Question

A random sample of 16 students is taken from a larger normal population of students in a multi-lecture course and the sample mean (average) of their final grades is found to be 62.25. Suppose it is known that the sample standard deviation is 3.2. You wish to create a 98% confidence interval for the true population mean. Which of the following statements is most correct?

a. The tα/2,n-1 (found from R) used to calculate the 98% confidence interval is 2.946713.

b. The tα/2,n-1 (found from R) used to calculate the 98% confidence interval is 2.24854.

c. The tα/2,n-1 (found from R) that is needed to calculate the 98% confidence interval is 2.60248.

d. The tα/2,n-1 (found from R) used to calculate the 98% confidence interval is 0.01.

Answer 1

The most correct statement for calculating the 98% confidence interval for the population mean, with a sample size of 16 and a sample standard deviation of 3.2, is option c, which states that the tα/2,n-1 value is 2.60248.

Step-by-step explanation:

To create a 98% confidence interval for the true population mean of final grades based on a sample mean of 62.25 and a sample standard deviation of 3.2 with a sample size of 16 students, we need to find the t-score corresponding to a 98% confidence level with degrees of freedom (df) equal to n-1, where n is the sample size.

The degrees of freedom (df) in this case would be 16 - 1 = 15. Looking up a t-distribution table or using statistical software, we find the t-score that corresponds to a 98% confidence interval and 15 degrees of freedom. The correct t-score (tα/2,n-1) required to calculate the 98% confidence interval from the options given is 2.60248 (answer c). Thus, statement c is most correct among the options provided.

The formula for a confidence interval is given by:

Sample mean ± tα/2,n-1 * (Sample standard deviation / √n)

Using this formula along with the correct t-score, we can calculate the confidence interval for the population mean.