Question

A class contains 200 students. The teacher wants to test whether the mean IQ in the class exceeds 120. He chooses a random sample of 16 students and finds that the mean IQ in the sample is 122.8 and the standard deviation of the IQ’s in the sample is 10.9.

Let alpha = 0.05. You may assume that IQ's are Normally distributed. Which one of the following is the correct conclusion for this hypothesis test?

Group of answer choices

A Do not reject Ho. The p-value is less than the significance level.

B Reject Ho. The p-value is less than the significance level.

C Reject Ho. The p-value is not less than the significance level.

D Reject Ho simply because the sample mean is greater than 120.

E Do not reject Ho. The p-value is not less than the significance level.

Answer 1

Here we need to test whether are mean IQ of the class is greater than 120 or not

Additionally, we can assume that IQ's are normally distributed so we will apply t-test to check the hypothesis

The provided sample mean is

Xbar= 122.8

and the sample standard deviation is

s = 10.9

and the sample size is

n = 16

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: \μ = 120

Ha: μ > 120

This corresponds to a right-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is tc​=1.753.

Here critical t value is calculated using t distribution table for 15 df freedom at alpha equal to 0.05

The rejection region for this right-tailed test is R = {t:t>1.753}

(3) Test Statistics

The t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that t=1.028≤tc​=1.753, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

We will calculate p-value first

Here p-value is calculated using t distribution having 15 df and t value as 1.028

p-value = P[t>1.028] (using t distribution table we get

p-value = P[t>1.028] = 0.1601

The p-value is p=0.1601, and since p=0.1601≥0.05, it is concluded that the null hypothesis is not rejected.

Option E

Do not reject Ho. The p-value is not less than the significance level.