Answer:
The formula for calculating a confidence interval for the mean is:
CI = X ± Zα/2 * (s/√n)
where X is the sample mean, s is the sample standard deviation, n is the sample size, and Zα/2 is the z-score for the desired confidence level.
In this case, n = 49, x = 405, s = 21, and CL = 99.7%. The corresponding z-score for a 99.7% confidence level is 2.967.
Plugging in the values, we get:
CI = 405 ± 2.967 * (21/√49)
CI = 405 ± 9.67
CI = (395.33, 414.67)
Therefore, the confidence interval is (396, 414). Answer choice (D) is the correct one.