Question

The quantitative reasoning EX1127 scores on the Graduate Record Examination (GRE) \(^{2}\) were recorded for students admitted to three different graduate programs at a local university. a. Do these data provide sufficient evidence to indicate a difference in the mean GRE scores for applicants admitted to the three programs? b. Find a \(95 \%\) confidence interval for the difference in mean GRE scores for Life Sciences and Physical Sciences. c. If you find a significant difference in the average GRE scores for the three programs, use Tukey's method for paired comparisons to determine which means differ significantly from the others. Use \(\alpha=.05 .\)

Short answer

If yes, calculate a 95% confidence interval for the difference in mean GRE scores for Life Sciences and Physical Sciences, and find which means differ significantly using Tukey's method.

Step-by-step solution

State the Null and Alternative Hypotheses

For the first part of the problem, we'll state the null and alternative hypotheses: - Null Hypothesis (H\(_0\)): There is no difference in the mean GRE scores for applicants admitted to the three graduate programs. Mathematically, we can state this as: \(\mu_1 = \mu_2 = \mu_3\). - Alternative Hypothesis (H\(_A\)): There is a difference in the mean GRE scores for applicants admitted to the three graduate programs. Mathematically, this can be stated as: Not all \(\mu_1, \mu_2, \mu_3\) are equal.

Perform the ANOVA test

To perform the ANOVA test, we need the program's sample sizes, the sum of the squares between groups (SSB), and the sum of the squares within groups (SSW). Since the data is not provided, we'll assume these values are calculated from the given data. With these values, we can then calculate the F-statistic, which is used to determine if there's a significant difference between the means. F-statistic = \(\frac{MS_B}{MS_W}\), where \(MS_B = \frac{SSB}{\mathrm{Degree\:of\: Freedom\: Between\: Groups (df_B)}}\) and \(MS_W = \frac{SSW}{\mathrm{Degree\:of\: Freedom\: Within\: Groups (df_W)}}\). Once calculated, compare the F-statistic to the critical F-value at the given significance level (0.05) and corresponding degrees of freedom. If the calculated F-statistic is greater than the critical F-value, we reject the null hypothesis, indicating that there is a significant difference between the mean GRE scores for applicants admitted to the three graduate programs.

Calculate the Confidence Interval for the Difference in Means

If the ANOVA test resulted in the rejection of the null hypothesis, we can proceed to calculate the 95% confidence interval for the difference in mean GRE scores between Life Sciences and Physical Sciences. Let \(d\) represent the difference in means, \(t_{\alpha/2}\) be the critical t-value, and \(SE_d\) be the standard error of the difference. 95% Confidence Interval: \(d \pm t_{\alpha/2} * SE_d\) After calculating the difference in means and standard error, find the critical t-value at the given confidence level (95%). The t-value can be found from a t-distribution table or using statistical software. Calculate the confidence interval using the formula above.

Conduct Tukey's Pairwise Comparison

If a significant difference was found in the ANOVA test, we can use Tukey's method to determine which means differ significantly. Tukey's method involves calculating a statistic, called the Tukey HSD (Honestly Significant Difference), for each pair of groups. If the absolute difference between group means (\(|\mu_i - \mu_j|\)) is greater than the Tukey HSD, we conclude that there is a significant difference between the means. Calculate the Tukey HSD: \(HSD = q_{\alpha; k, n-k} * \sqrt{\frac{MS_W}{n}}\), where \(q_{\alpha; k, n-k}\) is the critical q-value at a given significance level, \(k\) is the number of groups, and \(n\) is the total sample size. Compare the absolute differences between the means (\(|\mu_1 - \mu_2|\), \(|\mu_1 - \mu_3|\), and \(|\mu_2 - \mu_3|\)) to the calculated Tukey HSD value. If the absolute difference between any two group means is greater than the Tukey HSD value, there's a significant difference between those groups. Conclusions about the differences between the means can then be drawn according to the results of the Tukey HSD test.

Key concepts

Null Hypothesis

In the realm of statistics, the null hypothesis serves as a default statement that there is no effect or difference in the given situation. In our context with the GRE scores, the null hypothesis (denoted as H\(_0\)) posits that all the mean scores of students admitted to distinct graduate programs are the same. Essentially, it claims that the variations in GRE scores occur by random chance and not because of a real difference among the programs.

For example, if we denote the mean scores of the three programs as \(\mu_1\), \(\mu_2\), and \(\mu_3\), the null hypothesis can be mathematically expressed as: \(\mu_1 = \mu_2 = \mu_3\).

If this null is to be rejected, there must be strong statistical evidence suggesting that at least one program has a distinct mean GRE score. This rejection would lead us to the alternative hypothesis, where not all the average scores are equal. The process of testing this often involves ANOVA, a robust method to compare multiple group means simultaneously to identify any significant differences.

Confidence Interval

A confidence interval offers a range of values, derived from sample data, that is believed to contain the true population parameter. In the context of comparing GRE scores between Life Sciences and Physical Sciences, a 95% confidence interval means we're 95% confident that the true difference in mean scores lies within the calculated range.

To form this interval, you calculate the difference in sample means and apply a margin of error, determined by the standard error of the difference and a critical value from the t-distribution. The critical value is typically found using a statistical table or software.

Here's a simplified view of the formula for a confidence interval:

• Difference in means \(d\)
• \(t_{\alpha/2}\) as the critical t-value
• Standard Error of the difference \(SE_d\)

The 95% Confidence Interval is then \(d \pm t_{\alpha/2} \times SE_d\).

Interpreting this interval requires understanding that if it does not contain zero, it suggests a statistically significant difference in the mean scores between the two programs.

Tukey's Method

Tukey's Method, or Tukey's Honest Significant Difference (HSD) test, is a procedure used to find which means are significantly different following an ANOVA test. When an ANOVA test results in the rejection of the null hypothesis, implying that not all group means are equal, Tukey's Method helps pinpoint the differences.

This approach calculates the Tukey HSD value, which is a specific value that groups the mean differences by significance. The method considers all possible pair comparisons between groups and checks if their mean difference is larger than the Tukey HSD. If so, the difference is statistically significant.

The formula for Tukey HSD is:

• \(q_{\alpha; k, n-k}\) denotes the critical q-value found in statistical tables
• \(MS_W\) stands for Mean Square Within Groups
• \(n\) is the total sample size

The calculation is \(HSD = q_{\alpha; k, n-k} \times \sqrt{\frac{MS_W}{n}}\).

After calculating the HSD, compare differences like \(|\mu_1 - \mu_2|\) and \(|\mu_2 - \mu_3|\) with this HSD value. If the difference surpasses the Tukey HSD, it suggests that those specific means are indeed different. This clear step-by-step comparison method offers a straightforward decision-making framework for identifying significant differences post-ANOVA.