Samples of DNA are collected, and the four DNA bases of A, G, C, and T are coded as 1,2,3, and 4, respectively. The results are listed below. Construct a $95 \%$ confidence interval estimate of the mean. What is the practical use of the confidence interval?
$1,1,2,4,3,4,3,4,4,2$ 만
What is the confidence interval for the population mean $\mu$ ?
$< \mu< \square$ (Round to one decimal place as needed.)
What is the practical use of the confidence interval? Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. The given numbers are just substitutes for the four DNA base names, so the numbers do not measure or count anything, and they are at the nominal level of measurement. The confidence interval has no practical use.
B. The confidence interval can be used to estimate that, with $95 \%$ confidence, the interval from to t actually contains the true mean DNA base of all people. (Round to one decimal place as needed.)
C. The confidence interval can be used to estimate that $95 \%$ of all people have DNA bases between [ and (Round to one decimal place as needed.)
Answer
The practical use of the confidence interval is to estimate that, with 95% confidence, the interval from 2.0 to 3.6 actually contains the true mean DNA base of all people.
Step 1 ：The given DNA base codes are 1, 1, 2, 4, 3, 4, 3, 4, 4, 2. We are asked to find the 95% confidence interval for the mean of these codes.
Step 2 ：First, we calculate the sample mean (\(\bar{x}\)) of the DNA base codes, which is 2.8.
Step 3 ：Next, we calculate the sample standard deviation (s) of the DNA base codes, which is approximately 1.23.
Step 4 ：The sample size (n) is 10, as there are 10 DNA base codes.
Step 5 ：We then use the formula for the confidence interval, which is \(\bar{x} \pm z \frac{s}{\sqrt{n}}\), where z is the z-score corresponding to the desired level of confidence. For a 95% confidence interval, the z-score is approximately 1.96.
Step 6 ：Substituting the values into the formula, we get the 95% confidence interval for the population mean \(\mu\) as \(2.0 < \mu < 3.6\).
Step 7 ：Final Answer: The 95% confidence interval for the population mean \(\mu\) is \(\boxed{2.0 < \mu < 3.6}\).
Step 8 ：The practical use of the confidence interval is to estimate that, with 95% confidence, the interval from 2.0 to 3.6 actually contains the true mean DNA base of all people.
