Stats 11 (Fall 2004) Lecture Note Instructor: Hongquan XuIntroduction to Statistical Methods for Business and EconomicsFinal Exam Review — Chapters 1–10, 12NOTE: For additional review problems, see the review lectures for the midterm exams.Final Exam• Time: Tuesday December 14, 3-6pm.• Place: TBA• Material: Chapters 1–10, 12 (including all lectures, all homework , all labs)• It will be a closed-book exam.• Bring your photo ID, calculator, ruler, pen, pencil, eraser.NOTE: The following formulas (plus formulas for midterm exams 1 and 2) will be provided. Tables will beprovided if needed.Formulas.Hypotheses Testing for Population Means and ProportionsTest for mean µ [H0: µ = µ0] : test statistic z =¯x−µ0σ /√nor t =¯x−µ0s/√n.Test for proportion p [H0: p = p0]: test statistic z =bp−p0√p0(1−p0)/n.Correlation coe ffic ientr =1n−1Pxi−¯xsxyi−¯ysy.Simple Linear Regress ionModel: yi= β0+ β1xi+ i, where iare as sumed to be independent and have N(0, σ) distribution.Estimates: slopeˆβ1= rsysx, interceptˆβ0= ¯y −ˆβ1¯x, and regression line ˆy =ˆβ0+ˆβ1x.Estimate of σ : s =√s2=√MSECI for s lope β1:ˆβ1± t∗se(ˆβ1) (note: df = n − 2 for the t∗value)Test for slope [ H0: β1= 0 ] : test statistic t =ˆβ1se(ˆβ1)(note: df = n − 2)11. A service station has both self-service and full-service islands. On each island, there is a single regularunleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island ata particular time, and let Y denote the number of hoses on the full-service island in use at that time. Thejoint probability distribution of X and Y appears as follows:yf(x, y) 0 1 20 .10 .04 .02x 1 .08 .20 .062 .06 .14 .30(a) What is the probability of X < 2 and Y = 1?(b) What is the probability of X < 2 given Y = 1?(c) What is the marginal probability distribution of X?(d) What are the means of X and Y ?(e) Find the covariance of X and Y ?(f) Are X and Y independent? Explain.2. Suppose X and Y are two discrete random variables.(a) Show that cov (X, Y ) = E(XY ) − E(X)E(Y ).(b) Show that cov (X, Y ) = 0 if X and Y are independent.23. A national survey of restaurant employees found that 75% reported that work stress had a negativeimpact on their personal lives. A manager of a chain of restaurants finds that 68 of a random sample of 100of the chain’s employees say that work stress has a negative impact on their personal lives.(a) Is there evidence at 5% level to conclude that the proportion for this chain of restaurants differs fromthe value given for the national survey?(b) Give a 95% CI for the proportion of e mployees who work for this chain restaurants who believe thatwork s tress has a negative impact on their personal lives.(c) How many employees are needed to have a 95% CI with a width at most 5% ?4. A 95% CI for a population mean is (28, 35).(a) Can you reject the null hypothesis that µ = 34 at the 5% significance level? Why?(b) Can you reject the null hypothesis that µ = 36 at the 5% significance level? Why?35. Degree of Reading Power (DRP) scores for a random sample of 44 third-grade students in a suburbanschool district have a mean of 35.1 and standard deviation 11.2. Assume DRP scores are approximatelynormal. The researcher believes that the mean score µ of all third graders in this district is higher than thenational mean, which is 32.------------------------------------------------------------------------------Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]---------+--------------------------------------------------------------------DRP | 44 35.09091 1.686853 11.18932 31.68905 38.49277------------------------------------------------------------------------------Degrees of freedom: 43Ho: mean(DRP) = 32Ha: mean < 32 Ha: mean != 32 Ha: mean > 32t = 1.8324 t = 1.8324 t = 1.8324P < t = 0.9631 P > |t| = 0.0738 P > t = 0.0369(a) State the appropriate hypotheses to test this suspicion.(b) What are observed test statistic, degrees of freedom, and p-value?(c) Give the decision and conclusion at the 5% significance level.(d) Give the 95% CI for the mean DRP score µ of all thirdgraders in this district.(e) We assume DRP scores are approximately normal.Comment on the validity of this assumption based on thenormal quantile plot for the DRP scores.46. We wish to assess the effectiveness of a diet program. The weights before and after treatment wererecorded for a random sample of 6 people on the program. The data are provided below.Before 175 188 192 200 186 162After 172 190 186 199 180 163diff -3 2 -6 -1 -6 1Is there evidence (at the 10% significance level) to conclude that the program is effective at reducing weightoverall?(a) Ass ume the differences are normally distributed. Perform an appropriate test.(b) Do not assume the difference s are normally distributed. Perform an appropriate test.57. Amount of money (thousands of dollars) sp ent on education by states. T wo variables are Pay (averagesalary paid to teachers) and Spend (expenditures per pupil). We are interested in how the teachers’ salaryis related to the spending per pupil. Data analysis is done with Stata as follows.. summarizeVariable | Obs Mean Std. Dev. Min Max-------------+-----------------------------------------------------Pay | 51 35.03529 6.268551 26 50Spend | 51 5.942549 1.409631 3.67 9.93. corr(obs=51)| Pay Spend-------------+------------------Pay | 1.0000Spend | 0.7931 1.0000Scatterplot Residual plot(a) Interpret the scatterplot. What do you see?(b) What are the response variable (y) and explanatory variable (x)?6(c) What is the regress ion line?(d) What is the expected te achers’ salary if the spending per pupil is $6,000?(e) What proportion of the variation in teachers’ salary is explained by its linear relationship with thespending pe r pupil?(f) Interpret the residual plot. What do you see?(g) What is the correlation betwee n the residuals and spending?8. A study on the average daily gas consumption for each month and the average number of heating degree -days per day during the month. Here is the regression analysis from Stata.. regress gasconsumption degreedaysSource | SS df MS Number of obs = 9-------------+------------------------------ F( 1, 7) = 311.97Model | 58.9071348 1 58.9071348 Prob > F = 0.0000Residual | 1.32175382 7 .188821975 R-squared = 0.9781-------------+------------------------------ Adj R-squared = 0.9749Total | 60.2288886 8 7.52861108 Root MSE