Question

Suppose we want to conduct a hypothesis test relating to the mean of a certain population where the population variance is unknown. To conduct the test, we collect data from a random sample of 27 members of the population.

a. Explain why we should use a t-test instead of a z-test in this situation. What is the distribution of the t-statistic in this case? (Recall that the t-statistic is defined as ? = ?̅−? ?/√? )

b. Find the following values using the t-distribution specified in part a: i. ?(|?| ≥ 1.058) ii. ? such that ?(? ≤ ?) = 0.05 iii. ? such that ?(−? ≤ ? ≤ ?) = 0.80

Answer 1

a) Since the population variance is unknown, so we will use t-distribution.

Since the sample size is less than 30, so we cannot use the Central Limit Theorem.

So we must assume that the population is normally distributed.

The distribution of the t-statistic is normal.

b)i) P(|T| > 1.058)

= 1 - P(|T| < 1.058)

= 1 - P(-1.058 < T < 1.058)

= 1 - (P(T < 1.058) - P(T < -1.058))

= 1 - (0.8501 - 0.1499)

= 1 - 0.7002

= 0.2998

ii) P(T < k) = 0.05

or, k = -1.706

iii) P(-k < T < k) = 0.80

or, P(T < k) - P(T < -k) = 0.80

or, P(T < k) - (1 - P(T < k)) = 0.80

or, 2P(T < k) = 1.80

or, P(T < k) = 0.9

or, k = 1.315