Question

Question 3 This question concerns some concepts about hypothesis testing and confidence interval. For each part below, you must explain your answer.

(a) Suppose we are doing a one-sample t test at the 5% level of significance where the hypotheses are H0 : µ = 0 vs H1 : µ > 0. The number of observations is 8. What is the critical value? [2 marks]

(b) Suppose we are doing a hypothesis test and we can reject H0 at the 1% level of significance, can we reject the same H0 (with the same H1) at the 10% level of significance? [2 marks]

(c) Suppose we are doing a hypothesis test and we cannot reject H0 at the 5% level of significance, can we reject the same H0 (with the same H1) at the 1% level of significance? [2 marks]

(d) Suppose we are doing a two-sample proportion test at the 5% level of significance where the hypotheses are H0 : p1 − p2 = 0 vs H1 : p1 − p2 6= 0. The calculated test statistic is −1.21. Can we reject H0? [2 marks] (e) Based on the data, we obtain (0.53, 0.87) as the 95% confidence interval for the true mean. Can we reject H0 : µ = 0.5 against H1 : µ 6= 0.5 at the 5% level of significance? [2 marks]

Answer 1

a.

We know that:

- This is a one tailed test. Hence the entire rejection region will be on the right side.

- The number of observations is 8. Thus, the degrees of freedom are 7.

- The significance level is 5% or 0.05.

Thus, t_{0.05, 7} = 1.895

b.

Since we can reject the null hypothesis at 1% level of significance, we definitely reject the same hypothesis at 10% level of significance.

c.

We can reject the null hypothesis at the 5% level of significance, but this does not necessarily mean that we can reject the same hypothesis at 1% level of significance.

d.

At the 5% level of significance, the critical z value is + or - 1.96. Since -1.21 lies within this interval, we do not have enough evidence to reject the null hypothesis.

e.

Since the confidence interval does not contain the hypothesized population mean value, (0.50 is not within 0.53 and 0.87), we will reject the null hypothesis.