In a study conducted to determine the role that sleep disorders play in academic performance, researcher Jane Gaultney conducted a survey of 1845 college students to determine if they had a sleep disorder (such as narcolepsy, insomnia, or restless leg syndrome). Of the 503 students with a sleep disorder, the mean grade point average was 2.65 with a standard deviation of 0.87. Of the 1342 students without a sleep disorder, the mean grade point average was 2.82 with a standard deviation of 0.83. Source: SLEEP 2010: Associated Professional Sleep Societies 24th Annual Meeting. 
b. Is there evidence to suggest sleep disorders adversely affect one’s GPA at the α=0.05 level of significance? 

Verified step by step guidance

1. Step 1: Define the hypotheses for the test. The null hypothesis \(H_0\) assumes that there is no difference in mean GPA between students with and without sleep disorders, i.e., \(\mu_1 = \mu_2\). The alternative hypothesis \(H_a\) suggests that students with sleep disorders have a lower mean GPA, i.e., \(\mu_1 < \mu_2\).
2. Step 2: Identify the sample statistics and sizes. For students with sleep disorders: sample size \(n_1 = 503\), mean \(\bar{x}_1 = 2.65\), standard deviation \(s_1 = 0.87\). For students without sleep disorders: sample size \(n_2 = 1342\), mean \(\bar{x}_2 = 2.82\), standard deviation \(s_2 = 0.83\).
3. Step 3: Choose the appropriate test. Since we are comparing the means of two independent samples with known sample standard deviations, and the sample sizes are large, a two-sample z-test or t-test for the difference of means can be used. Typically, a two-sample t-test is appropriate here.
4. Step 4: Calculate the test statistic using the formula for the difference between two means: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \] This formula measures how many standard errors the observed difference in means is away from zero under the null hypothesis.
5. Step 5: Determine the degrees of freedom for the t-test using the Welch-Satterthwaite equation: \[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} \] Then, compare the calculated t-statistic to the critical t-value from the t-distribution table at \(\alpha = 0.05\) for a one-tailed test. If the test statistic falls in the rejection region, reject the null hypothesis, indicating evidence that sleep disorders adversely affect GPA.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In this context, it involves testing if the mean GPA differs between students with and without sleep disorders, using a significance level (α) of 0.05 to control the probability of a Type I error.

Two-Sample t-Test for Means

A two-sample t-test compares the means of two independent groups to determine if they are statistically different. Here, it assesses whether the average GPA of students with sleep disorders is significantly lower than that of students without, accounting for sample sizes, means, and standard deviations.

Significance Level and p-Value

The significance level (α) is the threshold for rejecting the null hypothesis, commonly set at 0.05. The p-value measures the probability of observing the data if the null hypothesis is true. If the p-value is less than α, it suggests strong evidence against the null, indicating sleep disorders may adversely affect GPA.