A. To determine the upper limit for a 95% confidence interval for the average age of students:
1. We begin by defining the sample size which is 18.
2. We also know the mean age which is 19.1 years, and the standard deviation which is 1.5 years.
3. Next, we calculate the degrees of freedom. This is done by subtracting 1 from the sample size of 18 yielding 17.
4. In order to find the t-value that corresponds to the 95% confidence level and 17 degrees of freedom, we refer to a t-distribution table. The t-value is approximately 2.110.
5. Thereafter, we calculate the standard error of the mean by dividing the standard deviation by the square root of the sample size (1.5 / √18). This equals approximately 0.354.
6. Finally, we calculate the upper limit of the confidence interval. This is found by adding the product of the t-value and the standard error of the mean to the mean age (19.1 + (2.110 * 0.354)). The upper limit rounded to two decimal places would be approximately 19.85.
B. To determine the margin of error for a 90% confidence interval for the average age of students:
1. As before, our sample size is 18, the mean age is 19.1 years and the standard deviation is 1.5 years.
2. The degrees of freedom remains 17.
3. The t-value that corresponds to the 90% confidence level and 17 degrees of freedom can be found in a t-distribution table. This t-value would be approximately 1.740.
4. The standard error of the mean remains approximately 0.354.
5. Next, we calculate the margin of error by multiplying the t-value by the standard error of the mean (1.740 * 0.354). The margin of error rounded to two decimal places would be about approximately 0.62.