The random sample shown below was selected from a normal distribution.
$$3,8,10,4,6,5$$
Complete parts $a$ and $b$
a. Construct a $99\mathrm{\%}$ confidence interval for the population mean $\mu$
$(1.71,10.29)$ (Round to two decimal places as needed)
b. Assume that sample mean $\overline{x}$ and sample standard deviation s remain exactly the same as those you just calculated but that are based on a sample of $n=25$ observations. Repeat part a. What is the effect of increasing the sample size on the width of the confidence intervals?
The confidence interval is (Round to two decimal places as needed)

Step 1 ：Given a random sample from a normal distribution: $3,8,10,4,6,5$

Step 2 ：Calculate the sample mean $\overline{x}$ and the sample standard deviation $s$

Step 3 ：The sample mean $\overline{x}$ is 6.0 and the sample standard deviation $s$ is approximately 2.61

Step 4 ：Construct a 99% confidence interval for the population mean $\mu$ using the formula: $\overline{x}\pm Z\frac{s}{\sqrt{n}}$, where $Z$ is the Z-score (which is 2.33 for a 99% confidence interval), $s$ is the sample standard deviation, and $n$ is the sample size

Step 5 ：The 99% confidence interval for the population mean $\mu$ is approximately $(3.26,8.74)$

Step 6 ：Assume that the sample mean $\overline{x}$ and sample standard deviation $s$ remain exactly the same as those you just calculated but that are based on a sample of $n=25$ observations

Step 7 ：Repeat the calculation of the confidence interval with a sample size of 25

Step 8 ：The 99% confidence interval for the population mean $\mu$ with a sample size of 25 is approximately $(4.66,7.34)$

Step 9 ：Compare the results of the confidence intervals with sample sizes of 6 and 25

Step 10 ：Increasing the sample size decreases the width of the confidence interval, making our estimate of the population mean more precise

Step 11 ：Final Answer: The 99% confidence interval for the population mean $\mu$ with a sample size of 6 is $\boxed{{\displaystyle (3.26,8.74)}}$. When the sample size is increased to 25, the confidence interval becomes $\boxed{{\displaystyle (4.66,7.34)}}$