Question

suppose you want to test the claim that $mu<65.4$ and the population standard deviation is known. given a sample size of $n = 35$ and a level of significance of $alpha=0.1$, when should you reject $h_0$?

a. reject $h_0$ if the standardized test is less than $-1.645$

b. reject $h_0$ if the standardized test statistic is less than $-2.33$

c. reject $h_0$ if the standardized test statistic is less than $-1.96$

d. reject $h_0$ if the standardized test statistic is less than $-1.28$

Explanation:

Step1: Determine the type of test

Since the claim is \( \mu<65.4 \), this is a left - tailed test.

Step2: Find the critical value

For a left - tailed test with \( \alpha = 0.1 \), we look up the \( z \) - value in the standard normal distribution table. The critical \( z \) - value for \( \alpha=0.1 \) (left - tailed) is \( z_{0.1}\).
Using the standard normal distribution table or a calculator, \( z_{0.1}=- 1.28 \) (because the area to the left of \( z = - 1.28 \) is approximately \( 0.1 \)).
In a hypothesis test, for a left - tailed test with population standard deviation known ( \( z \) - test), we reject the null hypothesis \( H_0 \) when the standardized test statistic \( z \) is less than the critical value. So we reject \( H_0 \) if \( z<-1.28 \).

Answer:

D. Reject \( H_0 \) if the standardized test statistic is less than \( - 1.28 \)