91Ó°ÊÓ

Paying for college College financial aid offices expect students to use summer earnings to help pay for college. But how large are these earnings? 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:\(^{33}\) $$\begin{array}{llll}{\text { Group }} & {n} & {\overline{x}} & {s_{x}} \\\ \hline \text { Males } & {675} & {\$ 1884.52} & {\$ 13688.37} \\ {\text { Females }} & {621} & {\$ 1360.39} & {\$ 1037.46}\end{array}$$ (a) How can you tell from the summary statistics that the distribution of earnings in each group is strongly skewed to the right? A graph of the data reveals no outliers. 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

Identify Skewness Indicators

The presence of high standard deviations compared to the means in both groups suggests a right-skewed distribution. In right-skewed distributions, the mean is greater than the median, and since these are summary statistics, large standard deviations relative to the means imply a long right tail.

02

Two-Sample t Procedures Justification

Despite the skewness, the large sample sizes (675 males and 621 females) allow us to use the Central Limit Theorem (CLT). The CLT implies that the sampling distribution of the sample means will be approximately normal even if the population distributions are skewed. This justifies the use of two-sample t procedures.

03

Determine the Difference Between Means

Calculate the difference between the mean earnings of males and females: \[\overline{x}_m - \overline{x}_f = 1884.52 - 1360.39 = 524.13\].

04

Compute the Standard Error for the Difference of Means

Use the formula for the standard error (SE) of the difference between two independent means: \[\text{SE} = \sqrt{\frac{s_m^2}{n_m} + \frac{s_f^2}{n_f}} = \sqrt{\frac{13688.37^2}{675} + \frac{1037.46^2}{621}}\].Calculate each part separately and find their square root.

05

Find the Critical t-value for 90% Confidence Interval

For a 90% confidence interval with a large sample size, use a z-distribution as an approximation. Find the critical z-value (1.645) for 90% confidence, since the sample sizes are large enough to justify this approximation.

06

Calculate the Confidence Interval

The 90% confidence interval is given by:\[(\overline{x}_m - \overline{x}_f) \pm z \times \text{SE} = 524.13 \pm 1.645 \times \text{SE}\].Substitute the SE from Step 4 to compute this interval.

07

Interpretation of the Confidence Interval

If the confidence interval for the difference does not include zero, it indicates a significant difference between the mean earnings of male and female students. The direction (positive or negative) of the interval shows which gender has higher earnings on average.

08

Explanation of the Confidence Level

The 90% confidence level means that if we were to take many samples and compute their confidence intervals, about 90% of these intervals would contain the true difference in mean earnings between male and female students.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Two-Sample t Procedures

The two-sample t procedure is a statistical method used to determine if there is a significant difference between the means of two independent groups. In this context, it helps us compare the average summer earnings of male and female students at a university. The important assumption here is that both samples are independent and each comes from a normally distributed population or has a large enough sample size to rely on the Central Limit Theorem.

For this exercise, even though the data are skewed, we are justified in using two-sample t procedures because of the large sample sizes: 675 for males and 621 for females. Large samples make the two-sample t procedure robust against violations of normality due to the Central Limit Theorem, allowing us to proceed with the analysis confidently.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics that states that the distribution of the sample means approximates a normal distribution as the sample size becomes large, regardless of the shape of the population distribution. This theorem is crucial for understanding why we can use the two-sample t procedures in our exercise.

In the case of the university’s financial aid study, even though the earnings distribution appears skewed to the right, the large sample sizes (\(n \geq 30\) is often considered large) make it possible for the sampling distribution of the mean earnings to approximate a normal distribution. This is why the skewness of the original data doesn't substantially affect the validity of our statistical tests, allowing us to use this powerful theorem to support our analysis.

Confidence Interval Interpretation

A confidence interval gives us a range within which we expect the true difference in means to lie with a certain degree of confidence, which is set at 90% in our case. The confidence interval calculated is for the difference in average earnings between male and female students.

To interpret it, if the interval does not include zero, it suggests a significant difference in earnings based on gender. If zero is not in the interval, the mean earnings between the groups statistically differ. The direction of the interval (positive or negative) indicates which group, males or females, tends to earn more on average.

• A positive interval implies males earn more on average than females.
• If the interval were negative, it would suggest females earn more.

Earnings Distribution Analysis

Analyzing the distribution of earnings, especially using standard deviation in relation to the mean, provides insights into the spread and skewness of data.

In the university study, the large standard deviations relative to the means for both males and females suggest a right-skewed distribution. A right-skewed distribution has a long tail on the higher end, indicating that while most students earn less, some earn significantly more, pulling the mean upwards.

This understanding of the skewness helps set realistic expectations when interpreting the average earnings. It also indirectly highlights the variance within each group, which is important when considering factors such as financial decisions students or the financial aid office may make.