Question

For each of the following, assume that the sample is a random sample from a distribution that is reasonably normally distributed and that we are doing inference for a population mean.

a.)Find endpoints of a t-distribution with 10% beyond them in each tail if the sample has size n = 5.

b.)Find the area in a t-distribution to the left of -2.728 if the sample has size n = 35.

c.)Find the area in a t-distribution to the right of 2.602 if the sample has size n = 16.

Answer 1

a) The endpoints of a t-distribution with 10% beyond them in each tail for a sample size of 5 are -2.776 and 2.776. b) The area in a t-distribution to the left of -2.728 for a sample size of 35 is approximately 0.005. c) The area in a t-distribution to the right of 2.602 for a sample size of 16 is approximately 0.0078.

Step-by-step explanation:

a) To find the endpoints of a t-distribution with 10% beyond them in each tail for a sample of size n = 5, we need to find the critical values for a two-tailed test. Using a t-table or a statistical software, we can find the critical value for a 10% significance level with 4 degrees of freedom (n-1 = 5-1 = 4) to be approximately ±2.776. Therefore, the endpoint of the t-distribution with 10% beyond them in each tail for a sample size n = 5 is -2.776 and 2.776.

b) To find the area in a t-distribution to the left of -2.728 for a sample of size n = 35, we need to use the cumulative distribution function (CDF) of the t-distribution. We can find this using a t-table or a statistical software. The area to the left of -2.728 is approximately 0.005 (or 0.5%).

c) To find the area in a t-distribution to the right of 2.602 for a sample of size n = 16, we can subtract the area to the left of 2.602 (found using the CDF of the t-distribution) from 1. The area to the left of 2.602 is approximately 0.9922, so the area to the right of 2.602 is approximately 1 - 0.9922 = 0.0078 (or 0.78%).