Task 1 (35 marks) For Task 1 you will be using data on the ages of...

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Task 1 (35 marks) For Task 1 you will be using data on the ages of all Master’s students at a mid-size regional university; thus, these data represent the population of the Master’s students at that university. Using the provided data, implement the following steps and answer the following questions: i. What are the population mean and the population variance of the students’ ages equal to? (5 marks) ii. Obtain 30 sample values of two estimators of the population mean (𝑋̅ 1 and 𝑋̅ 2): a. Draw 30 random samples of size 𝑛 = 100 from the population. b. For each of the 30 samples drawn, calculate: o Sample mean 𝑋̅ 1 = βˆ‘ 𝑋𝑖 100 𝑖=1 ⁄100, where 𝑋𝑖 are all 100 observations of student age in the sample (𝑖 = 1, 2, . . . ,100). o An alternative estimator 𝑋̅ 2 = βˆ‘ 𝑋𝑖 20 𝑖=1 ⁄20 where 𝑋𝑖 are the first 20 observations of student age in the sample (𝑖 = 1, 2, . . . ,20). c. Present your calculations in a table: o The first column of this table should contain the 30 values of 𝑋̅ 1 you have obtained. o The second column of this table should contain the 30 values of 𝑋̅ 2 you have obtained. (10 marks) iii. Compute the (approximate) bias of the estimator 𝑋̅ 1 and the (approximate) bias of the estimator 𝑋̅ 2. Is one of the two estimators more biased than the other? Explain why or why not. (10 marks) iv. Compute the (approximate) variance of the estimator 𝑋̅ 1 and the (approximate) variance of the estimator 𝑋̅ 2. Compute the relative efficiency of the two estimators and interpret it. Is one of the two estimators more efficient than the other? (10 marks) Notes: o Include the list of steps and Excel functions you used to answer each question (at the bottom of an answer to each question). o In the parts iii and iv, show your calculations and how you obtained your answers. 4 Task 2 (55 marks) For Task 2 you will be using data on house prices in the UK (in year 2002). These data represent a random sample of 500 properties from the population of all properties sold in 2002 in the UK. Besides the price of each property (in thousands GBP), your also know whether the property is located in or outside London. Assume that: 1. House prices are normally distributed both in London and outside London. 2. The population variance of house prices in London (𝜎1 2 ) is equal to the population variance of house prices outside London (𝜎2 2 ): 𝝈𝟏 𝟐 = 𝝈𝟐 𝟐 . The data are stored in Stata file β€˜cwkdata_task2.dta’ on Moodle. Open this data file in Stata (version 12 or higher; if using an older version of Stata contact the lecturer). Explore the data by using commands describe, summarize, codebook, tab, and/or browse. Using the provided data, implement the following steps and answer the following questions: i. Using Stata, calculate the 99% confidence intervals for: a. the mean price of all properties in London and b. the mean price for all properties outside London Interpret these confidence intervals. (10 marks) ii. Confirm that you get (approximately) the same answers using the appropriate formula for the confidence interval from the lecture notes. (10 marks) iii. What would the sample size need to be, if you wanted the margin of error for the mean price of all properties in London to be GBP5,000 (at the 99% confidence level)? (5 marks) iv. Calculate a 99% confidence interval for the difference in the mean prices of properties in London and properties outside London using Stata. Interpret this confidence interval. (5 marks). v. Confirm that you get (approximately) the same answer using the appropriate formula for the confidence interval from the lecture notes. (10 marks) 5 vi. A newspaper article (published in 2002) claimed that an average family could no longer afford a house in London, because the mean house price in London had increased to more than GBP150,000. a. How would you define the null and alternative hypotheses if you wanted to verify this claim (that the mean house price in London is more than GBP150,000)? b. Test this set of hypotheses at the 5% significance level using Stata. c. Interpret the test result. (7 marks) vii. Confirm that you reach the same conclusion using the test-statistic approach using the appropriate formula for the test-statistic from the lecture notes. (8 marks) Notes: o Include the list of Stata commands you used to answer each question (at the bottom of an answer to each question). o In parts ii, iii, v, and vii, show your calculations and how you obtained your answers. o When testing hypotheses, remember to specify the test-statistic and its distribution, state the decision rule and make a decision (whether to reject or fail to reject the null hypothesis).

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