A newspaper reports that the population mean annual salary of a full-time college professor in a certain region is exactly $127,000. Scott wants to test the report by claiming that the population mean annual salary of a full-time college professor in the region is not $127,000. He selects a random sample of 160 full-time college professors in the region and calculates the sample mean salary to be $126,092 with a sample standard deviation of $8,509. The test statistic t for a hypothesis test of H0:μ=127,000 versus Ha:μ≠127,000 is t≈−1.35 with 159 degrees of freedom. If 0.10
Select all that apply:

A) Reject the null hypothesis that the true population mean annual salary of full-time college professors in the region is equal to $127,000.

B) Fail to reject the null hypothesis that the true population mean annual salary of full-time college professors in the region is equal to $127,000.

C) There is enough evidence at the α=0.01 level of significance to support the claim that the true population mean annual salary of a full-time college professor in the region is not equal to $127,000.

D) There is not enough evidence at the α=0.01 level of significance to suggest that the true population mean annual salary of a full-time college professor in the region is not equal to $127,000.

Expert Answer

Option B) Fail to reject the null hypothesis that the true population mean annual salary of full-time college professors in the region is equal to $127,000.

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = $127,000

Sample mean, [tex]\bar{x}[/tex] = $126,092

Sample size, n = 160

Alpha, α = 0.10

Sample standard deviation, σ = $8,509

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 127000\text{ dollars}\\H_A: \mu \neq 127,000\text{ dollars}[/tex]

We use Two-tailed t test to perform this hypothesis.

Formula:

[tex]t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]

[tex]t_{stat} = -1.35[/tex]

Now, [tex]t_{critical} \text{ at 0.10 level of significance, 159 degree of freedom } = \pm 1.654[/tex]

Since,

The calculated t-statistic lies in the acceptance region, we fail to reject and accept the null hypothesis.

Option B) Fail to reject the null hypothesis that the true population mean annual salary of full-time college professors in the region is equal to $127,000.