Bank of America's Consumer Spending Survey collected data on...

Bank of America's Consumer Spending Survey collected data on annual credit card charges in seven different categories of expenditures: transportation, groceries, dining out, household expenses, home furnishings, apparel, and entertainment. Using data from a sample of 42 credit card accounts, assume that each account was used to identify the annual credit card charges for groceries (population 1) and the annual credit card charges for dining out (population 2). Using the difference data, the sample mean difference was $$\bar{d}=\$ 850$$, and the sample standard deviation was $$s_d=\$ 1123$$. a. Formulate the null and alternative hypotheses to test for no difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out. b. Use a .05 level of significance. Can you conclude that the population means differ? What is the $p$-value? c. Which category, groceries or dining out, has a higher population mean annual credit card charge? What is the point estimate of the difference between the population means? What is the $95 \%$ confidence interval estimate of the difference between the population means?

Bank of America's Consumer Spending Survey collected data on annual credit card charges in seven different categories of expenditures: transportation, groceries, dining out, household expenses, home furnishings, apparel, and entertainment. Using data from a sample of 42 credit card accounts, assume that each account was used to identify the annual credit card charges for groceries (population 1) and the annual credit card charges for dining out (population 2). Using the difference data, the sample mean difference was $$\bar{d}=\$ 850$$, and the sample standard deviation was $$s_d=\$ 1123$$.
a. Formulate the null and alternative hypotheses to test for no difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out.
b. Use a .05 level of significance. Can you conclude that the population means differ? What is the $p$-value?
c. Which category, groceries or dining out, has a higher population mean annual credit card charge? What is the point estimate of the difference between the population means? What is the $95 \%$ confidence interval estimate of the difference between the population means?

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from the sample data, within which the true population parameter is expected to fall with a specified level of confidence (e.g., 95%). This interval gives a sense of the precision of the point estimate and helps assess the practical significance of the findings.

p-value

The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It is used to assess the strength of the evidence against the null hypothesis; if the p-value is less than the significance level, the null hypothesis is rejected.

Significance Level

The significance level, often denoted by alpha (?), is the probability threshold used to determine whether the observed data are sufficiently extreme to reject the null hypothesis. In this case, a 0.05 significance level indicates that there is a 5% chance of rejecting the null hypothesis when it is actually true, guiding the interpretation of the test results.

Point Estimate

A point estimate is a single value metric that provides the best estimate of a population parameter based on sample data. In the context of this analysis, the sample mean difference serves as the point estimate for the difference between the population means of the two types of expenditures.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. For comparing paired data, null hypotheses typically state that there is no difference between the two population means, while alternative hypotheses state that a difference exists.

Paired Sample Design

This concept pertains to situations where the same subjects are measured under two different conditions, or paired observations are collected. In this context, each credit card account provides two related measurements (groceries and dining out), allowing the analysis to control for variability across different accounts, which is a central idea when using paired sample test methods.

Null and Alternative Hypotheses

This concept involves the formal setting up of the hypothesis test. The null hypothesis (H0) usually states that there is no difference between the population means (i.e., their difference is zero), and the alternative hypothesis (H1) is that there is a nonzero difference. These hypotheses form the basis of the statistical test conducted in the analysis.

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