A simple random sample of size $\mathrm{n}$ is drawn. The sample mean, $\bar{x}$, is found to be 18.3 , and the sample standard deviation, $\mathrm{s}$, is found to be 4.6 . E Click the icon to view the table of areas under the t-distribution. (a) Construct a $95 \%$ confidence interval about $\mu$ if the sample size, $n$, is 34 . Lower bound: 16.69 ; Upper bound: 19.91 (Use ascending order. Round to two decimal places as needed.) (b) Construct a $95 \%$ confidence interval about $\mu$ if the sample size, $n$, is 51 . Lower bound: 17.03 ; Upper bound: 19.57 (Use ascending order. Round to two decimal places as needed.) How does increasing the sample size affect 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. (c) Construct a $99 \%$ confidence interval about $\mu$ if the sample size, $n$, is 34 . Lower bound: 16.14 '; Upper bound: 20.46 ' (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 increases. B. The margin of error decreases. C. The margin of error does not change. (d) If the sample size is 17 , what conditions must be satisfied to compute the confidence interval? A. The sample data must come from a population that is normally distributed with no outliers. B. The sample must come from a population that is normally distributed and the sample size must be large. C. The sample size must be large and the sample should not have any outliers.
Solution
Step 1 ：Given that the sample mean, \( \bar{x} \) is 18.3, the sample standard deviation, \( s \) is 4.6, and the sample size, \( n \) is 34.
Step 2 ：The formula for the confidence interval is \( \bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \), where \( t_{\alpha/2} \) is the t-score for the desired level of confidence.
Step 3 ：Look up the t-score for a 95% confidence interval with 33 degrees of freedom (which is \( n - 1 \)) in a t-distribution table or calculate it using a statistical function. The t-score is approximately 2.034515297449338.
Step 4 ：Calculate the margin of error, which is \( t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \). The margin of error is approximately 1.6050158768314189.
Step 5 ：Calculate the lower bound of the confidence interval, which is \( \bar{x} - \text{margin of error} \). The lower bound is approximately 16.69498412316858.
Step 6 ：Calculate the upper bound of the confidence interval, which is \( \bar{x} + \text{margin of error} \). The upper bound is approximately 19.90501587683142.
Step 7 ：The 95% confidence interval for the population mean when the sample size is 34 is approximately \(\boxed{[16.69, 19.91]}\).
