(Ans: Q=9.9256; χ(α = 0.05, df = 6) = ; accept hypothesis of independence)

Transcription

1 22S:39 Fall 2000 Extra Problems for Final Exam Preparation (1.2-5, 2.1-5, 3.2, 3.4, 4.1-5, 5.1-2, 6.1-3, 7.1-2, 8.3-4, 9.1-4, Multivariate Distributions) M.Larson December A random sample of n = 1362 persons were classified according to the respondent s education level and whether the respondent was Protestant, Catholic, or Jewish. Use these data to test at an α = 0.05 significance level the hypothesis that these attributes of classification are independent. Education Level Protestant Catholic Jewish Less than high school High school or junior college Bachelor s degree Graduate degree (Ans: Q=9.9256; χ(α = 0.05, df = 6) = ; accept hypothesis of independence) 2. A random sample of size four is taken from a normal distribution with unknown mean µ and variance σ 2 > 0. To test H 0 : µ = 0 versus H 1 : µ < 0 the following test is used: Reject H 0 if and only if X 1 + X 2 + X 3 + X 4 < 20. For what value of σ is the significance level of this test closest to 0.14? (Ans: 9.1) 3. Let X denote the number of heads that occur when four coins are tossed at random. Under the assumptions that the four coins are independent and the probability of heads on each coin is 1/2, X is binomial(n = 4, p = 0.5). One hundred repetitions of this experiment resulted in 0, 1, 2, 3, and 4 heads being observed on 7, 18, 40, 31, and 4 trials, respectively. Do these results support the assumptions? Use a significance level of 1%. (Ans: Q = 4.47; χ(α = 0.01, df = 4) = ; the data support the hypothesis that binomial(4, 0.5) is a reasonable model) 4. A random sample of 121 observations from a Poisson distribution has a mean equal to Construct an approximate 95% confidence interval for the mean of the distribution. (Ans: (5.8, 6.7)) 5. A machine shop that manufactures toggle levers has both a day and a night shift. A toggle lever is defective if a standard nut cannot be screwed onto the threads. Let p 1 and p 2 be the proportion of defective levers among those

2 manufactured by the day and night shifts, respectively. Test the hypothesis, H 0 : p 1 = p 2, against a two-sided alternative hypothesis based on two random samples, each of 1000 levers taken from the production of each of the respective shifts. You wish to determine if the proportion of defective levers is the same for the two shifts. The number of defectives in the day shift sample is 37 and the number of defectives from the night shift sample is 53. What is your conclusion at the α = 0.05 significance level? (Ans: test statistic = 1.73; z(0.025) = 1.960; conclude there is no difference between the shifts) 6. Let X denote the number of alpha particles emitted by barium-133 in 1/10 of a second. The following 50 observations of X were taken with a Geiger counter in a fixed position: The experimenter is interested in determinig whether X has a Poisson distribution. Fit a Poisson distribution. Then find the estimated expected value of each cell after combining the outcomes into the following sets: {0, 1, 2, 3}, {4}, {5},{6},{7}, and {8, 9, 10,...}. Compute the test statistic, Q. Do we accept or reject the Poisson distribution at the 5% significance level? (Ans: ˆλ = 5.4; Q=2.763; χ(α = 0.05; df = = 4) = Since < 9.488, accept the hypothesis of the Poisson distribution) 7. A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 with a variance of 260. If a tax of 20% is introduced on all items associated with the maintenance and repair of cares (i.e. everything is made 20% more expensive), what will be the variance of the annual cost of maintaining and repairing a car? (Ans: ) 8. A company manufactures rope whose breaking strengths have a mean of 300 lb and standard deviation of 24 lb. It is believed that by a newly developed process the mean breaking strength can be increased. We decide to test 64 ropes and use a significance level of (a) What is the critical value for this test? (Ans: reject H 0 : µ = 300 lbs, if X > 307 lbs) (b) What is the probability of accepting the old process when in fact the new process has increased the mean breaking strength to 310 lb? (Ans: )

3 (c) Assuming that we reject the hypothesis, H 0 : µ = 300 lbs, if X > 307 lbs, construct an OC curve. 9. The Federal Trade Commission measured the number of milligrams of tar and carbon monoxide (CO) per cigarette for all domestic cigarettes. Let x and y equal the measurements of tar and CO, respectively, for 100-millimeter filtered and mentholated cigarettes. A sample of 12 brands yielded the following data: Brand x y Brand x y Capri 9 6 Now 3 4 Carlton 4 6 Salem Kent Triumph 6 8 Kool Milds True 7 8 Marlboro Lights Vantage 8 13 Merit Ultras 5 7 Virginia Slims You wish to calculate the least squares regression line Y i = β 0 + β 1 x i + ɛ i, i = 1, 2,..., n to this data. Use the following output from MINITAB to answer the questions. Regression Analysis: CO versus tar The regression equation is CO = tar Predictor Coef SE Coef T P Constant tar S = R-Sq = 79.3% R-Sq(adj) = 77.2% Analysis of Variance Source DF SS MS F P Regression Residual Error Total (a) What is the sample correlation coefficient? (Ans: 0.891)

4 (b) Find the point estimates for the parameters β 0 and β 1. (Ans: 2.575; ) (c) Find 95% confidence intervals for β 0 and β 1 under the usual assumptions. (Ans: ( 0.415, 5.565); (0.524, 1.114)) (d) What is the fitted value for the Marlboro Lights brand? (Ans: 10.77) (e) What is the residual value for True cigarettes? (Ans: ) (f) Test H 0 : β 1 = 0 against H 1 : β 0 at the α = 0.05 significance level. (Ans: since p-value = α, reject H 0 ) 10. An insurance company issues 1250 vision care insurance policies. The number of claims filed by a policyholder under a vision care insurance policy during one year is a Poisson random variable with mean 2. Assume the number of claims filed by distinct policyholders are independent of one another. What is the approximate probability that there is a total of between 2450 and 2600 claims during a one-year period? (Ans: ) 11. Packages of candies were inspected. The inspector recorded the number of candies in the package (pieces) and the total weight of the package (weight). Use the given Minitab output to answer the questions. Regression Analysis: weight versus pieces The regression equation is weight = pieces Predictor Coef SE Coef T P Constant pieces S = R-Sq = 74.8% R-Sq(adj) = 73.4% Analysis of Variance Source DF SS MS F P Regression Residual Error Total (a) What is the sample correlation coefficient? (Ans: 0.865)

5 (b) What proportion of the weight can be explained by the number of pieces? (Ans: 74.8%) (c) Construct a 99% confidence interval for the slope of the regression line. (Ans: (0.376, 0.864)) (d) At the 5% significance level, is the null hypothesis of zero slope accepted or rejected? (Ans: rejected) (e) What are the degrees of freedom used to check the F-statistic? (Ans: 1,18) 12. Message bursts are transmitted through a microwave relay at a mean rate of 10 per second. Five percent of these are garbled by radio interference and must be retransmitted. (a) What distribution is appropriate for finding the probability that there will be exactly 3 messages transmitted in a particular 1-second span? Find that value. (Ans: Poisson; 0.007) (b) What distribution is appropriate for finding the probability that there will be exactly 3 garbled messages in the next 15 transmissions? Find that value. (Ans: Binomial; ) 13. The random variable X had the density function f(x) = 5e 5x for 0 x. Find the following (a) E(X) (Ans: 0.2) (b) Var(X) (Ans: 0.04) (c) P (0.5 X 1.5) (Ans: ) 14. If a machine is working correctly, it fills boxes of cereal with amounts in grams according to the following distribution: x < x < x < x The actual frequencies of 40 boxes of cereal were: 14, 12, 8, and 6, respectively. (a) What is the value of the χ 2 statistic, Q, used in testing whether or not the machine is operating correctly? (Ans: 4) (b) What is the correct degrees of freedom to be used in this test? (Ans: 3) (c) Would you conclude that the machine is working correctly? Why or why not? (Note: You must choose your own significance level) (Ans: Yes, since Q = 4 < χ 2 (0.05, 3) = 7.815)

6 15. You are to use simple linear regression to assess the impact of the unemployment rate on the duration of disability for injured workers. You have collected data on the unemployment rate, X i, and the duration of disability, expressed in days, Y i for 1967 through You have determined: 1987 i=1967 X i = , 1987 i=1967 Y i = , 1987 i=1967 (X i X) 2 = , 1987 i=1967 (Y i Y ) 2 = , 1987 i=1967 (X i X)(Y i Y ) = (a) Determine the sample correlation coefficient. (b) Compute ˆβ 1. (c) Compute SSR. (d) What is the value of n? (e) Compute the F statistic. (f) What conclusion can you draw? 16. An urn contains 7 balls, θ of which are red. A sample of size 2 is drawn without replacement to test H 0 : θ 1 against H 1 : θ > 1. If the null hypothesis is rejected if one or more red balls are drawn, (a) find the probability of a type II error when θ = 2. (Ans: ) (b) What is the significance level of the test? (Ans: 2 7 ) (c) Suppose that two red balls are drawn. What is the p-value associated with this observation? (Ans: p-value=0) 17. The following numbers of positions have been held by a random sample of aerospace engineers during the 10 to 15 years since their graduation: Calculate (a) the sample mean. (Ans: 3.1) (b) the sample median. (Ans: 3) (c) the sample mode. (Ans: 2)

7 (d) the sample variance. (Ans: 2.2) 18. The probability that a randomly chosen male has a circulation problem is Males who have a circulation problem are twice as likely to be smokers as those who do not have a circulation problem. What is the conditional probability that a male has a circulation problem, given that he is a smoker? (Ans: 0.40) 19. A satellite range prediction error has the standard normal distribution with mean 0 NM and standard deviation 1 NM. Find the probability that the prediction error is greater than NM. (Ans: ) 20. The elongation of a steel bar under a particular tensile load may be assumed to be normally distributed with a mean µ = 0.06 inches and a standard deviation σ = inches. A sample of n = 100 bars are subjected to the test. Find the probability that the sample mean elongation falls between inches and inches. (Ans: ) 21. Consider the hypothesis-testing application: H 0 : Memory chips are satisfactory. Indicate for each outcome whether it is a correct decision, a type I error, or a type II error. (a) Reject satisfactory shipment. (b) Accept satisfactory shipment. (c) Reject unsatisfactory shipment. (d) Accept unsatisfactory shipment. 22. The following data (minutes) apply for a random sample of processing times for a chemical reaction: Can you conclude that the mean of all processing times exceeds 4 minutes? use α = 0.05 (Ans: Yes) 23. A bridge player has been dealt 9 spades and 4 hearts. What is the probability that his partner has the remaining 4 spades? (Ans: ) 24. Before leaving the office each workday, a certain executive calls, or is called by, his family. The times of the executive s calls are approximately normally distributed with mean 4:30 p.m. and standard deviation 20 minutes. The times

8 of his family s calls are independently and approximately normally distributed with mean 4:00 p.m. and standard deviation 15 minutes. What is the probability that, on a given day, the executive will be the first to call? (Ans: ) 25. What is the probability that a 3-card hand drawn at random and without replacement from an ordinary deck consists entirely of face cards (i.e. King, Queen, or Jack)? (Ans: 33/3315 = 0.01) 26. A bag contains two coins, coin A for which the probability of heads is equal to 1 and coin B for which the probability of heads is equal to 3/5. A coin is selected at random and two tosses of this coin result in two heads. (a) What is the conditional probability that coin A was selected? ) (b) What is the conditional probability that coin B was selected? ) (Ans: (Ans: (c) What is the probability that a third toss of the selected coin will produce a head? (Ans: ) 27. A production process yields 90% satisfactory items. The quality event pertaining to successive items are independent. Find the probability of finding (a) all satisfactory when the number of selected items is 3. (Ans: 0.729) (b) none satisfactory when the number of selected items is 5. (Ans: ) (c) at least one satisfactory when the number of selected items is 3. (Ans: ) (d) at least 6 satisfactory when the number of selected items is ) (Ans: 28. An oil wildcatter has assigned a 0.30 probability for striking gas (G) under a particular leasehold. He has ordered a seismic survey that has a 90% positive reliability (given gas it confirms gas [C] with probability 0.90), but only 70% negative reliability (given no gas it denies gas [D] with probability 0.70). (a) Draw a tree diagram showing all possible outcomes. (b) What is the probability of gas, given that gas was confirmed? i.e. P (G C) (c) What is the probability of no gas, given gas was denied? 29. The following probability distribution applies to the high temperature X to be experienced in a city when the morning reading is 24 C:

9 x f(x) 30 C C C C C C.05 Calculate the expected value, variance, and standard deviation for the day s high temperature as expressed in (a) Celsius degrees. (Ans: 32.56; ; 1.177) (b) Fahrenheit degrees. Use your results from part (a) and the transformation F = C. (Ans: 90.61; 4.49; 2.12) 30. For students in a particular class, their scores on the midterm and final were collected. A simple linear regression model was fit to the data using the final exam score as the response variable. The Minitab output is given below. Compute the missing items (a), (b), (c), (d), (e), (f), (g), (h), (i), (j). Regression Analysis: Final versus Midterm The regression equation is Final = (g) Midterm Predictor Coef SE Coef T P Constant (d) Midterm S = R-Sq = (e)% R-Sq(adj) = 61.4% Analysis of Variance Source DF SS MS F P Regression (c) Residual Error (a) (b) Total 9 (f) 31. Data was collected from 15 piolot batches tested at a chemical plant. The variables measured were: Final Product Yield (Yield), Settling Time (Set Time),

10 Solidy Catalyst (Solid) and Liquid Catalyst (Liquid). Use the given Minitab output to answer the questions. Regression Analysis: Yield versus Set Time, Solid, Liquid The regression equation is Yield = Set Time Solid Liquid Predictor Coef SE Coef T P Constant Set Time Solid Liquid S = R-Sq = 90.3% R-Sq(adj) = 87.6% Analysis of Variance Source DF SS MS F P Regression Residual Error Total (a) Identify the coefficient estimates and their standard errors. (b) Calculate 95% confidence intervals for the regression coefficients. (c) Explain why the degrees of freedom for Regression is 3. (d) What is the null hypothesis being tested by the F statistic? (e) Interpret R 2. (f) Test the hypothesis that the coefficient for Set Time is equal to zero.

11 32. The following ANOVA table and sample means were obtained by an industrial engineer studying queue discouragement. The percentage of arriving customers who left before receiving service was determined for five sample time periods each under three different queue disciplines. One-Way Analysis of Variance Source DF SS MS F Treatment Error Total You are also given: X 1 = 25%, X 2 = 35%, X 3 = 39%, X 1 = 33%. (a) Are there differences among the queue disciplines? Why or why not. (b) Find a 90% confidence interval for µ 1 µ 3, the difference of means of the first and third queue disciplines. Can you conclude that they are different?