Question

a simple random sample of size n is drawn. the sample mean, x, is found to be 17.6, and the sample standard deviation, s, is found to be 4.9
click the icon to view the table of areas under the t - distribution
(c) construct a 99% confidence interval about μ if the sample size, n, is 35
lower bound 15.34; upper bound 19.86
(use ascending order. round to two decimal places as needed.)
compare the results to those obtained in part (a). how does increasing the level of confidence affect the size of the margin of error, e?
a. the margin of error decreases
b. the margin of error increases
c. the margin of error does not change
(d) if the sample size is 16, what conditions must be satisfied to compute the confidence interval?
a. the sample size must be large and the sample should not have any outliers
b. the sample data must come from a population that is normally distributed with no outliers
c. the sample must come from a population that is normally distributed and the sample size must be large

Explanation:

When the sample size \(n = 16\) (which is small, as \(n<30\)), to compute a confidence interval for the population mean \(\mu\) using the \(t -\)distribution (since the population standard deviation \(\sigma\) is unknown and we use the sample standard deviation \(s\)), the sample data must come from a population that is normally distributed with no outliers. This is because the \(t -\)distribution's assumption for small - sample confidence interval construction (when \(\sigma\) is unknown) is that the population from which the sample is drawn is normal. Outliers can significantly affect the sample mean \(\bar{x}\) and sample standard deviation \(s\), violating the underlying assumptions of the confidence - interval formula.

Answer:

B. The sample data must come from a population that is normally distributed with no outliers