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mathematics
statistics

Questions and Answers of Statistics

In a survey of400 likely voters, 215 responded that they would vote for the incumbent and 185 responded that they would vote for the challenger. Let p denote the fraction of all likely voters who
A survey of 1055 registered voters is conducted, and the voters are asked to choose between candidate A and candidate B. Let p denote the fraction of voters in the population who prefer candidate A,
In a given population, 11 % of the likely voters are African American. A survey using a simple random sample of 600 landline telephone numbers finds 8% African Americans. Is there evidence that the
Suppose that a light-bulb manufacturing plant produces bulbs with a mean life of 2000 hours and a standard deviation of 200 hours. An inventor claims to have developed an improved process that
Suppose that a researcher, using data on class size (CS) and average test scores from 100 third-grade classes, estimates the OLS regression(a) A classroom has 22 students. What is the regression's
Consider the regression model Yi = β0 + β1Xi + ui.(a) Suppose you know that β0 = 0. Derive a formula for the least squares estimator of β1.(b) Suppose you know that β0 = 4. Derive a formula for
Suppose that Yi Î²0 + Î²1Xi + kui where K is a non-zero constant and (Yi, Xi) satisfy the three least squares assumptions. Show that the large sample variance of 1 is given by
A regression of average weekly earnings (AWE, measured in dollars) on age (measured in years) using a random sample of college-educated fulltime workers aged 25-65 yields the following:(a) Explain
A professor decides to run an experiment to measure the effect of time pressure on final exam scores. He gives each of the 400 students in his course the same final exam, but some students have 90
Show that 0 is an unbiased estimator of β0.
(a) A linear regression yields 1 = 0. Show that R2 = 0.(b) A linear regression yields R2 = 0. Does this imply that 1 = 0?
Suppose that a researcher, using data on class size (CS) and average test scores from 100 third-grade classes, estimates the OLS regressiona. Construct a 95% confidence interval for ph the regression
A random sample of workers contains nm = 120 men and nw = 131 women. The sample average of men's weekly earnings is $523.10, and the sample standard deviation is $68.1. The corresponding values for
Suppose that (Yi, Xi) satisfy the assumptions in Key Concept 4.3 and, in addition, ui is N(0, σ2u) and is independent of Xi. (a) Is pi conditionally unbiased? (b) Is pi the best linear conditionally
A researcher has two independent samples of observations on (Yi -, Xi). To be specific, suppose that Yi denotes earnings, Xi denotes years of schooling, and the independent samples are for men and
In the 1980s, Tennessee conducted an experiment in which kindergarten students were randomly assigned to "regular" and "small" classes, and given standardized tests at the end of the year. (Regular
Suppose that (Yi, Xi) satisfy the assumptions in Key Concept 4.3. A random sample of size n = 250 is drawn and yieldsŶ = 5.4 + 3.2X, R2 = 0.26, SER = 6.2.(3.1) (1.5)(a) Test H0: βi = 0 vs. H1: β1
Consider the regression model Yi = βXi + ui, where ui and Xi{ satisfy the assumptions in Key Concept 4.3. Let denote an estimator of β that is constructed as = /, where and are the sample
Compute for each of the regressions.
(Requires calculus) Consider the regression modelYi, = Î²1X1i + Î²2X2i + uifor i = 1,..., n. (Notice that there is no constant term in the regression.) Following analysis like that used in
Using the regression results in column (2): (a) Is age an important determinant of earnings? Explain. (b) Sally is a 29-year-old female college graduate. Betsy is a 34-year-old female college
Data were collected from a random sample of 220 home sales from a community in 2003. Let Price denote the selling price (in $1000), BDR denote the number of bedrooms, Bath denote the number of
Critique each of the following proposed research plans. Your critique should explain any problems with the proposed research and describe how the research plan might be improved. Include a discussion
(Yi, Xi, X2i) satisfy the assumptions in Key Concept 6.4. You are interested in β1, the causal effect of X1 on Y. Suppose that X1 and X2 are uncorrelated. You estimate by β1 regressing Y onto X1
A school district undertakes an experiment to estimate the effect of class size on test scores in second grade classes. The district assigns 50% of its previous year's first graders to small
Using the regression results in column (2):(a) Is age an important determinant of earnings? Use an appropriate statistical test and/or confidence interval to explain your answer.(b) Sally is a
The regression shown in column (2) was estimated again, this time using data from 1992 (4000 observations selected at random from the March 1993 CPS, converted into 1998 dollars using the consumer
Question 6.5 reported the following regression (where standard errors have been added):a. Is the coefficient on BDR statistically significantly different from zero?b. Typically five-bedroom houses
Consider the regression model Yi = β0 + β1X1i + β2X2i + ui. Use Approach #2 from Section 7.3 to transform the regression so that you can use a t-statistic to test(a) β1 = β2;(b) β1 + aβ2 = 0,
Sales in a company are $196 million in 2009 and increase to $198 million in 2010. (a) Compute the percentage increase in sales using the usual formula 100 × (sales2010 - Sales2009) / Sales2009
Derive the expressions for the elasticities given in Appendix 8.2 for the linear and log-log models.
After reading this chapter's analysis of test scores and class size, an educator comments, "In my experience, student performance depends on class size, but not in the way your regressions say.
Read the box "The Demand for Economics Journals" in Section 8.3. (a) The box reaches three conclusions. Looking at the results in the table, what is the basis for each of these conclusions? (b) Using
This problem is inspired by a study of the "gender gap" in earnings in top corporate jobs [Bertrand and Hallock (2001)]. The study compares total compensation among top executives in a large set of
Explain how you would use "Approach #2" of Section 7.3 to calculate the confidence interval discussed below Equation (8.8).
Suppose that you have just read a careful statistical study of the effect of advertising on the demand for cigarettes. Using data from New York during the 1970s, the study concluded that advertising
Read the box "The Demand for Economics Journals" in Section 8.3. Discuss the internal and external validity of the estimated effect of price per citation on subscriptions.
Assume that the regression model Yt = β0 + β1Xi + ui satisfies the least squares assumptions in Key Concept 4.3 in Section 4.4. You and a friend collect a random sample of 300 observations on Y and
Labor economists studying the determinants of women's earnings discovered a puzzling empirical result. Using randomly selected employed women, they regressed earnings on the women's number of
The demand for a commodity is given by Q = β0 + β1P + u, where O denotes quantity, P denotes price, and u denotes factors other than price that determine demand. Supply for the commodity is given
Consider the linear regression of Test Score on Income and the nonlinear regression in Equation (8.18). Would either of these regressions provide a reliable estimate of the effect of income on test
This exercise refers to the drunk driving panel data regression summarized in Table 10.1.a. New Jersey has a population of 8.1 million people. Suppose that New Jersey increased the tax on a case of
Let Î²1DM denote the entity-demeaned estimator given in Equation (10.22), and let Î²1BA denote the "before and after" estimator without an intercept, so thatShow that, if
Section 9.2 gave a list of five potential threats to the internal validity of a regression study. Apply this list to the empirical analysis in Section 10.6 and thereby draw conclusions about its
Consider the model with a single regressor Yit = Î²1X1,it + Î±i + Î»t + uit. This model also can be written aswhere B2i = 1 if t = 2 and 0 otherwise, D2i = 1 if i = 2 and 0 otherwise, and so
A researcher believes that traffic fatalities increase when roads are icy and so states with more snow will have more fatalities than other states. Comment on the following methods designed to
a. In the fixed effects regression model, are the fixed entity effects, αi, consistently estimated as n → ∞ with T fixed?b. If n is large (say, n = 2000) but T is small (say, T = 4), do you
Using the results in column (1):a. Does the probability of passing the test depend on Experience? Explain.b. Matthew has 10 years of driving experience. What is the probability that he will pass the
(Requires Appendix 11.3) Which model would you use for: a. A study explaining the number of minutes that a person spends talking on a cell phone during the month? b. A study explaining grades (A
a. Answer (a) through (c) from Exercise 11.1 using the results in column (3).b. Sketch the predicted probabilities from the probit and linear probability in columns (1) and (3) as a function of
Using the results in column (7):a. Akira is a man with 10 years of driving experience. What is the probability that he will pass the test?b. Jane is a woman with 2 years of driving experience. What
Repeat Exercise 11.6 using the logit model in Equation (11.10). Are the logit and probit results similar? Explain. Exercise 11.6 Use the estimated probit model in Equation (11.8) to answer the
Use the estimated linear probability model shown in column (1) of Table 11.2 to answer the following: a. Two applicants, one white and one black, apply for a mortgage. They have the same values for
This question refers to the panel data regressions summarized in Table 12.1. a. Suppose that the federal government is considering a new tax on cigarettes that is estimated to increase the retail
A classmate is interested in estimating the variance of the error term in Equation (12.1).a. Suppose that she uses the estimator from the second-stage regression of TSLS:where Xi is the fitted value
Consider the instrumental variable regression modelYi = β0 + β1Xi + β2Wi + ui,where Xi is correlated with ui and Zi is an instrument. Suppose that the first three assumptions in Key Concept 12.4
In an instrumental variable regression model with one regressor, Xi, and two instruments, Z1i and Z2i, the value of the J-statistic is J = 18.2.a. Does this suggest that E(ui|Zli, Z2i) ≠ 0?
A researcher is interested in the effect of military service on human capital. He collects data from a random sample of 4000 workers aged 40 and runs the OLS regression Yi = β0 + β1Xi + ui, where
Using the results in Table 13.1, calculate the following for each grade: an estimate of the small class treatment effect, relative to the regular class; its standard error; and its 95% confidence
In Chapter 12, state-level panel data were used to estimate the price elasticity of demand for cigarettes, using the state sales tax as an instrumental variable. Consider in particular regression (1)
Suppose that, in a randomized controlled experiment of the effect of an SAT preparatory course on SAT scores, the following results are reported:a. Estimate the average treatment effect on test
Consider a study to evaluate the effect on college student grades of dorm room Internet connections. In a large dorm, half the rooms are randomly wired for high-speed Internet connections (the
Suppose that you have panel data from an experiment with T = 2 periods (so t = 1, 2). Consider the panel data regression model with fixed individual and time effects and individual characteristics Wi
Derive the final equality in Equation (13.10).
Consider the AR(1) model Yt = β0 +β1Yt-1 + ut. Suppose that the process is stationary.a. Show that E(Yt) = E(Yt-1).b. Show that E(Yt) = β0/(1 - β1).
Using the same data as in Exercise 14.2, a researcher tests for a stochastic trend in In(IPt) using the following regression:where the standard errors shown in parentheses are computed using the
Prove the following results about conditional means, forecasts, and forecast errors:(a) Let W be a random variable with mean Î¼W and variance Ïƒ2W and let c be a constant. Show that(b) Consider
Suppose that Yt follows the stationary AR(1) model Yt = 2.5 + 0.7Yt-1 + ut, where ut is i.i.d. with E(ut) = 0 and var(ut) = 9.(a) Compute the mean and variance of Yt.(b) Compute the first two auto
The moving average model of order q has the formwhere et is a serially uncorrelated random variable with mean 0 and variance Ïƒ2e.a. Show that E(Yt) = Î²0.b. Show that the variance of Yt isc.
Increases in oil prices have been blamed for several recessions in developed countries. To quantify the effect of oil prices on real economic activity, researchers have done regressions like those
Consider two different randomized experiments. In experiment A, oil prices are set randomly and the central bank reacts according to its usual policy rules in response to economic conditions,
Derive Equation (15.7) from Equation (15.4) and show that δ0 = β0, δ1 = β1, δ2 = β1 + β2, δ3 = β1 + β2 + β3 (etc.).
Consider the regression model Yt = Î²0 + Î²1Xt + ut, where ut follows the stationary AR(1) modelwith mean 0 and variancea. Suppose that Xt is independent of uj for all t and j. Is Xt exogenous
Consider the "constant-term-only" regression model Yt = Î²0 + ut, where ut follows the stationary AR(1) modelwith mean 0 and variancea. Show that the OLS estimator isb. Show that the (infeasible)
Suppose that Yt follows a stationary AR(1) model, Yt = Î²0 + Î²1Yt-1 + ut.a. Show that the h-period ahead forecast of Yt is given by Yt+h|t = Î¼Y + Î²1h(Yt - Î¼Y), where Î¼Y = Î²0/(1 -
Suppose that ut follows the ARCH process,a. Let E(ut2) = var(ut) be the unconditional variance of ut. Show that var(ut) = 2.b. Suppose that the distribution of ut conditional on lagged values of ut
Verify Equation (16.20).Useto show that and solve for
Suppose that ˆ†Yt = ut, where ut is i.i.d. N(0, 1), and consider the regression Yt = Î²Xt + error, where Xt = ˆ†Yt+1 and error is the regression error. Show that
a. Suppose that E(ut|ut-1, ut-2,...) = 0, that var(ut|ut-1, ut-2,...) follows the ARCH(l) modeland that the process for ut is stationary. Show that var(ut) = Î±0/(1 - Î±1).b. Extend the result in
Consider the regression model without an intercept term, Yi = Î²1Xi + ui (so the true value of the intercept, Î²0, is zero).a. Derive the least squares estimator of Î²1 for the restricted
Suppose that X and Y are distributed bivariate normal with density given in Equation (17.38).a. Show that the density of Y given X = x can be written aswhereb. Use the result in part (a) to show
Consider the heterogeneous regression model Yi = Î²0i + Î²1iXi + ui, where Î²0i and Î²1i are random variables that differ from one observation to the next.
This exercise fills in the details of the derivation of the asymptotic distribution of Î²1 given in Appendix 4.3.a. Use Equation (17.19) to derive the expressionwhere vi = (Xi - Î¼X)ui.b. Use the
Suppose that W is a random variable with E(W4) < ∞. Show that E(W2) < ∞.
Suppose that X and u are continuous random variables and (Xi, ui), i = 1,..., n are i.i.d.a. Show that the joint probability density function (p.d.f.) of (ui, uj, Xi, Xj) can be written as f(ui,
Prove Equation (17.16) under Assumptions #1 and #2 of Key Concept 17.1 plus the assumption that Xi and ui have eight moments.
Consider the population regression of test scores against income and the square of income in Equation (8.1).a. Write the regression in Equation (8.1) in the matrix form of Equation (18.5). Define Y,
Suppose that C is an n Ã— n symmetric idempotent matrix with rank r and leta. Show that C = AA€², where A is n Ã— r with A€²A = Ir.b. Show that A€²V ~ N(0r, Ir).c. Show that V€²CV ~
Consider the problem of minimizing the sum of squared residuals subject to the constraint that Rb = r, where R is q Ã— (k + l) with rank q. Let Î² be the value of b that solves the constrained
(Consistency of clustered standard errors.) Consider the panel data model Yit = Î²Xit + Î±i + uit, where all variables are scalars. Assume that Assumptions #1, #2, and #4 in Key Concept 10.3 hold
Let W be an m × 1 vector with covariance matrix ∑W, where ∑W is finite and positive definite. Let c be a nonrandom m × 1 vector, and let Q = c′W.a. Show that var(Q) = c′∑Wc.b. Suppose
Let Px and Mx be as defined in Equations (18.24) and (18.25). a. Prove that PXMX = 0n×n and that Px and Mx are idempotent. b. Derive Equations (18.27) and (18.28).
Consider the regression model, Yi = Î²1Xi + Î²2Wi + ui, where for simplicity the intercept is omitted and all variables are assumed to have a mean of zero. Suppose that Xi is distributed
This exercise shows that the OLS estimator of a subset of the regression coefficients is consistent under the conditional mean independence assumption stated in Appendix 7.2. Consider the multiple
Consider the following organizations. How would you define quality in each context? Specify attributes/variables that may measure quality. How do you integrate these measures? Discuss the ease or
Explain how it is feasible to increase productivity, reduce costs, and improve market share at the same time.
Explain why it is possible for external failure costs to go up even if the first-pass quality level of a product made by a company remains the same.
Discuss the impact of technological breakthrough on the prevention and appraisal cost and failure cost functions.
As natural resources become scarce, discuss the role of ISO 14000 in promoting good environmental management practices.
Discuss the processes through which supply chain quality may be monitored.

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