Question 1. Four sub-questions worth five points each; 20 points in total.Consider the standard setup of randomly sampling X1, X2, . . . , Xn from a normal distribution with meanµ and variance σ2. The following four statements related to that setup are all false (but they have beenencountered in exam papers nonetheless). Explain what is wrong with each of them.1.a. The distribution of µ is normal.1.b. E[¯x] = µ.1.c. VarX¯= σ2.1.d. The distribution of X¯ is normal, with mean µ and variance s2/n.Question 2. Three sub-questions worth six, seven, and seven points, respectively; 20 points in total.Assume that we have a data set with n data points x1, x2, . . . , xn. They have sample mean x¯ and samplestandard deviation sx. We have also calculated y1, y2, . . . , yn according to the formula yi = 5xi − 2.The sample mean and standard deviation of this new variable are called y¯ and sy. Use the definitions ofsample means and standard deviations (in summation notation) to prove the following statements.2.a. 1nXni=1(xi − x¯) = 0.2.b. y¯ = 5¯x − 2.2.c. sy = 5sx.Question 3. Five sub-questions worth four points each; 20 points in total.We have asked 41 Australians to perform an IQ test. Stata summarizes their scores as follows:Variable | Obs Mean Std. Dev. Min Max————+——————————————————–iq | 41 109.4051 20.5052 60.83574 158.6759For this question, feel free to round numbers to one decimal place if it simplifies your calculations.3.a. Construct a 95% confidence interval for the population mean.3.b. Explain carefully how your answer to question 3.a. should be interpreted.3.c. How would your answer to question 3.a. change if the sample mean would be higher?3.d. How would your answer to question 3.a. change if the sample standard deviation would be higher?3.e. The population mean of IQ scores should be 100 by definition. Test this hypothesis at the 5% level.There is one more question on the next page.Question 4. Eight sub-questions worth five points each; 40 points in total.The data set salary.dta contains data on weekly earnings for a random sample of 35 people workingin New South Wales. For each person in the survey, we also know their gender (coded by female==1for women and female==0 for men), and whether they work in Sydney (sydney==1) or elsewhere inthe state (sydney==0). We are interested in testing whether the mean salary is equal for both genders.4.a. Describe carefully what a Type I error would be for this particular test.4.b. Describe carefully what a Type II error would be for this particular test.4.c. Perform the test. (This can be done in Stata using the ttest command with the by option.Alternatively, this is called a “two-group mean-comparison test” in the drop-down menu.) What do youconclude, at the 5% significance level?4.d. Repeat the test from question 4.c., this time using the natural logarithms of the salaries rather thanthe salaries themselves.4.e. Which of the tests in questions 4.c. and 4.d. is preferable from a statistical point of view? Why?4.f. Despite the non-rejections in questions 4.c. and 4.d., we still observe that in this sample, the averageman makes more than the average woman. Perform a one-sided test to assess the claim that women arebeing paid less than men on average.4.g. Statistically speaking, what is wrong with the procedure we followed in question 4.f.?4.h. One might suspect that labour market discrimination is a more common issue in rural areas thanit is in a modern city like Sydney. Repeat your preferred two-sided test, this time using only data frompeople whose workplace is outside Sydney. What can you conclude in this case?