Question

1. Assume that the population variance is unknown. We test the hypothesis that Ho: µ=5 against the alternative that it is not at a level of significance of 5% and a sample size of n=151. We calculate a test statistic = -1.976. The p-value of this hypothesis test is approximately ______%, Write your answer in percent form. In other words, write 5% as 5.0

2. When choosing between two unbiased estimators, we prefer to use the one that is most efficient.

True False

3. If a null hypothesis is rejected at the 5% significance level but not at the 1% significance level, then the p-value of the test is less than 1%.

True False

Answer 1

1. The p-value is approximately 5%.

For one-sample t-test (where population variance is unknown), we have the null hypothesis as and alternate as . The test-statistic is , which follows the t-distribution with df=n-1=150 (s is sample standard deviation).

For the test-statistic be , the p-value would be as (for x be t-distribution with df=n-1=150) or (since t-distribution is symmetrical around the mean) or .

2. The correct option would be

• True

The estimator having less variance would be more efficient than the other, and hence chances of obtaining (estimating) the true value increases if the estimator have less variance, ie if estimator is more efficient.

3. The correct option would be

• False

The null would be rejected at 5% if p-value is less than 5% or 0.05, ie . The null would fail to be rejected at 1% is p-value is more than 1% or 0.01, ie . Hence, if we have and , that means that we have , ie the p-value is between 1% and 5%. Hence, the p-value can not be less than 1%.