Explain the following questions.

Question 1 (understanding mean and variance of linear combinations of random variables) Let T_15 be the percentage of adult males who used tobacco products in 2015 in a country and T_10 be this percentage in 2010 in the same country. Define the random variable Z in the following way: Z =T_15 -T_10. We do not observe T_15 and T_10 for all countries of the world. We can only hope to get data from a random sample of n countries, where n is much smaller than the number of countries in the world. We want to estimate the E (Z) for the distribution of countries in the world. Each group member should attempt one of the following questions. The group can consult and improve the answer and only submit the improved answer, but the original person who attempted each part must be named. 1. What does the hypothesis E (Z) = 0 mean? After explaining what this hypothesis means, describe whether or not E (Z) = 0 implies T_15 = T_10 in every country in the world. Then, describe whether or not E (Z) = 0 implies -Er_15; = -Er_10; 1= 1 1= 1 for the n countries in the sample [Note that "Yes it does" or "No it doesn't" are not sufficient, you are expected to justify your answer.] 2. Using the result that sample average is an unbiased estimator of the population mean, show that iz = MET_15; - MELT_10; is an unbiased estimator of E (Z) . 3. Using the result that the variance of the sample average of a random sample of n observations from a distribution with mean / and variance o' is , compute the variance of /z = > >_,T_15; - " Ein T_10;, for a random sample of n = 40 countries, when Var (T_15) = Var(T_10) = 100, and p the correlation coefficient between 7_15 and T_10 is 0.8. 4. Suppose that we have obtained data on T_15 and T_10 for a sample n countries and computed Z; =T_15; -T_10; for i = 1, ..., n. Using the matrix formula for the OLS estimator, show that if we regress this variable on a constant only, the OLS estimate of the constant will be ! )_, T_15;- = ELIT_10.Question 1 (understanding mean and variance of linear combinations of random variables) Let T_15 be the percentage of adult males who used tobacco products in 2015 in a country and T_10 be this percentage in 2010 in the same country. Define the random variable Z in the following way: Z =T_15 -T_10. We do not observe T_15 and T_10 for all countries of the world. We can only hope to get data from a random sample of n countries, where n is much smaller than the number of countries in the world. We want to estimate the E (Z) for the distribution of countries in the world. Each group member should attempt one of the following questions. The group can consult and improve the answer and only submit the improved answer, but the original person who attempted each part must be named. 1. What does the hypothesis E (Z) = 0 mean? After explaining what this hypothesis means, describe whether or not E (Z) = 0 implies T_15 = T_10 in every country in the world. Then, describe whether or not E (Z) = 0 implies -Er_15; = -Er_10; 1= 1 1= 1 for the n countries in the sample [Note that "Yes it does" or "No it doesn't" are not sufficient, you are expected to justify your answer.] 2. Using the result that sample average is an unbiased estimator of the population mean, show that iz = MET_15; - MELT_10; is an unbiased estimator of E (Z) . 3. Using the result that the variance of the sample average of a random sample of n observations from a distribution with mean / and variance o' is , compute the variance of /z = > >_,T_15; - " Ein T_10;, for a random sample of n = 40 countries, when Var (T_15) = Var(T_10) = 100, and p the correlation coefficient between 7_15 and T_10 is 0.8. 4. Suppose that we have obtained data on T_15 and T_10 for a sample n countries and computed Z; =T_15; -T_10; for i = 1, ..., n. Using the matrix formula for the OLS estimator, show that if we regress this variable on a constant only, the OLS estimate of the constant will be ! )_, T_15;- = ELIT_10.You may need to use the appropriate appendix table or technology to answer the question. Last year, 43:4 of business owners gave a holiday gift to their employees. A survey of business owners conducted thes plan to provide a holiday gift to their employees. Suppose the survey results are based on a sample of do business owners. (2) How many business owners in the survey plan to provide a holiday gift to their employees this year? * business owners (6) Suppose the business owners in the sample did as they plan. Compute the p-value for a hypothesis test that can be used to determine if the proportion of providing holiday gifts has decreased from last year. Find the value of the test statistic. (Round your answer to two decimal places,) Find the p-value. (Round your answer to four decimal places.) X (c)Using a 0.05 level of significance, would you conclude that the proportion of business owners providing gifts decreased? O Reject:. There is Insufficient evidence to conclude that the proportion of business owners providing holiday gifts has decreased from last year O Reject Me There is sufficient evidence to conclude that the proportion of business owners providing holiday gifts has decreased from last year, Q Do not reject N.. There is insufficient evidence to conclude that the proportion of business owners providing holiday gifts has decreased from last Year. O Do not reject H. There is sufficient evidence to conclude that the proportion of business owners providing holiday gifts has decreased from last roar What is the smallest level of significance for which you could draw such a conclusion? (Round your answer to four dodmal places.) Need Help? Can1. Towards the end of the 20th century, the U.S. government wanted to save money by closing a small portion of its domestic military installations. While many people agreed that saving money was a desirable goal, people in areas potentially affected by a closing soon reacted negatively. Congress finally selected a panel whose task was to develop a list of installations to close, with the legislation specifying that Congress could not alter the list. Since the goal was to save money, why was this problem so hard to solve? 2. Your car gets 29 miles per gallon (mpg) at 60 miles per hour (mph) and 25 mpg a 70 mph. At what speed should you make a 525-mile trip: a. If gas costs $3 per gallon and your time is worth $18 per hour b. If gas costs $4 per gallon and your time is worth $12 per hour C. If gas costs $5 per gallon and your time is worth $9 per hour 3. A firm is planning to manufacture a new product. As the selling price is increased, the quantity that can be sold decreases. Numerically the sales department estimates: P = $475-0.250 Where P = selling price per unit and Q = quantity sold On the other hand, management estimates that the average unit cost of manufacturing and selling the product will decrease as the quantity sold increases. They estimate C = $480 + $22,500 Where C = cost to produce and sell Q per year The firm's management wishes to maximize profit. What quantity should be sold? How much profit will be made