Question

a random sample of 15 statistics textbooks has a mean price of $105 with a standard deviation of $30.25. determine whether a normal distribution or a t - distribution should be used or whether neither of those can be used to construct a confidence interval. assume the distribution of statistics textbook prices is not normally distributed.

a. use the t - distribution.
b. use normal distribution.
c. cannot use normal distribution or t - distribution.

Explanation:

Step1: Check sample size and distribution

The sample size \(n = 15\) (small, \(n<30\)) and the population distribution is not normal.

Step2: Recall conditions for normal and t - distributions

• For normal distribution: Population is normal or \(n\geq30\) (Central Limit Theorem).
• For t - distribution: Population is normal or \(n\) is small but we assume some underlying symmetry (not the case here as population is non - normal).

Since the population is not normally distributed and \(n = 15<30\), we cannot use the Central Limit Theorem (which requires \(n\geq30\) for non - normal populations) and we also do not meet the normality assumption for the t - distribution (when \(n\) is small, the t - distribution assumes the population is approximately normal).

Answer:

C. Cannot use normal distribution or t - distribution