Econ 423 Econometrics I – Recall that the random variable X

Econ 423: Econometrics ISchroeder 1 Problem Set 7 Problem 1Refer to Problem 8 from Problem Set 6. (a) Recall that the random variable X ? µx is the sampling error of the sample mean. What is the probability distribution of X ? µx ? Compute E(X ? µx ) and Var(X ? µx ). 2(b) The random variable Sx ? ?x2 is the sampling error of the sample variance. What is the probability distribution of Sx2 ? ?x2 ? Compute E(Sx2 ? ?x2 ) and Var(Sx2 ? ?x2 ).Problem 2Suppose we ask students in Econ 423 how many siblings they have and how far away fromCorvallis they grew up.(a) Assume the population of interest is all students at OSU, and the sample means from thesurvey questions are used to estimate the respective population means. Discuss whether nonsampling error is a major problem with the estimates.(b) Now assume the population of interest is all American adults. Discuss whether non-samplingerror is likely a major problem with the estimates.(c) Explain why the presence of non-sampling error depends on whether the population ofinterest is all OSU students or all American adults.Problem 3Suppose you have a random sample of size n on a random variable X, and you are interestedin estimating the population mean, µx . Consider two different estimators: (i) the sample mean,nnPe = 1 X1 + 1 P Xi .X= n1Xi , and (ii) another estimator, X22(n?1)i=1 i=2 e are unbiased estimators of the population mean.(a) Show that both X and Xe ). Which one has the lower variance?(b) Calculate Var(X) and Var(XnHint: You will use the fact that n1 < 14 n?1for n>2.e is not consistent.(c) Show that X is consistent, and show that Xe(d) Explain why it is incorrect to say,“X is more efficient than X.”(e) Give an example of an estimator that has a lower variance than X. 1 Problem 4 Suppose you have a random sample of size n= 50 for a random variable, X, withobserved x? =2.4.(a) Assuming you know ?x2 =2, construct an approximate 95% Confidence Interval estimate ofthe population mean, µx .(b) Assuming you know ?x2 =2, construct an approximate 99% Confidence Interval estimate ofthe population mean, µx .(c) Assuming you only have an estimate of the variance of x, s2x =2, construct an approximate95% Confidence Interval estimate of the population mean, µx .(d) Assuming you only have an estimate of the variance of x, s2x =2, construct an approximate99% Confidence Interval estimate of the population mean, µx .(e) Which of your intervals were shorter? What is the intuitive reason for this difference? Problem 5 (Weiss 9th edition 8.31)Venture Capital Investments. Data on investments in the high-tech industry by venturecapitalists are compiled by VentureOne Corporation and published in America’s Network TelecomInvestor Supplement. A random sample of 18 venture-capital investments in the fiber optics business sector yielded the following data, in millions of dollars.5.60 6.27 5.96 10.51 2.04 5.48 5.74 5.58 4.13 8.63 5.95 6.67 4.21 7.71 9.21 4.98 8.64 6.66 (a) Determine a 95% confidence interval for the mean amount, µ, of all venture-capital investments in the fiber optics business sector. Assume that the population standard deviation is $2.04million. (Note: the sum of the data is $113.97 million.)(b) Interpret your answer from part (a).Problem 6 (Weiss 9th edition 8.37)Refer to Problem 5.(a) Find a 99% confidence interval for µ.(b) Why is the confidence interval you found in part (a) longer than the one in Problem 5?(c) Draw a graph similar to Weiss Figure 8.3 (below) to display both confidence intervals.(d) Which confidence interval yields a more precise estimate of µ? Explain your answer. 2 Figure 1: Weiss Figure 8.3Problem 7Set up the rejection region, compute the p-value, and interpret the result of the following tests.The observed x is denoted x? .(a) H0 : µx = 1000, H1 : µx 6= 1000, ?x = 200, n = 100, x? = 980, ? = 0.01(b) H0 : µx = 50, H1 : µx > 50, ?x = 5, n = 9, x? = 51, ? = 0.01(c) H0 : µx = 15, H1 : µx < 15, ?x = 2, n = 25, x? = 14.3, ? = 0.10(d) H0 : µx = 50, H1 : µx < 50, ?x = 15, n = 100, x? = 52, ? = 0.05 Problem 8 (Weiss 9th edition 9.28)Teacher Salaries. The Educational Resource Service publishes information about wages andsalaries in the public schools system in National Survey of Salaries and Wages in Public Schools.The mean annual salary of (public) classroom teachers is $49.0 thousand. A hypothesis test is tobe performed to decide whether the mean annual salary of classroom teachers in Hawaii is greaterthan the national mean.(a) Determine the null and alternative hypothesis. Is this a two-tailed, left-tailed or right-tailedtest?(b) Explain what each of the following would mean: (i) Type I error. (ii) Type II error. (iii)Correct decision. Problem 9 Weiss 9th edition 9.77 and 9.85Serving time. According to the Bureau of Crime Statistics and Research of Australia, as reported on Lawlink, the mean length of imprisonment for motor-vehicle-theft offenders in Australiais 16.7 months. One hundred randomly selected motor-vehicle-theft offenders in Sydney, Australia,had a mean length of imprisonment of 17.8 months. 3 (a) At the 5% significance level, do the data provide sufficient evidence to conclude that themean length of imprisonment for motor-vehicle-theft offenders in Sydney differs from the nationalmean in Australia? Assume that the population standard deviation of the lengths of imprisonmentfor motor-vehicle-theft offenders in Sydney is 6.0 months.(b) Construct a 95% confidence interval for µ.This relationship holds in general: For a two-tailed hypothesis test at significance level ?, thenull hypothesis H0 : µ = µ0 will be rejected in favor of the alternative hypothesis H1 : µ 6= µ0 ifand only if µ0 lies outside the (1 ? ?) confidence interval for µ.Problem 10Set up the rejection region and interpret the result of the following tests at the 5% significancelevel. The observed x is denoted x? .(a) H0 : µx = 22, H1 : µx > 22, s = 4; n = 15 x? = 24(b) H0 : µx = 22, H1 : µx > 22, s = 4; n = 15 x? = 23 4