Math 533 ( applied managerial statistics ) final exam answers by Elsievogel

MATH 533 ( Applied Managerial Statistics ) Final Exam Answers

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MATH 533 ( Applied Managerial Statistics ) Final Exam Answers MATH 533 Final Exam Set 1 1.

(TCO D) PuttingPeople2Work has a growing business placing out-of-work MBAs. They claim they can place a client in a job in their field in less than 36 weeks. You are given the following data from a sample. Sample size: 100 Population standard deviation: 5 Sample mean: 34.2 Formulate a hypothesis test to evaluate the claim. (Points : 10) Ho: µ = 36; Ha: µ ≠ 36 Ho: µ ≥ 36; Ha: µ < 36 Ho: µ ≤ 34.2; Ha: µ > 34.2 Ho: µ > 36; Ha: µ ≤ 36 Ans. b. H0 must always have equal sign, < 36 weeks 2. (TCO B) The Republican party is interested in studying the number of republicans that might vote in a particular congressional district. Assume that the number of voters is binomially distributed by party affiliation (either republican or not republican). If 10 people show up at the polls, determine the following: Binomial distribution 10 n 0.5 p cumulative probability 0 0.00098 0.00098 1 0.00977 0.01074 2 0.04395 0.05469 3 0.11719 0.17188 4 0.20508 0.37695 5 0.24609 0.62305 6 0.20508 0.82813 7 0.11719 0.94531 8 0.04395 0.98926 9 0.00977 0.99902 10 0.00098 1.00000 What is the probability that no more than four will be republicans? (Points : 10) 38% 12% 21% X

P(X)

62% Ans. a look at x=4, cumulative probability 3. (TCO A) Company ABC had sales per month as listed below. Using the Minitab output given, determine: (A) Range (5 points); (B) Median (5 points); and (C) The range of the data that would contain 68% of the results. (5 points). Raw data: sales/month (Millions of $) 23 45 34 34 56 67 54 34 45 56 23 19 Descriptive Statistics: Sales Variable Total Count Mean StDev Variance Minimum Maximum Range Sales 12 40.83 15.39 236.88 19.00 67.00 48.00 Stem-and-Leaf Display: Sales Stem-and-leaf of Sales N = 12 Leaf Unit = 1.0 1 1 9 3 2 33 3 2 6 3 444 6 3 6 4 6 4 55 4 5 4 3 5 66 1 6 1 6 7 Reference: (TCO A) Company ABC had sales per month as listed below. Using the MegaStat output given, determine: (A) Range (5 points) (B) Median (5 points) (C) The range of the data that would contain 68% of the results. (5 points) Raw data: sales/month (Millions of $) 19 34

23 34 56 45 35 36 46 47 19 23 count 12 mean 34.75 sample variance 146.20 sample standard deviation 12.09 minimum 19 maximum 56 range 37 Stem and Leaf plot for # 1 stem unit = 10 leaf unit = 1 count 12.00000 mean 34.75000 sample variance 146.20455 sample standard deviation 12.09151 minimum 19.00000 maximum 56.00000 range 37.00000 1st quartile median 3rd quartile interquartile range mode

23.00000 34.50000 45.25000 22.25000 19.00000

4. (TCO C, D) Tesla Motors needs to buy axles for their new car. They are considering using Chris Cross Manufacturing as a vendor. Teslaâ&#x20AC;&#x2122;s requirement is that 95% of the axles are 100 cm Âą 2 cm. The following data is from a test run from Chris Cross Manufacturing. Should Tesla select them as a vendor? Explain your answer. Descriptive statistics

count mean sample variance sample standard deviation minimum maximum range population variance population standard deviation standard error of the mean

16 99.850 4.627 2.151 96.9 104 7.1 4.338 2.083 0.538

tolerance interval 95.45% lower 95.548 tolerance interval 95.45% upper 104.152 margin of error 4.302 1st quartile 98.850 median 99.200 3rd quartile 100.550 interquartile range 1.700 mode 103.000 (Points : 25) Reference: Chegg Tesla Motors needs to buy axles for their new car. They are considering using Chris Cross Manufacturing as a vendor. Tesla’s requirement is that 95% of the axles are 100 cm ± 5 cm. The following data is MegaStat output from a test run from Chris Cross Manufacturing. Descriptive statistics count: 16 mean: 99.938 sample variance: 2.313 sample standard deviation: 1.521 minimum: 97 maximum: 102.9 range: 5.9 population variance: 2.169 population standard deviation: 1.473 standard error of the mean: 0.380 tollerance interval 95.45% lower: 96.896 tolerance interval 95.45% upper: 102.979 half-width: 3.042 1st quartile: 98.900 median: 99.850 3rd quartile: 100.475 interquartile range: 1.575 mode: 98.900 Question: Should Tesla select them as a vendor? Explain your answer. Answers (1) · Given that, Tesla Motors needs to buy axles for their new car. They are considering using Chris Cross Manufacturing as a vendor. Tesla’s requirement is that 95% of the axles are 100 cm ± 5 cm. The following data is MegaStat output from a test run from Chris Cross Manufacturing: Descriptive statistics count: 16 mean: 99.938 sample variance: 2.313 sample standard deviation: 1.521 minimum: 97

maximum: 102.9 range: 5.9 population variance: 2.169 population standard deviation: 1.473 standard error of the mean: 0.380 tollerance interval 95.45% lower: 96.896 tolerance interval 95.45% upper: 102.979 half-width: 3.042 1st quartile: 98.900 median: 99.850 3rd quartile: 100.475 interquartile range: 1.575 mode: 98.900 Now, we have to construct 95% confidence interval for the data from the Chris Cross Manufacturing 5.

(TCO D) A PC manufacturer claims that no more than 2% of their machines are defective. In a random sample of 100 machines, it is found that 4.5% are defective. The manufacturer claims this is a fluke of the sample. At a .02 level of significance, test the manufacturer’s claim, and explain your answer. Test and CI for One Proportion Test of p = 0.02 vs p > 0.02 Sample X N Sample p 98% Lower Bound Z-Value P-Value 1 4 100 0.040000 0.000000 1.43 0.077 Reference: Set up the hypotheses: H0: p <= 0.02 Ha: p > 0.02 This is a one tailed test, since we will only reject for high proportions. Since we are using a 0.02 level of significance (it’s just chance that the hypotheses happen to have the same value as this), we’ll reject the null hypothesis if our P Value is less than 0.02. The computed P value from Megastat was 0.0371. This is higher than the significance level. Therefore, we do not reject H0:. We can say that the proportion is still less than or equal to 2%, and this was a fluke. Final Page 2 1. (TCO B) The following table gives the number of a. Total South probability 1773/2459 b. Total Midwest Divide: visits to recreational facilities by kind and people =+ — divide = approx Midwest local geographical region. 2459 West these: 0.721 = 298 park = (Points : 30) = 29 Ans. 1368 East South Midwest West Totals + 405 Local = 55 328 29 52 464 1773 Park

National 233 514 204 Park State Park100 526 65 Totals 388 1368 298

251 1202 102 793 405 2459

(A) Referring to the above table, if a visitor is chosen at random, what is the probability that he or she is either from the South or from the West? (15 points) (B) Referring to the above table, given that the visitor is from the Midwest, what is the probability that he or she visited a local park? (15 points) 2.

(TCO B, F) The length of time Americans exercise each week is normally distributed with a mean of 15.8 minutes and a standard deviation of 2.2 minutes X P(X≤x) P(X≥x) Mean Std dev 11 .0146 .9854 15.8 2.2 15 .3581 .6419 15.8 2.2 21 .9910 .0090 15.8 2.2 24 .9999 .0001 15.8 2.2 p(lower) p(upper) (A) Analyze the output above to determine what percentage of Americans will exercise between 11 and 21 minutes per week. (15 points) (B) What percentage of Americans will exercise less than 15 minutes? If 1000 Americans were evaluated, how many would you expect to have exercised less than 15 minutes? (15 points) (Points : 30) MATH 533 Final Exam Set 2

(TCO A) Seventeen salespeople reported the following number of sales calls completed last month. 93 82 81 82 97 102 107 119 88 91 83 93 73 100 102 Compute the mean, median, mode, and standard deviation, Q1, Q3, Min, and Max for the above sample data on number of sales calls per month. b. In the context of this situation, interpret the Median, Q1, and Q3. (Points : 33) 72 86

(TCO B) Cedar Home Furnishings has collected data on their customers in terms of whether they reside in an urban location or a suburban location, as well as rating the customers as either “good,” “borderline,” or “poor.” The data is below. Urban Suburban Total Good 60 168 228 Borderline 36 72 108 Poor 24 40 64 Total 120 280 400 If you choose a customer at random, then find the probability that the customer a. is considered “borderline.”

(TCO B) Historically, 70% of your customers at Rodale Emporium pay for their purchases using credit cards. In a sample of 20 customers, find the probability that a. exactly 14 customers will pay for their purchases using credit cards.

(TCO C) An operations analyst from an airline company has been asked to develop a fairly accurate estimate of the mean refueling and baggage handling time at a foreign airport. A random sample of 36 refueling and baggage handling times yields the following results. Sample Size = 36 Sample Mean = 24.2 minutes Sample Standard Deviation = 4.2 minutes a. Compute the 90% confidence interval for the population mean refueling and baggage time.

(TCO C) The manufacturer of a certain brand of toothpaste claims that a high percentage of dentists recommend the use of their toothpaste. A random sample of 400 dentists results in 310 recommending their toothpaste. a. Compute the 99% confidence interval for the population proportion of dentists who recommend the use of this toothpaste.

(TCO D) A Ford Motor Company quality improvement team believes that its recently implemented defect reduction program has reduced the proportion of paint defects. Prior to the implementation of the program, the proportion of paint defects was .03 and had been stationary for the past 6 months. Ford selects a random sample of 2,000 cars built after the implementation of the defect reduction program. There were 45 cars with paint defects in that sample. Does the sample data provide evidence to conclude that the proportion of paint defects is now less than .03 (with a = .01)? Use the hypothesis testing procedure outlined below. a. Formulate the null and alternative hypotheses.

(TCO D) A new car dealer calculates that the dealership must average more than 4.5% profit on sales of new cars. A random sample of 81 cars gives the following result. Sample Size = 81 Sample Mean = 4.97% Sample Standard Deviation = 1.8% Does the sample data provide evidence to conclude that the dealership averages more than 4.5% profit on sales of new cars (using a = .10)? Use the hypothesis testing procedure outlined below. a. Formulate the null and alternative hypotheses.

(TCO E) Bill McFarland is a real estate broker who specializes in selling farmland in a large western state. Because Bill advises many of his clients about pricing their land, he is interested in developing a pricing formula of some type. He feels he could increase his business significantly if he could accurately determine the value of a farmerâ&#x20AC;&#x2122;s land. A geologist tells Bill that the soil and rock characteristics in most of the area that Bill sells do not vary much. Thus the price of land should depend greatly on acreage. Bill selects a sample of 30 plots recently sold. The data is found below (in Minitab), where X=Acreage and Y=Price ($1,000s). PRICE ACREAGE PREDICT 60 20.0 50 130 40.5 250 25 10.2 300 100.0 85 30.0

182 115 24 60 92 77 122 41 200 42 60 20 145 61 235 250 278 118 46 69 220 235 50 25 290

56.5 41.0 10.0 18.5 30.0 25.6 42.0 14.0 70.0 13.0 21.6 6.5 45.0 19.2 80.0 90.0 95.0 41.0 14.0 22.0 81.5 78.0 16.0 10.0 100.0

Correlations: PRICE, ACREAGE Pearson correlation of PRICE and ACREAGE = 0.997 P-Value = 0.000 Regression Analysis: PRICE versus ACREAGE The regression equation is PRICE = 2.26 + 2.89 ACREAGE Predictor Coef SE Coef T P Constant 2.257 2.231 1.01 0.320 ACREAGE 2.89202 0.04353 66.44 0.000 S = 7.21461 R-Sq = 99.4% R-Sq(adj) = 99.3% Analysis of Variance Source DF SS MS F P Regression 1 229757 229757 4414.11 0.000 Residual Error 28 1457 52 Total 29 231215 Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI 1 146.86 1.37 (144.05, 149.66) (131.82, 161.90) 2 725.26 9.18 (706.46, 744.06) (701.35, 749.17)XX XX denotes a point that is an extreme outlier in the predictors.

Values of Predictors for New Observations New Obs ACREAGE 1 50 2 250 Analyze the above output to determine the regression equation.

(TCO E) An insurance firm wishes to study the relationship between driving experience (X1, in years), number of driving violations in the past three years (X2), and current monthly auto insurance premium (Y). A sample of 12 insured drivers is selected at random. The data is given below (in MINITAB): Y X1 X2 Predict X1 Predict X2 74 5 2 8 1 38 14 0 50 6 1 63 10 3 97 4 6 55 8 2 57 11 3 43 16 1 99 3 5 46 9 1 35 19 0 60 13 3 Regression Analysis: Y versus X1, X2

The regression equation is Y = 55.1 â&#x20AC;&#x201C; 1.37 X1 + 8.05 X2 Predictor Coef SE Coef T P Constant 55.138 7.309 7.54 0.000 X1 -1.3736 0.4885 -2.81 0.020 X2 8.053 1.307 6.16 0.000 S = 6.07296 R-Sq = 93.1% R-Sq(adj) = 91.6% Analysis of Variance Source DF SS MS F P Regression 2 4490.3 2245.2 60.88 0.000 Residual Error 9 331.9 36.9 Total 11 4822.3 Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI 1 52.20 2.91 (45.62, 58.79) (36.97, 67.44) Values of Predictors for New Observations New Obs X1 X2 1 8.00 1.00 Correlations: Y, X1, X2 Y

X1 -0.800 0.002 X2 0.933 -0.660 0.000 0.020 Cell Contents: Pearson correlation P-Value a. Analyze the above output to determine the multiple regression equation.

MATH 533 Final Exam Set 3 MATH 533 Final Exam Set 4