Final Exam Math 114 - Statistics for Business Fall 2009 Name: Section: (circle one) 1 2 3 4 5 6 INSTRUCTIONS: This exam contains 24 problems. The first 16 are multiple-choice, and are 3 points each. Record the answers to problems 1 to 16 on the cover sheet. Clearly mark one answer only. If two or more answers are marked, no credit will be given. No partial credit will be given if a wrong answer is marked. Problems 17 to 24 are to be worked in the space provided. Show all your work clearly for these problems, because partial credit will be given. Calculators are allowed. Good luck! ANSWER SHEET: DO NOT SEPARATE the answer sheet from the rest of the test. Work and CIRCLE the answer to each problem INSIDE the test. QUESTION ANSWER QUESTION ANSWER 1. a b c d 9. a b c d 2. a b c d 10. a b c d 3. a b c d 11. a b c d 4. a b c d 12. a b c d 5. a b c d 13. a b c d 6. a b c d 14. a b c d 7. a b c d 15. a b c d 8. a b c d 16. a b c d 1
MULTIPLE CHOICE: Circle one answer for problems 1-10 below: (1) According to the Bureau of Justice Statistics, 73.5% of all licensed drivers who are stopped by police are 25 year or older. Give a percentile ranking, p, for the age of 25 years or above in the distribution of all ages of licensed drivers stopped by the police. (i.e. a ranking such that p% of drivers fall below it.) (a) 33.3rd percentile (b) 26.5th percentile (c) 73.5th percentile (d) 99.5th percentile (2) Consider two independent events A and B with P (A) = 0.7 and P (B) = 0.3. What is P (A B)? (a) 0.21 (b) 0.50 (c) 0.79 (d) 1.00 (3) Suppose in a population of 1,000 athletes, 200 are illegally using testosterone. Of the users, suppose 80 would test positive. Of the nonusers, suppose 11 would test positive. If an athlete tests positive for testosterone, find the probability is really using testosterone. (a) 0.845 (b) 0.872 (c) 0.879 (d) 0.901 For problems 4 and 5, consider the following distribution for a random variable X Value of X 4 8 15 16 23 p(x) 0.1 0.2 0.5 0.15 0.05 (4) Find the expected value E(X) (a) 12.85 (b) 13.05 (c) 13.20 (d) 14.00 (5) Find the variance of X (a) 21.45 (b) 21.47 (c) 21.57 (d) 21.63 2
(6) A bowl contains 8 red balls and 5 green balls. Suppose you randomly draw 4 balls and after each draw, you replace the ball. What is the probability that 3 of the balls are red? (a) 0.359 (b) 0.392 (c) 0.418 (d) 0.608 (7) A poll by the Gallop Organization found that 40% of employees have missed work due to a back injury. In a sample of fifteen workers, what is the probability that more than four workers missed work due to a back injury? (a) 0.091 (b) 0.217 (c) 0.783 (d) 0.909 (8) Suppose we have selected a random sample of n = 36 observations from a population with mean µ = 78.25 and a standard deviation of σ = 5.43. Find the probability that x will be larger than 80. (a) 0.37342 (b) 0.12658 (c) 0.02657 (d) 0.00452 (9) Suppose in a previous sample of 75 new DVDs total, 15 were defective. If we want to estimate p, the population proportion of new DVDs that are defective, how many new DVDs should be randomly sampled to estimate p to within 0.02 of its true value with 99% confidence? (a) 1,083 (b) 1,145 (c) 2,653 (d) 10,609 (10) Consider the following test of hypothesis: H 0 : µ = 5, H a : µ 5. For a sample, you found the test statistic z = 2.04. For which values of α can we reject the alternative hypothesis? (a) α > 0.0414 (b) α > 0.0207 (c) α < 0.0414 (d) α < 0.0207 3
(11) Consider the lower-tailed test of hypothesis with H 0 : µ = 12 tested against the alternative hypothesis H a : µ < 12. The observed value of the test statistic was z = 2.91. The observed significance level of this test is (a) 0.0018 (b) 0.0058 (c) 0.0036 (d) 0.0072 For problems 12 and 13, consider the following Independent samples selected from two populations which produced the following results: Sample 1 Sample 2 n 1 = 20 n 2 = 10 x 1 = 73.01 x 2 = 67.93 s 2 1 = 13 s 2 2 = 17 (12) Assuming that σ 1 = σ 2 = σ, calculate the pooled estimator of σ 2 (a) 14.286 (b) 15.000 (c) 2.350 (d) 4.138 (13) Find the 95% confidence interval for (µ 1 µ 2 ) (a) 5.08 ± 3.00 (b) 5.08 ± 4.20 (c) 5.08 ± 2.86 (d) 5.08 ± 2.50 (14) A paired difference experiment yielded n d = 12 pairs of observations. What is the rejection region for testing H a : µ d > 2 against H 0 : µ d = 2, using α = 0.05? (a) t > 1.782 (b) t > 1.796 (c) t > 2.201 (d) t > 2.201 4
(15) Which of the following is NOT an assumption made about the probability distribution of the random error ɛ in a probabilistic linear model? (a) µ ɛ = 0 (b) The probability distibution of ɛ is normal (c) The variance of the probability distribution of ɛ is constant for all x (d) The values of ɛ associated with any two observed values of y are dependent (16) Consider the following set of data with independent variable x: x 1 2 3 4 y 1 1.6 2.45 3.3 The slope of the least squares line, ˆβ 1 is given by: (a) 0.775 (b) 0.925 (c) 1.290 (d) -0.100 5
SHORT ANSWER: Problems 17-24 (17) Consider the following small sample (a) (2 points) Find the sample mean 1.24 1.35 0.89 0.77 0.74 1.11 0.89 0.64 (b) (2 points) Find the sample standard deviation (c) (3 points) Construct a 95% confidence interval for µ (d) (3 points) Use a test of hypothesis to see if µ < number for α = 0.05 6
(18) Independent random samples were selected from two populations and produced the following results Sample 1 Sample 2 n 1 = 110 n 2 = 120 x 1 = 41.22 x 2 = 42.68 s 1 = 3.12 s 2 = 3.23 (a) (3 points) Find a 99% confidence interval for µ 1 µ 2. (b) (3 points) Use a test for hypothesis with α = 0.01 to determine if there is a difference between the two means. 7
(19) The murder rate from 8 different cities from 2005 and 2007 are recorded in the table below City 2005 Murder Rate 2007 Murder Rate Baltimore 42.0 44.7 Boston 12.9 11.2 Chicago 15.6 16.3 Detroit 41.4 45.9 Nashville 17.1 13.2 Atlanta 20.9 26.4 San Francisco 12.8 14.3 Seattle 4.3 4.3 (a) (3 points) Find a 90% confidence interval for µ 1 µ 2, where µ 1 and µ 2 denote mean murder rates from 2005 and 2007, respectively. (b) (3 points) Use a test for hypothesis with α = 0.1 to determine if there is a difference between the two means. 8
(20) A county welfare agency employs 11 welfare workers who interview prospective food stamp recipients. Periodically, the supervisor randomly selects 2 forms filled out by two workers to audit for illegal deductions. Unknown to the supervisor, 3 of the workers have been giving illegal deductions to applicants. (a)(1 point) How many different ways can the supervisor select 2 forms at random? (b) (2 points) Find the probability of the event that both selected workers have been giving illegal deductions (c) (2 points) Find the probability of the event that at least one of the selected workers have been giving illegal deductions 9
(21) (3 points) A sample is selected from one of two populations, S 1 and S 2 with probabilities P (S 1 ) = 0.80 and P (S 2 ) = 0.20. If the sample has been selected from S 1, The probability of observing an event A is P (A S 1 ) = 0.15. Similary, if the sample selected from S 2, the probability of observing A is P (A S 2 ) = 0.10. If a sample if randomly selected from one of the 2 populations, what is the probability that event A occurs? (22) According to the Journal of Business Venturing, 27% of all small businesses owned by non-hispanic whites nationwide are women-owned firms. (a) (3 points) If 6 small businesses owned by non-hispanic whites are selected at random, what is the probability that at least 3 of them are women-owned? (b) (2 points) Assume we are considering a sample of n = 6 non-hispanic owned smallbusinesses, and let x=the number owned by women. What is the expected value and standard deviation of x? 10
(23) Consider the following data on the independent variable x and the response variable y (a) (2 points) Draw a scatterplot for this data x 1 3 4 6 8 10 y 8.6 8.2 7.3 8.1 4.3 0.3 (b) (2 points) From the scatterplot, do we seem to have a positive trend, a negative trend, or no clear trend (c) (2 points) Find the least squares regression line y = ax + b (d) (2 points) Back up your answer to (b) with your answer from (c) 11
(24) We have data on 20 cases for the independent variables x 1, x 2, x 3 and the reponse variable y. We assume that for some fixed unknown parameters β 0, β 1, β 2, β 3 the response variable y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + ɛ where the error term ɛ is normally distributed, with mean 0, standard deviation equal for all values of x, and independent for all values of x. Below is a MS Excel regression analysis of our data set. Regression Statistics Multiple R 0.890849811 R Square 0.793613385 Adjusted R Square 0.754915895 Standard Error 6.424470937 Observations 20 ANOVA df SS M S F Signif icancef Regression 3 2539.34773 846.4492433 20.50813575 9.93016E 06 Residual 16 660.3812291 41.27382682 Total 19 3199.728959 Coef f icients StandardError tstat P value Intercept 62.76622836 7.055254986 8.896379859 1.36197E 07 X Variable 1 0.439468354 0.119501207 3.677522309 0.002036799 X Variable 2 0.132145553 0.168739967 0.78313132 0.444987864 X Variable 3 1.244279557 0.168689128 7.376169243 1.56391E 06 (a) (3 points) Which independent variable has little predictive value for our model? Why? (b) (3 points) Of the true parameters β 1, β 2, β 3, which one(s) are we sure are positive? Negative? 12
(c) (3 points) Give a 95 % confidence interval for β 1 13