Question

a study done in 2018 by the college board was conducted by randomly selecting 40 public college students and found that the mean cost these students paid for tuition and fees was $9,970 with a standard deviation of $358. construct and interpret a 99% confidence interval for the mean amount spent by public college students on tuition and fees. a boxplot verifies that there are no outliers. (make sure to verify the requirements before constructing the confidence interval) we are 99% confident that the mean amount spent on tuition and fees by public college students will lie between $9784 and $10156. we are 99% confident that the mean amount spent on tuition and fees by public college students will lie between $9737 and $10203. we are 99% confident that the mean amount spent on tuition and fees by public college students will lie between $9817 and $10123. we are 99% confident that the mean amount spent on tuition and fees by public college students will lie between $9755 and $10185.

Explanation:

Step1: Check requirements

We have a random sample ($n = 40$). The box - plot shows no outliers. Since $n\geq30$, by the Central Limit Theorem, the sampling distribution of $\bar{x}$ is approximately normal.

Step2: Find the critical value

For a $99\%$ confidence interval, $\alpha=1 - 0.99=0.01$, and $\frac{\alpha}{2}=0.005$. The degrees of freedom $df=n - 1=40-1 = 39$. Using a $t$-distribution table or technology, $t_{\frac{\alpha}{2}}\approx 2.708$.

Step3: Calculate the margin of error

The formula for the margin of error $E=t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}$. Given $\bar{x}=9970$, $s = 358$, $n = 40$.

$$
LATEXBLOCK0
$$

Step4: Calculate the confidence interval

The confidence interval is $\bar{x}-E<\mu<\bar{x} + E$.

$$
LATEXBLOCK1
$$

Answer:

We are 99% confident that the mean amount spent on tuition and fees by public college students will lie between $9817 and $10123.