A sociologist develops a test to measure attitudes about public transportation, and a simple random sample of 27 subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.6. Assume that the population of these scores is normally distributed. Construct the 95% confidence interval estimate of the population mean.
A) $64.2

Question

A sociologist develops a test to measure attitudes about public transportation, and a simple random sample of 27 subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.6. Assume that the population of these scores is normally distributed. Construct the 95% confidence interval estimate of the population mean. 
A) $64.2 < \mu < 88.2$
B) $69.2 < \mu < 83.2$
C) $67.7 < \mu < 84.7$
D) $74.6 < \mu < 77.8$

Quizplus · Accepted Answer

To construct a 95% confidence interval for the population mean when the population standard deviation is unknown and the sample size is small (n < 30), we use the t-distribution. The formula for the confidence interval is $\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$, 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 corresponding to the desired confidence level and degrees of freedom (df = n - 1). Given: $\bar{x} = 76.2$, $s = 21.6$, $n = 27$, and for a 95% confidence interval with 26 degrees of freedom, the t-score (from a t-table or calculator) is approximately 2.056.The margin of error (ME) is calculated as: $ME = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} = 2.056 \cdot \frac{21.6}{\sqrt{27}} \approx 8.5$.Thus, the confidence interval is: $76.2 \pm 8.5$, which gives $67.7 < \mu < 84.7$.

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A Sociologist Develops a Test to Measure Attitudes About Public $64.2 < \mu < 88.2$