A statistician believes that the standard deviation of the weights of firemen is less than 25 pounds. A sample of 20 firemen had a standard deviation of 23.2 pounds. Assume that the variable is normally
Distributed with $a = 0.05$ . What is the critical value?
A) 31.410
B) 30.144
C) 10.851
D) 10.117

Question

Quizplus · Accepted Answer

Since the population standard deviation is unknown, we use the t-distribution with degrees of freedom n-1 (where n is the sample size). We want to test the null hypothesis that the population standard deviation is greater than or equal to 25 pounds, versus the alternative hypothesis that the population standard deviation is less than 25 pounds. This is a one-tailed test with alpha = 0.05. Looking up the critical value in a t-table with 19 degrees of freedom and a one-tailed alpha of 0.05, we get a critical value of 1.729. Multiplying this by the sample standard deviation of 23.2 gives a critical value of 40.144, which is higher than the hypothesized population standard deviation of 25 pounds. Therefore we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the standard deviation of the weights of firemen is less than 25 pounds. The closest choice to 1.729*23.2 is 30.144, so the answer is B.

A Statistician Believes That the Standard Deviation of the Weights a=0.05a = 0.05a=0.05

A Statistician Believes That the Standard Deviation of the Weights $a = 0.05$