A sociologist develops a test to measure attitudes about public transportation, and 27 randomly selected subjects are given the test.Their mean score is 76.2 and their standard
Deviation is 21.4.Construct the 95% confidence interval for the mean score of all such
Subjects.
A)(64.2, 83.2)
B)(64.2, 88.2)
C)(74.6, 77.8)
D)(69.2, 83.2)
E)(67.7, 84.7)

Question

Quizplus · Accepted Answer

The 95% confidence interval for the mean is calculated using the formula: $\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $t_{\alpha/2}$ is the t-score for the desired confidence level with $n-1$ degrees of freedom. Given $\bar{x} = 76.2$, $s = 21.4$, and $n = 27$, and knowing that the t-score for 95% confidence with 26 degrees of freedom is approximately 2.056 (this value might slightly vary depending on the t-distribution table used, but it's generally around 2.056), the confidence interval is calculated as follows:$$76.2 \pm 2.056 \cdot \frac{21.4}{\sqrt{27}}$$$$76.2 \pm 2.056 \cdot 4.12$$$$76.2 \pm 8.48$$This results in a confidence interval of approximately (67.7, 84.7), which matches choice E.

A Sociologist Develops a Test to Measure Attitudes About Public