91Ó°ÊÓ

Putting It Together: Time to Complete a Degree A researcher wanted to determine if the mean time to complete a bachelor's degree was different depending on the selectivity of the first institution of higher education that was attended. The following data represent a random sample of 12 th-graders who earned their degree within eight years. Probability plots indicate that the data for each treatment level are normally distributed. $$ \begin{array}{ccccc} \text { Highly } & & & \text { Not } \\ \text { Selective } & \text { Selective } & \text { Nonselective } & \text { Open-door } & \text { Rated } \\ \hline 2.5 & 4.6 & 4.7 & 5.9 & 5.1 \\ \hline 5.1 & 4.3 & 2.3 & 4.2 & 4.3 \\ \hline 4.3 & 4.4 & 4.3 & 6.4 & 4.4 \\ \hline 4.8 & 4.0 & 4.2 & 5.6 & 3.8 \\ \hline 5.5 & 4.1 & 5.1 & 6.0 & 4.7 \\ \hline 2.6 & 3.1 & 4.7 & 5.0 & 4.5 \\ \hline 4.1 & 3.8 & 4.2 & 7.3 & 5.5 \\ \hline 3.4 & 5.2 & 5.7 & 5.6 & 5.5 \\ \hline 4.3 & 4.5 & 2.3 & 6.9 & 4.6 \\ \hline 3.9 & 4.0 & 4.5 & 5.1 & 4.1 \\ \hline \end{array} $$ (a) What type of observational study was conducted? What is the response variable? (b) Find the sample mean for each treatment level. (c) Find the sample standard deviation for each treatment level. Using the general rule presented in this chapter, does it appear that the population variances are the same? (d) Use the time to degree completion for students first attending highly selective institutions to construct a \(95 \%\) confidence interval estimate for the population mean. (e) How many pairwise comparisons are possible among the treatment levels? (f) Consider the null hypothesis \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}=\mu_{5}\). If we test this hypothesis using \(t\) -tests for each pair of treatments, use your answer from part (e) to compute the probability of making a Type I error, assuming that each test uses an \(\alpha=0.05\) level of significance. (g) Use the one-way ANOVA procedure to determine if there is a difference in the mean time to degree completion for the different types of initial institutions. If the null hypothesis is rejected, use Tukey's test to determine which pairwise differences are significant using a familywise error rate of \(\alpha=0.05\)

Step by step solution

01

Title - Identify Study Type and Response Variable

The type of observational study conducted is a comparative study. The response variable is the mean time to complete a bachelor's degree.

02

Title - Calculate Mean Time for Each Treatment Level

Calculate the mean for each treatment level: Highly Selective: Selective: Nonselective: Open-door: Rated: Use the formula for the average: \ \( \bar{x} = \frac{ \text{sum of all values} }{ \text{number of values} } \)

03

Title - Calculate Sample Standard Deviation

Calculate the sample standard deviation for each treatment level using the formula: \ \( s = \frac{ 1 }{ n-1 } \times \text{sum of squared deviations from the mean} \) Observe if the variances appear similar.

04

Title - Construct 95% Confidence Interval for Highly Selective Institutions

Use the formula for a confidence interval: \( \bar{x} \pm z \times \frac{s}{ \sqrt n } \) where \( z \) is the z-score for a 95% confidence interval.

05

Title - Calculate Pairwise Comparisons

Count the number of unique pairs of treatment levels to find multiple comparisons.\( \text{Number of comparisons} = \frac{k(k-1)}{2} \)

06

Title - Compute Probability of Type I Error

Compute using, \(1 - (1 - \alpha)^c \) where \(c\) is the number of comparisons and \( \alpha = 0.05 \).

07

Title - Perform One-Way ANOVA

Use ANOVA F-test to compare variances. If null hypothesis is \( \mu_1 = \mu_2 = \mu_3 = \mu_4 = \mu_5\) is rejected, follow up with Tukey's HSD test.

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.

t-test

A t-test is a statistical test used to compare the means of two groups and determine if they are significantly different from each other. This test is extremely useful when you have two sample groups and you want to see if there is a significant difference between them, often under the assumption that the data follows a normal distribution.
When conducting a t-test, it's vital to start with the null hypothesis, which states that there is no difference between the means of the groups being compared. The alternative hypothesis suggests that there is a difference.
To perform a t-test, follow these basic steps:

• Calculate the means of both groups.
• Determine the standard deviations of both groups.
• Use the t-test formula: \( t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{2}{n}}} \), where \(s_p\) is the pooled standard deviation and \(n\) is the number of observations.
• Compare this calculated t value with the critical t value from the t-distribution table based on your desired significance level (often \( \alpha = 0.05 \)).

If the calculated t-value is greater than the critical t-value, you reject the null hypothesis. This indicates that the means of the two groups are significantly different. Otherwise, you fail to reject the null hypothesis.

confidence interval

A confidence interval (CI) is a range of values that is likely to contain a population parameter with a certain level of confidence, usually expressed as a percentage like 95% or 99%. Confidence intervals are crucial in inferential statistics as they provide a range of plausible values for the parameter being estimated.
For example, if you're estimating the mean time to complete a bachelor's degree and want to construct a 95% confidence interval, you would proceed as follows:

• Calculate the sample mean \( \bar{x} \).
• Determine the standard deviation \( s \).
• Find the appropriate z-score for a 95% confidence level, which is typically 1.96.
• Compute the margin of error using the formula: \( MOE = z \cdot \frac{s}{\sqrt{n}} \), where \(n\) is the sample size.
• Finally, create the confidence interval: \( \bar{x} \pm MOE \).

This interval tells us that we are 95% confident that the true mean time to complete a bachelor's degree lies within this range. By providing a confidence interval, you allow for some degree of uncertainty in your estimate, but it also gives a more comprehensive picture of what the true mean might be.

pairwise comparison

Pairwise comparisons are used to analyze and compare the means of multiple groups to determine which pairs of groups are significantly different. This method is particularly useful when dealing with more than two groups, as it extends the principles of the t-test to multiple comparisons.
In the context of ANOVA, after determining that there is a significant difference among group means, pairwise comparisons can be employed to understand which specific group pairs are different. Here’s what to do:

• First, conduct a one-way ANOVA to establish if there are any overall differences between the groups.
• If the ANOVA indicates significant differences, proceed with pairwise comparison tests like Tukey's HSD (Honestly Significant Difference).
• For each pair of groups, compute the difference in their means and then use a post-hoc test to see if this difference is significant.

By using pairwise comparisons, you can identify which specific groups differ from each other while controlling for the familywise error rate, thus providing a more detailed understanding of the differences within your data.