look back on adolescent romance? Investigators interviewed 40 couples in their midtwenties. The female and male partners were interviewed separately. Each was asked about his or her current relationship and also about a romantic relationship that lasted at least two months when they were aged 15 or \(16 .\) One response variable was a measure on a numerical scale of how much the attractiveness of the adolescent partner mattered. You want to find out how much men and women differ on this measure. (b) Are more than \(75 \%\) of Toyota owners generally satisfied with their vehicles? Let's design a study to find out. We'll select a random sample of 400 Toyota owners. Then we'll ask each individual in the sample: "Would you say that you are generally satisfied with your Toyota vehicle?" (c) Are male college students more likely to binge drink than female college students? The Harvard School of Public Health surveys random samples of male and female undergraduates at four-year colleges and universities about whether they have engaged in binge drinking. (d) A bank wants to know which of two incentive plans will most increase the use of its credit cards and by how much. It offers each incentive to a group of current credit card customers, determined at random, and compares the amount charged during the following six months.

Step by step solution

01

Identify Type of Comparison for Men and Women

For part (a), the goal is to compare how men and women differ on the importance of attractiveness of an adolescent partner. This involves comparing means from two separate groups (males and females), leading to the use of a Two-sample t-test for comparing two means.

02

Determine Proportion Test for Satisfaction

In part (b), we want to determine if more than \(75\%\) of Toyota owners are satisfied. This is a single proportion test, so we would use a One-sample z-test for a population proportion to determine if the proportion of satisfied owners is significantly greater than \(75\%\).

03

Evaluate Binge Drinking Difference

Part (c) concerns comparing proportions between two groups (male and female students), with respect to binge drinking. This requires a Two-sample z-test for the difference between two proportions to check if males have a higher proportion of binge drinking than females.

04

Compare Incentive Impact on Spending

In part (d), the bank is comparing the effect of two incentive plans on the spending of credit card users. This is a case of comparing two means from different groups, suggesting the use of a Two-sample t-test for comparing the mean spending under each incentive plan.

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Key Concepts

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

Two-sample t-test

Understanding the Two-sample t-test helps in comparing the means from two distinct groups. Imagine we want to determine if there's a significant difference between the average score of men and women regarding the importance of attractiveness in adolescent relationships. A Two-sample t-test is suitable here, especially when the groups we are comparing are independent of each other. When utilizing this test, it's critical to ensure that we check a few assumptions:

• The data should be approximately normally distributed in each group, especially if the sample size is small.
• Variability or variance in both groups should be similar, which is referred to as homogeneity of variance.

If the data satisfy these conditions, the test will compare the means of the two groups and indicate if any observed difference is statistically significant. With larger sample sizes, the test becomes robust even if data slightly deviates from normality.

One-sample z-test

The One-sample z-test is ideal for evaluating if the proportion of a single sample is different from a known population proportion. For instance, if we want to check whether more than 75% of Toyota owners are satisfied, this is where the One-sample z-test for a proportion comes into play. This test checks if our sample provides enough evidence to conclude that the proportion of satisfied Toyota owners is indeed greater than 75%. Before performing this test:

• Ensure the sample size is large enough, typically at least 30 individuals, to fulfill the normality condition due to large sample size.
• The sample should be representative of the population to yield valid results.

Once these requirements are satisfied, we calculate the z-score, which indicates how many standard deviations our sample proportion is from the hypothesized population proportion, validating our conclusions.

Two-sample z-test

The Two-sample z-test is used when comparing proportions between two independent groups. For example, determining if male college students have a higher proportion of binge drinkers compared to female students, is a classic use case. While setting up this test, you should focus on:

• The independence of the two groups, ensuring that one group's data does not influence the other.
• Sample sizes should be sufficiently large for normal approximation, which typically means having at least 30 observations in each group.

By verifying these conditions, the Two-sample z-test calculates the test statistic that compares the difference between two sample proportions to assume statistical significance. It helps us decide if the observed difference is likely due to random chance or an actual difference between the two groups.

Proportion comparison

Comparison of proportions is a strategic way to determine differences or similarities between groups. For instance, when evaluating the satisfaction rate among Toyota owners against the desired level, or assessing binge drinking rates between male and female students. This process typically involves both qualitative and quantitative assessments, using tests like the z-test for accurate comparison. Here are a few steps to keep in mind:

• Ensure a clear definition of the population and sample for accurate data collection and representation.
• Consider the margin of error when interpreting the results, especially with smaller sample sizes.
• Keep in mind potential biases that might influence both collected data and the final outcomes.

By meticulously following these guidelines, proportion comparison gives significant insights into differences across various domains, facilitating informed decisions.