91Ó°ÊÓ

The article "So Close, Yet So Far: Predictors of Attrition in College Seniors" (Journal of College Student Development \([1999]: 343-354\) ) attempts to describe differences between college seniors who disenroll before graduating and those who do graduate. Researchers randomly selected 42 nonreturning and 48 returning seniors, none of whom were transfer students. These 90 students rated themselves on personal contact and campus involvement. The resulting data are summarized here: \begin{tabular}{lcccc} & \multicolumn{2}{c} { Returning \((n=48)\)} & & Nonrefurning \((n=42)\) \\ \cline { 2 } \cline { 4 - 5 } & & Standard & & Standard \\ & Mean Deviation & & Mean & Deviation \\ \hline Personal & & & & \\ Contact & \(3.22\) & \(.93\) & & \(2.41\) & \(1.03\) \\ Campus & & & & \\ Involvement & \(3.21\) & \(1.01\) & \(3.31\) & \(1.03\) \\ & & & & \end{tabular} a. Construct and interpret a \(95 \%\) confidence interval for the difference in mean campus involvement rating for returning and nonreturning students. Does your interval support the statement that students who do not return are less involved, on average, than those who do? Explain. b. Do students who don't return have a lower mean personal contact rating than those who do return? Test the relevant hypotheses using a significance level of \(.01\).

Step by step solution

01

Calculate and Interpret the Confidence Interval

First, we calculate the confidence interval. The formula for a confidence interval for the difference between two independent means is: \[CI = (\bar{x}_1 - \bar{x}_2) ± Z \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\] where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes. From the data given, we can substitute the values into the formula: \[CI=(3.21-3.31) ± 1.96*sqrt{(\frac{(1.01)^2}{48}+(\frac{(1.03)^2}{42}}\] After solving it, one can get two limits of confidence interval. We interpret the confidence interval and check whether it contains zero. If it doesn't contain zero, there is evidence to suggest that there is a difference in mean campus involvement rating between the two types of students.

02

Perform the Hypothesis Test

The relevant hypotheses here are: \(H_0: \mu_{returning} = \mu_{nonreturning}\), and \(H_1: \mu_{returning} > \mu_{nonreturning}\), where \(\mu_{returning}\) and \(\mu_{nonreturning}\) represent the population mean of personal contact rating for returning and nonreturning students, respectively. The formula for the test statistic is: \[Z = (\bar{x}_1 - \bar{x}_2) / \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\] We substitute the values and calculate Z. Then we compare this to the z score at significance level 0.01 (which is approximately 2.33). If the test statistic is greater than the z score, we reject the null hypothesis.

03

Interpretation

After performing the test, we interpret the result in the context of the problem. If we reject the null hypothesis, it suggests that students who do not return have a lower mean personal contact rating than those who do return.

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Key Concepts

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

Confidence Interval

Understanding confidence intervals is crucial in statistical analysis. In the context of this exercise, they are used to estimate the range within which the true difference in mean campus involvement ratings between returning and nonreturning students lies.

A confidence interval is calculated using the formula:

• \[CI = (\bar{x}_1 - \bar{x}_2) \pm Z \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]

Here, \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(s_1\) and \(s_2\) are their respective standard deviations, and \(n_1\) and \(n_2\) are the sample sizes.

• **Z** is the Z-value that corresponds to the desired confidence level, like 1.96 for 95% confidence.

In simple terms, the confidence interval gives a span of plausible values for the difference in means.
If the interval doesn't contain zero, it suggests there's a significant difference between the two groups. In our exercise, by checking if zero falls within the interval, we determine the nature of that difference.

Hypothesis Testing

Hypothesis testing is a key method used to make decisions based on data. In this exercise, we're testing whether nonreturning students have lower personal contact ratings than returning students.

We define **two hypotheses**:

• Null Hypothesis (\(H_0\)): There is no difference in mean personal contact ratings \((\mu_{returning} = \mu_{nonreturning})\).
• Alternative Hypothesis (\(H_1\)): Returners have a higher mean rating \((\mu_{returning} > \mu_{nonreturning})\).

To test these, we calculate a test statistic using:

• \[Z = (\bar{x}_1 - \bar{x}_2) / \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]

We compare this **Z-value** with the critical value from the standard normal distribution table, typically at a chosen significance level like 0.01 (critical Z of about 2.33). If our calculated Z is larger, it suggests that the difference is significant enough to reject the null hypothesis.
This means students not returning potentially have lower engagement, as shown by the lower personal contact ratings.

Population Mean

In statistics, the population mean is an average calculated for all members of a group that we are interested in. However, since it's often impractical to measure everyone, samples are used.

In real-world studies like this one, sample means estimate the population mean.
For returning students and nonreturning students, we have specific data that represent their average ratings on various factors like personal contact and campus involvement. The means derived from these ratings provide insights into overall tendencies and behaviors of each student group in the larger population.
Using these sample means, we can infer back to what the true population means might be. This is possible thanks to methods like confidence intervals and hypothesis testing, which allow us to make probabilistic statements about the population based on sample data.
Thus, understanding how sample data reflects the broader population helps researchers draw meaningful conclusions.