QUESTION 1
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 an one-sample t test at the 10% level of significance where the hypotheses are H0 : µ = 0 vs H1 : µ > 0. The number of observations is 14. What is the critical value? [2 marks]
(b) Suppose we are doing an hypothesis test and we can reject H0 at the 5% level of significance, can we reject the same H0 at the 10% level of significance? [2 marks]
(c) Suppose we are doing an hypothesis test and we can reject H0 at the 5% level of significance, can we reject the same H0 at the 1% level of significance? [2 marks]
(d) Suppose we are doing a two-sample proportion test at the 10% level of significance where the hypotheses are H0 : p1 - p2 = 0 vs H1 : p1 - p2 6= 0. The calculated test statistic is 0.34. Can we reject H0? [2 marks]
(e) Based on the data, we obtain (0.43, 0.67) as the 95% confidence interval for the true proportion. Can we reject H0 : p = 0.4 against H1 : p 6= 0.4 at the 5% level of significance? [2 marks]
Question 2
Before the 2016 U.S. president election, 1,000 Americans were surveyed and 516 of them said they will vote for Clinton.
(a) Calculate the sample proportion of Americans who support Clinton. [1 mark]
(b) Using the sample proportion, calculate the sample size needed to construct a 95% confidence interval for a similar survey so that the margin of error is no greater than 0.02. [3 marks]
(c) Conduct an hypothesis test to determine if the true proportion of Americans supporting Clinton is greater than 0.5 at the 5% level of significance. [6 marks]
(d) Suppose the 1,000 Americans came from either California (CA) or Florida (FL) and assume the following figures:
Number of Respondents Supporting Clinton: 348 (CA), 168 (FL)
Total Number of Respondents 580 (CA), 420 (FL)
(i) Denote by pc and pf the true proportions of Americans supporting Clinton in CA and FL respectively. Calculate the sample proportions ˆpc and ˆpf . [2 marks]
(ii) Conduct an hypothesis test to determine if the true proportions of Americans supporting Clinton is the same in CA and FL at the 5% level of significance. [6 marks]