A typical month, men spend 178 and women spend 96 on...

A typical month, men spend $\$ 178$ and women spend $\$ 96$ on leisure activities," according to the results of an International Communications Research (ICR) for American Express poll, as reported in a USA Today Snapshot found on the Internet June 25,2005 Suppose random samples were taken from the population of male and female college students. Each student was asked to determine his or her expenditures for leisure activities in the prior month. The sample data results had a standard deviation of $\$ 75$ for the men and a standard deviation of $\$ 50$ for the women. a. If both samples were of size $20,$ what is the standard error for the difference of two means? b. Assuming normality in leisure activity expenditures, is the difference found in the ICR poll significant at $\alpha=0.05$ if the samples in part a are used? Explain.

A typical month, men spend $\$ 178$ and women spend $\$ 96$ on leisure activities," according to the results of an International Communications Research (ICR) for American Express poll, as reported in a USA Today Snapshot found on the Internet June 25,2005
Suppose random samples were taken from the population of male and female college students. Each student was asked to determine his or her expenditures for leisure activities in the prior month. The sample data results had a standard deviation of $\$ 75$ for the men and a standard deviation of $\$ 50$ for the women.
a. If both samples were of size $20,$ what is the standard error for the difference of two means?
b. Assuming normality in leisure activity expenditures, is the difference found in the ICR poll significant at $\alpha=0.05$ if the samples in part a are used? Explain.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure that involves formulating a null hypothesis and an alternative hypothesis. Through the calculation of a test statistic and its associated p-value, one determines whether the observed data provide sufficient evidence to reject the null hypothesis in favor of the alternative.

Standard Error

The standard error is a measure of the variability of a sample statistic from sample to sample. It quantifies how much the statistic—such as a mean—is expected to fluctuate due to random sampling of a population and is computed by dividing the standard deviation by the square root of the sample size.

Difference of Two Means

The difference of two means represents the numerical difference between the average values of two independent groups drawn from different populations. This concept is important for comparing groups and serves as the basis for inferential tests that determine if observed differences are statistically significant.

t-Distribution

The t-distribution is used instead of the normal distribution when sample sizes are small and/or the population variances are unknown. It accounts for the additional uncertainty in estimating the population standard deviation from the sample, and it is central to constructing confidence intervals and conducting hypothesis tests on means.

Significance Level

The significance level, denoted as alpha (?), is the threshold probability used in hypothesis testing to decide whether to reject the null hypothesis. For instance, an ? of 0.05 indicates that there is a 5% chance of rejecting the null hypothesis when it is actually true, balancing the risk of Type I error.

Normality Assumption

The normality assumption refers to the requirement that the data (or the sampling distribution of the statistic) are approximately normally distributed. This assumption is particularly important for small sample sizes, where parametric tests like the t-test assume that the underlying data follow a normal distribution in order to produce reliable results.

Transcript

00:01 In this table, i have summarized the information given in the question about the study on men's and women's average spending habits on leisure activities.

00:09 So its sample sizes are both 20.

00:12 Average expenditure by men on leisure activities is $178, $96 for women, and the sample standard deviations are $75 and $50 for men and women, respectively.

00:24 So for part a, we're asked to calculate the estimate on standard error.

00:28 So this is equal, the standard error is calculated like this, and then substituting in the numbers from the table.

01:01 This comes out to 20 .16.

01:04 So that is the estimate for a standard error.

01:09 And for b, we're asked if the difference in men's and women's average spending habits on leisure activities is significant at a significance level of alpha equals 0 .05.

01:23 So we have alpha equals 0 .05.

01:27 So the claim that we're testing is that there is some difference in spending habits.

01:32 So the mean for man minus the mean for women is not equal to zero.

01:41 So the no hypothesis would be that the same.

01:44 So mean for man minus mean for women is equal to zero.

01:51 So we can use the classic method where we compare a test statistic to a critical value.

01:58 So first let's calculate the test statistic...