ter 6 Homework
Question 13, 6.2.9-T
HW Score: $27.66 \%, 13$ of 47 points
Points: 0 of 2
Construct the indicated confidence interval for the population mean $\mu$ using the $t$-distribution. Assume the population is normally distributed.
\[
c=0.90, \bar{x}=14.2, s=3.0, n=7
\]
(Round to one decimal place as needed.)

Step 1 ：Find the t-value corresponding to the given confidence level c=0.90 and degrees of freedom df=n-1=7-1=6. From the t-distribution table, the t-value is approximately \(1.943\).

Step 2 ：Calculate the standard error of the mean (SE), which is given by \(s/\sqrt{n} = 3.0/\sqrt{7} \approx 1.13\).

Step 3 ：Construct the confidence interval using the formula \(x̄ ± t*SE\), where \(x̄\) is the sample mean, \(t\) is the t-value, and \(SE\) is the standard error.

Step 4 ：Substitute the values into the formula to get the confidence interval: \(14.2 ± 1.943*1.13 = (14.2 - 2.2, 14.2 + 2.2) = (12.0, 16.4)\).

Step 5 ：\(\boxed{\text{Therefore, we are 90% confident that the population mean μ is between 12.0 and 16.4.}}\)