Among a simple random sample of 12 students selected from one college, the mean number of siblings is 1.3 with a standard deviation of 1.1. Construct the 95% confidence interval estimate of the population mean.
A) $1.1

Question

Among a simple random sample of 12 students selected from one college, the mean number of siblings is 1.3 with a standard deviation of 1.1. Construct the 95% confidence interval estimate of the population mean. 
A) $1.1 < \mu < 1.5$
B) $0.66 < \mu < 1.9$
C) $10.1 < \mu < 13.9$
D) $1.0 < \mu < 1.6$

Quizplus · Accepted Answer

To construct a 95% confidence interval for the population mean when 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 that corresponds to the desired level of confidence for a given degrees of freedom (df = n - 1). Here, $\bar{x} = 1.3$, $s = 1.1$, $n = 12$, and $df = 11$. For a 95% confidence interval and 11 degrees of freedom, the t-score (from a t-table or calculator) is approximately 2.201. Plugging these values into the formula gives:$$1.3 \pm 2.201 \cdot \frac{1.1}{\sqrt{12}}$$$$1.3 \pm 2.201 \cdot \frac{1.1}{3.464}$$$$1.3 \pm 0.7$$Thus, the 95% confidence interval is $0.6 < \mu < 1.9$, which matches choice B.

Among a Simple Random Sample of 12 Students Selected from One