Question

A random sample of 100 households and collect data on each household’s income. You find a sample income standard deviation of $160,000. You want to test the null hypothesis that the population mean income is $100,000, using a p-value of 0.05, against the alternative that it does not equal $100,000.

You will:

A. Reject the null hypothesis if the sample mean is $130,000

B. Fail to reject the null hypothesis if the sample mean is between $80,000 and $120,000

C. Fail to reject the null hypothesis if the sample mean is less than $120,000

D. Accept the null hypothesis

Answer 1

The answer is:

B. Fail to reject the null hypothesis if the sample mean is between $80,000 and $120,000

Explanation

Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is

tc​=1.984.

The rejection region for this two-tailed test is R={t:∣t∣>1.984}

A. FALSE

When the sample mean = 130000 then the t-statistic is computed as follows:

Since it is observed that ∣t∣=1.875≤tc​=1.984, it is then concluded that the null hypothesis is not rejected.

C. FALSE

If mean = 10000, then:

Since it is observed that ∣t∣=5.625>tc​=1.984, it is then concluded that the null hypothesis is rejected.

Hence the only viable option left is OPTION B which is true.

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