91Ó°ÊÓ

Paying for college College financial aid offices expect students to use summer eamings to help pay for college. But how large are these eamings? One large university studied this question by asking a random sample of 1296 students who had summer jobs how much they earned. The financial aid office separated the responses into two groups based on gender. Here are the data in summary form: \({ }^{30}\) $$ \begin{array}{lccc} \hline \text { Group } & n & \bar{\chi} & s_{\chi} \\ \text { Males } & 675 & \$ 1884.52 & \$ 1368.37 \\ \text { Females } & 621 & \$ 1360.39 & \$ 1037.46 \\ \hline \end{array} $$ (a) How can you tell from the summary statistics that the distribution of earnings in each group is strongly skewed to the right? The use of two-sample \(t\) procedures is still justified. Why? (b) Construct and interpret a \(90 \%\) confidence interval for the difference between the mean summer earnings of male and female students at this university. (c) Interpret the \(90 \%\) confidence level in the context of this study.

Step by step solution

01

Analyze Skewness of the Data

The summary statistics for both males and females show that the means are larger than the medians, and the standard deviations are quite large compared to the means. This is typically indicative of a right-skewed distribution, where a few individuals have very high earnings, pulling the mean upward. This skewness in the data suggests non-normality, but for large sample sizes like these (675 males and 621 females), the Central Limit Theorem justifies the use of two-sample \( t \) procedures even if the data is not perfectly normal.

02

State Method for Confidence Interval Calculation

To find a 90% confidence interval for the difference between the means of the two groups (males and females), we use the formula for the confidence interval for the difference between two independent means:\[CI = (\bar{x}_1 - \bar{x}_2) \pm t^* \cdot \sqrt{ \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} }\]Where \( \bar{x}_1 \) and \( \bar{x}_2 \) are the sample means, \( s_1 \) and \( s_2 \) are the standard deviations, \( n_1 \) and \( n_2 \) are the sample sizes, and \( t^* \) is the critical value from the \( t \) distribution for a 90% confidence level.

03

Calculate the Confidence Interval Components

Given: \( n_1 = 675 \), \( \bar{x}_1 = 1884.52 \), \( s_1 = 1368.37 \); \( n_2 = 621 \), \( \bar{x}_2 = 1360.39 \), \( s_2 = 1037.46 \).The difference of means: \( \bar{x}_1 - \bar{x}_2 = 1884.52 - 1360.39 = 524.13 \).To find the critical value \( t^* \), we refer to the \( t \) distribution table with degrees of freedom approximated using a method like the "Welch–Satterthwaite equation."For large samples, \( t^* \approx 1.645 \) for a 90% confidence level. Now compute the standard error:\[ \sqrt{ \frac{1368.37^2}{675} + \frac{1037.46^2}{621} } \approx 78.24\]

04

Compute the Confidence Interval

Using the values computed: \( 524.13 \pm 1.645 \times 78.24 \).Calculating the margin of error: \( 1.645 \times 78.24 \approx 128.68 \).The 90% confidence interval is:\[(524.13 - 128.68, 524.13 + 128.68) = (395.45, 652.81)\]This means we are 90% confident that the true difference in summer earnings between male and female students is between \(395.45 and \)652.81.

05

Interpret the Confidence Level

The 90% confidence level means that if we were to take many random samples of males and females from this university and compute a confidence interval for each sample, we would expect about 90% of those intervals to contain the true difference in mean earnings between male and female students.

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Key Concepts

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

Two-Sample T Procedures

Two-sample t procedures are used to compare the means of two independent groups. In our context, it involves the comparison of male and female students' summer earnings. This method utilizes the calculated sample means and standard deviations to estimate the difference between the population means. When applying two-sample t procedures, ensure that:

• Both samples are randomly selected.
• The data is collected independently from each group.
• The samples have normal or approximately normal distributions; however, larger samples can relax this condition due to the Central Limit Theorem.

We find the confidence interval using the formula for the difference between two means and accounting for variability within and between samples. The critical value from the t-distribution, often determined based on the desired confidence level (e.g., 90%), helps gauge the precision of our estimate.

Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of the sample mean approaches a normal distribution as the sample size becomes large, regardless of the distribution of the population itself. This principle is critical, especially when dealing with data that isn't perfectly normal or is skewed, as in earnings data. For large sample sizes, like the 675 male students and 621 female students here, the CLT lets us use parametric tests like the two-sample t procedures with confidence because it assures us that the sampling distribution of the mean will be approximately normal. Thus, the approximation becomes increasingly accurate as sample sizes grow. This makes statistical methods robust even for data with skewness or other non-normal features.

Right-Skewed Distribution

A distribution is right-skewed when the tail on the right side is longer or fatter than the left side. Most of the data values fall on the lower end, creating this skewed shape, usually because of a few high outliers pulling the mean upwards. In the context of the college earnings study, both males and females showed a right-skewed distribution due to summary statistics such as larger means compared to medians and relatively large standard deviations. These features indicate a few students with exceptionally high earnings. Analyzing such data involves careful consideration, especially for statistical calculations like means and standard deviations, that skew can distort. Nonetheless, for large samples, two-sample t procedures are still justified due to the robustness provided by the Central Limit Theorem.

Sample Mean Difference

The sample mean difference is a critical component when comparing two groups. It represents the average difference in outcomes, such as earnings, between the groups studied—here, male and female student earnings.Calculating this difference involves subtracting the mean of one sample from the other (\(ar{x}_ ext{male} - \bar{x}_ ext{female}\)). This straightforward calculation provides a preliminary estimate of how the two groups compare, either in study or in practical scenarios.Incorporating statistical measures like the confidence interval adds depth to this simple comparison. By calculating and interpreting a confidence interval for this mean difference, researchers can make informed inferences about the population parameter they are estimating, providing insight into factors such as wage disparities based on gender.