Sample_ Test1to3.doc

Sample Test 361B

Prof. Pandya

Practice Test #1 1. A professor of statistics refutes the claim that the average student spends 3 hours studying for the midterm exam. Which hypothesis is used to test the claim? a.

H0 :   3

vs. H 1 :   3

b. H 0 :   3 vs. H 1 :   3 c.

H0 :   3

vs. H 1 :   3

d. H 0 :   3 vs. H 1 :   3 ANSWER: b 2.

In hypothesis testing, whatever we are investigating or researching is specified as: a. the null hypothesis b. the alternative hypothesis c. either the null or alternative d. the p-value ANSWER: b

3.

A spouse stated that the average amount of money spent on Christmas gifts for immediate family members is above $1200. The correct set of hypotheses is: a. H 0 :   200 vs. H 1 :   1200 b. H 0 :   1200 vs. H 1 :   1200 c. H 0 :   1200 vs. H 1 :   1200 d. H 0 :   1200 vs. H 1 :   1200 ANSWER: c

4. Researchers determined that 60 Kleenex tissues is the average number of tissues used during a cold. Suppose a random sample of 100 Kleenex users yielded a mean number of 54 tissues used during a cold. Give the null and alternative hypotheses to determine if the number of tissues used during a cold is less than 60. a. H o :   60 and H1 :   60 b. H o :   60 and H1 :   60 c. H o : X  60 and H1 : X  60 d. H o : X  54 and H1 : X �54 ANSWER: b 5..

Formulate the null and alternative hypothesis for each of the following statements: a. The average American drinks 2.5 cups of coffee per day b. A researcher at the University of Michigan is looking for evidence to conclude that the average SAT score for entering freshmen is well over 1650

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c. The manager of the University of Iowa bookstore claims that the average student spends less than $400 per semester at the university's bookstore ANSWER: a. H 0 :   2.5 , H 1 :   2.5 b. H 0 :   1650 , H 1 :   1650 c. H 0 :   400 , H 1 :   400 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 6. Narrative: Production Filling Operation A production filling operation has a historical standard deviation of 6 ounces. When in perfect adjustment, the mean filling weight for the production process is 50 ounces. A quality control inspector periodically selects at random 36 containers and uses the sample mean filling weight to see if the process is in perfect adjustment. {Production Filling Operation Narrative} State the null and alternative hypotheses. ANSWER:

H 0 :   50 vs. H1 :  �50

{Production Filling Operation Narrative} Using a standardized test statistic, test the hypothesis at the 5% level of significance if the sample mean filling weight is 48.6 ounces. ANSWER: Test statistic: z = -1.40 Rejection region: |z| > z .025  1.96 Conclusion: Don’t reject H 0 . We can infer that the process is in perfect adjustment. {Production Filling Operation Narrative} Develop a 95% confidence interval and use it to test the hypothesis. ANSWER: LCL = 46.64 and UCL = 50.56. Since the hypothesized value 50 falls in the 95% confidence interval, we fail to reject H 0 at   0.05 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 7. For a given level of significance, if the sample size increases, the probability of a Type II error will: a. remain the same b. increase c. decrease d. be equal to 1.0 regardless of  ANSWER: c

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8. If the probability of committing a Type I error for a given test is to be decreased, then for a fixed sample size n a. the probability of committing a Type II error will also decrease b. the probability of committing a Type II error will increase c. the power of the test will increase d. a one-tailed test must be utilized ANSWER: b 9. A random sample of 10 observations was drawn from a normally distributed population. These are: 6, 4, 4, 7, 5, 5, 4, 5,  6, and 4. Test to determine if we can infer at = 0.05 that the population mean is less than 6. ANSWER: H 0 :   6 , H1 :   6 Rejection region: t  t0.05,9  1.833 Test statistic: t = -3.0 Conclusion: Reject H 0 . Yes, .we can infer at  = 0.05 that the population mean is less than 6. 10. During the past few weeks Laila stopped in Burger King fast food restaurant 5 times, and each time ordered large size French fries. Having nothing better to do, she counted how many French fries she received. The results follow: 73, 75, 83, 68, and 78. Assume that the number of French fries served at Burger King are normally distributed. Can we infer at the 10% significance level that the average number of large size orders of French fries served at Burger King is over 70? ANSWER: H 0 :   70 , H 1 :   70 Rejection region: t > t0.10,4 = 1.533 Test statistics: t = 2.158 Conclusion: Reject H 0 . Yes, we can infer at the 10% significance level that the average number of large size orders of French fries served at Burger King is over 70 11. Workers in a large plant are expected to complete a particular task in 60 seconds or less. The production manager believes that the average worker is satisfying that expectation. To examine the issue she watches eight workers perform the task and measures their times. The times, which are assumed to be normally distributed, are 58, 53, 63, 62, 57, 55, 53, and 55. Do these data provide sufficient evidence at the 5% significance level to support the production manager’s belief? ANSWER: H 0 :   60 , H 1 :   60 Rejection region: t < - t0.05,7 = -1.895 3

Test statistics: t = -2.223 Conclusion: Reject H 0 . Yes, these data provide sufficient evidence at the 5% level of significance to support the production manager’s belief 12. The owner of a service station wants to determine if owners of new cars (two years old or less) change their cars’ oil more frequently than owners of older cars (more than two years old). From his records he takes a random sample of ten new cars and ten older cars and determines the number of times the oil was changed in the last 12 months. The data follow. Do these data allow the service station owner to infer at the 10% significance level that new car owners change their cars’ oil more frequently than older car owners? Is this an example of Independent Sampling? Explain your answer. Frequency of Oil Changes in Past 12 Months New Car Owners Old Cars Owners 6 4 3 2 3 1 3 2 4 3 3 2 6 2 5 3 5 2 4 1

ANSWER:

H 0 : 1   2  0 , H 1 :  1   2  0

Rejection region: t > t0.10,18  1.33 Test statistic: t = 2.914 Conclusion: Reject the null hypothesis. Yes, new car owners change their cars’ oil more frequently than older car owners. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

13. Narrative: Auto Tires Wear

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To compare the wearing of two types of automobile tires, 1 and 2, an experimenter chose to “pair” the measurements, comparing the wear for the two types of tires on each of 7 automobiles, as shown below. Why is this an example of PAIRED SAMPLING? Give an explanation. Automobile Tire 1 Tire 2

1 8 12

2 15 18

3 7 8

4 9 9

5 10 12

6 13 11

7 11 10

{Auto Tires Wear Narrative} Determine whether these data are sufficient to infer at the 10% significance level that the two population means differ. ANSWER:

H 0 :  D  0 , H1 :  D  0

Rejection region: |t| > t0.05,6  1.943 Test Statistics: t = -1.225 Conclusion: Don’t reject the null hypothesis. No, these data are not sufficient to infer at the 10% significance level that the two population means differ. 14. Narrative: Promotional Campaigns The general manager of a chain of fast food chicken restaurants wants to determine how effective their promotional campaigns are. In these campaigns “20% off” coupons are widely distributed. These coupons are only valid for one week. To examine their effectiveness, the executive records the daily gross sales (in $1,000s) in one restaurant during the campaign and during the week after the campaign ends. The data is shown below. Why is this an example of PAIRED SAMPLING? Give an explanation. Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday

Sales During Campaign 18.1 10.0 9.1 8.4 10.8 13.1 20.8

Sales After Campaign 16.6 8.8 8.6 8.3 10.1 12.3 18.9

{Promotional Campaigns Narrative} Can they infer at the 5% significance level that sales increase during the campaign? ANSWER:

H 0 :  D  0 , H1 :  D  0

Rejection region: t > 1.943 5

Test statistic: t = 4.111 Conclusion: Reject the null hypothesis. Yes, they infer at the 5% significance level that sales increase during the campaign +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 15. Narrative: GMAT Scores A recent college graduate is in the process of deciding which one of three graduate schools he should apply to. He decides to judge the quality of the schools on the basis of the Graduate Management Admission Test (GMAT) scores of those who are accepted into the school. A random sample of six students in each school produced the following GMAT scores. Assume that the data are normally distributed. School 1 650 620 630 580 710 690

GMAT Scores School 2 105 550 700 630 600 650

School 3 590 510 520 500 490 530

{GMAT Scores Narrative} Set up the ANOVA Table. Use   0.10 to determine the critical value. 1. Is this a Single Factor ANOVA test? 2. How many levels are there to the FACTOR? 3. Is the FACTOR a nominal variable? 4. What is the response variable? 5. Is this an interval or ordinal variable? 6. How many TREATMENTS are there? 7. What is the test Statistics? 8. How many degrees of freedom does it have? 9. What are the degrees of freedom?

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10. How do you determine the degrees of freedom? 11. What does SST represent? 12. What does SSE represent? 13. How is the test statistics computed? 14. How do you determine the F critical value? 15. How do you determine the P-value?

ANSWER: Source of Variation Treatments Error Total

SS 47,511.11 41,400.00 88,911.11

df 2 15 17

MS 23,755.56 2,760.00

F 8.607

P-value 0.0032

F critical 2.695

{GMAT Scores Narrative} Can he infer at the 10% significance level that the GMAT scores differ among the three schools? ANSWER:

H 0 : 1   2   3 vs. H 1 : At least two means differ

Conclusion: Reject the null hypothesis. Yes, the GMAT scores differ in at least two of the three schools +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 16. Narrative: TV Viewing Habits A statistician employed by a television rating service wanted to determine if there were differences in television viewing habits among three different cities in California. She took a random sample of five adults in each of the cities and asked each to report the number of hours spent watching television in the previous week. The results are shown below.

Hours Spent Watching Television San Diego 25 31 18 23 27

Los Angeles 28 33 35 29 36

San Francisco 23 18 721 17 15

1. Is this a Single Factor ANOVA test? 2. How many levels are there to the FACTOR? 3. Is the FACTOR a nominal variable? 4. What is the response variable? 5. Is this an interval or ordinal variable? 6. How many TREATMENTS are there? 7. What is the test Statistics? 8. How many degrees of freedom does it have? 9. What are the degrees of freedom? 10. How do you determine the degrees of freedom? 11. What does SST represent? 12. What does SSE represent? 13. How is the test statistics computed? 14. How do you determine the F critical value? 15. How do you determine the P-value?

{TV Viewing Habits Narrative} Set up the ANOVA Table. Use   0.05 to determine the critical value. ANSWER: Source of Variation Treatments Error Total

SS 450.533 184.400 634.933

df 2 12 14

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MS 225.267 15.367

F 14.659

P-value 0.0006

F critical 3.885

{TV Viewing Habits Narrative}Can she infer at the 5% significance level that differences in hours of television watching exist among the three cities? ANSWER:

H 0 : 1   2   3 vs. H 1 : At least two means differ

Conclusion: Reject the null hypothesis. Yes, differences in mean hours of television watching exist in at least two of the three cities. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 17. Narrative: Automobile Repair Cost Automobile insurance appraisers examine cars that have been involved in accidental collisions and estimate the cost of repairs. An insurance executive claims that there are significant differences in the estimates from different appraisers. To support his claim he takes a random sample of six cars that have recently been damaged in accidents. Three appraisers then estimate the repair costs of all six cars. The data are shown below. Car 1 2 3 4 5 6

Appraiser 1 650 930 440 750 1190 1560

Estimated Repair Cost Appraiser 2 Appraiser 3 600 750 910 1010 450 500 710 810 1050 1250 1270 1450

{Automobile Repair Cost Narrative} Set up the ANOVA Table. Use determine the critical values. 1. Is this a Single Factor ANOVA test? 2. Is this an example of Blocking? 3. Which factor is of primary interest? 4. Why would you want to Block? 5. What is/are nuisance factors that might affect the measured result? 6. What are the homogeneous blocks? 7. How many levels are there to the FACTOR? 8. Is the FACTOR a nominal variable? 9. What is the response variable?

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

= 0.05 to

10. Is this an interval or ordinal variable? 11. How many TREATMENTS are there? 12. What is the test Statistics? 13. How many degrees of freedom does it have? 14. What are the degrees of freedom? 15. How do you determine the degrees of freedom? 16. What does SST represent? 17. What does SSB represent? 18. What does SSE represent? 19. How is the test statistics computed? 20. How do you determine the F critical value? 21. How do you determine the P-value? 22. Are there two test statistics, and explain their significance? 23. Are there two critical F-values? 24. Was the correct design chosen?

ANSWER: Source of Variation Treatments Blocks Error Total

SS 52,877.78 1,844,311.11 35,455.56 1,932,644.44

df 2 5 10 17

MS 26,438.889 368,862.222 3545.556

F 7.457 104.035

P-value 0.01042 0.00003

{Automobile Repair Cost Narrative} Can we infer at the 5% significance level that the executive’s claim is true? ANSWER:

H 0 : 1   2   3 vs. H 1 : At least two means differ

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F critical 4.103 3.326

Conclusion: Reject the null hypothesis. Yes, the insurance executive’s claim is true +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

18.

A randomized block design experiment produced the following data. Block 1 2 3 4 5

1 25 19 15 23 30

Treatment 2 27 18 20 27 31

3 25 17 16 20 28

a. Set up the ANOVA Table. Use  = 0.05 to determine the critical values. b. Test to determine whether the treatment means differ. (Use  = 0.05.) c. Test to determine whether the block means differ. (Use  = 0.05.) 1. Is this a Single Factor ANOVA test? 2. Is this an example of Blocking? 3. Which factor is of primary interest? 4. Why would you want to Block? 5. What is/are nuisance factors that might affect the measured result? 6. What are the homogeneous blocks? 7. How many levels are there to the FACTOR? 8. Is the FACTOR a nominal variable? 9. What is the response variable? 10. Is this an interval or ordinal variable? 11. How many TREATMENTS are there?

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12. What is the test Statistics? 13. How many degrees of freedom does it have? 14. What are the degrees of freedom? 15. How do you determine the degrees of freedom? 16. What does SST represent? 17. What does SSB represent? 18. What does SSE represent? 19. How is the test statistics computed? 20. How do you determine the F critical value? 21. How do you determine the P-value? 22. Are there two test statistics, and explain their significance? 23. Are there two critical F-values? 24. Is Independent Sampling design to be recommended?

ANSWER: a. Source of Variation Treatments Blocks Error Total

SS 29.733 336.933 18.267 384.933

df 2 4 8 14

MS 14.867 84.233 2.283

H 0 : 1   2   3

F 6.511 36.891

P-value 0.02097 0.00335

F critical 4.459 3.838

vs. H 1 : At least two means differ Conclusion: Reject the null hypothesis. Yes, at least two treatment means differ. H 0 : 1   2   3   4   5 vs. H 1 : At least two means differ c. Conclusion: Reject the null hypothesis. Yes, at least two block means differ. Sample Test 361B Prof. Pandya b.

Practice Test #2

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1. Bob Sandwich Shop would like to estimate its profit for the upcoming month. In order to carry out this, it must forecast future sandwich sales. Below are the number of Sandwiches sold each day of the five-day work week for each of the past four weeks.

Day Monday Tuesday Wednesday Thursday Friday

W 1 140 156 172 144 112

E 2 162 135 158 174 146

E 3 172 163 141 162 138

K 4 159 132 135 151 142

a. Graph the Time Series b. Determine whether the Time Series exhibits any trend. c. Using a five-day moving average, forecast the number of sandwiches that will be sold daily during the upcoming week. d. What is forecast of the number of sandwiches sold daily during the upcoming week if the shop uses exponential smoothing value of 0.1? e. Repeat the calculations with smoothing value of 0.3 f. Using the MSE performance measure for this data, which of the two smoothing values appear to give a reliable forecast so that the shop can plan to hire more workers? 2. Quarterly revenue sales ($1000s) of Pal Hotel chain over the past five years have been as follows: Quarter 1 2 3 4

Y 1 5108 6121 6394 4593

E 2 4871 5907 6485 4798

A 3 5248 6214 6582 4893

R 4 5365 6323 6647 5021

5 5423 6408 6586 5134

On the basis of these data, use the multiplicative forecasting model based on classical decomposition to forecast quarterly revenues for year 6.

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3. The quarterly sales (in millions of dollars) of a department store chain were recorded for the years 2001 – 2004 as listed below: Quarter 1 2 3 4

2001 21 36 28 44

2002 25 23 39 36

Year 2004 34 29 32 48

2003 30 41 47 55

a) Explain in your words the Seasonal Index (SI), and what purpose does it serve?

b) Then compute the adjusted SI (seasonal index or factor - assuming that the multiplicative model is suitable) using the Centered Moving Average method in Excel. Generate the trend line (use Regression) using the De-seasonalized data values. HINT: Follow the Excel spreadsheet Gasoline_Data per our discussion, and this spreadsheet is available on the Blackboard.

c) In the table below fill in the values for the SI, and explain what does SI mean about the sales in each of the seasons? Year 2001 2002 2003 2004

Quarter 1

Quarter 2

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Quarter 3

Quarter 4

HINT: Initially you compute the Period Seasonal Error Factor (PSEF), then you compute the Unadjusted Error Factor (USF), and finally you compute the Adjusted Error Factor (ASF) – These ASF values are the Seasonal Index.

d) Using Seasonal Index and the linear trend, forecast the values for periods 17, 18, 19, and 20 for the year 2005 for possible sales. Describe the seasonal fluctuations in the sales. Season

Period

estimate

SI

Forecast

4. Leather Problem Leather Co. manufactures belts and shoes. A belt requires 2 square yards of leather and 1 hour of skilled labor. A pair of shoes requires 3 sq. yd. of leather and 2 hours of skilled labor. As many as 25 sq. yd. of leather and 15 skilled labor can be purchased as a price of $5/sq. yd. of leather and $10/hour of skilled labor. A belt sells for $23, and a pair of shoes sells for $40. Leather Co. wants to maximize profits (revenue – costs). You are to formulate a Linear Programming model that can be used to maximize Leather CO’s profit. Solution: Let x be number of belts produced Let y be number of pairs of shoes produced Cost/belt = 2(5) + 1(10) = $20 Cost/pair of shoes = 3(5) + 2(10) = $30 Leather Co objective function: (23-20)x + (40- 35)y = 3x + 5y

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Objective Function to maximize is: 3x + 5y Constraint 1: 2x + 3y <= 25 Constraint 2:

x + 2y <= 15

Signed Constraints: x >=0 ,

y>=0

1) What is the optimal value of the objective function? 2) How many belts are produced? 3) How many shoes are produced? 4) Identify the binding and non-binding constraints and explain, and explain how is the optimal solution affected by the constraints?

5) Is there a shadow price in the analysis? Explain the role of the shadow price.

6) If there is a Reduced Cost on either belt or shoes, what action would you take, and explain why?

7) If you were to sell the belt at $23.00 – determine the optimal solution and check if there is a reduced cost, and shadow price.

8) Explain your optimal solution, and reduced cost. 9) Are the constraints binding, and if so explain? 5. Bakery: Mary Custard’s is a Pie shop that specializes in custard and fruit pies. It makes delicious pies and sells them at a reasonable price so that it can sell all the pies it makes in a day. Every dozen custard pies nets Mary Custard’s $15.00 and requires 12 pounds of flour, 50 eggs, 5 pounds of sugar and no fruit mixture. Every dozen fruit pies nets $25.00 profit and uses 10 pounds of flour, 40 eggs, 10 pounds of sugar, and 15 pounds of fruit mixture.

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On a given day, the bakers at Mary Custard’s found that they had 150 pounds of flour, 500 eggs, 90 pounds of sugar, and 120 pounds of fruit mixture with which to make pies. a) Formulate and solve a linear program that will give the optimal production schedule for the pies for the day.

b) If Mary Custard’s could double its profit on custard pies, should more custard be produced? Explain.

c) If Mary Custard’s raised the price (and hence the profit) on all pies by $0.25 ($3.00 per dozen), would the optimal production schedule for the day change? Would the profit change?

d) Suppose Mary Custard’s found that 10% of its fruit mixture had been stored in containers that were not air-tight. For quality and health reasons, it decided that it would be unwise to use any of this portion of the fruit mixture. How would this affect the optimal production schedule? Explain.

e) Mary Custard’s currently pays $2.50 for a five-pound bag of sugar from its bakery vendor. (The $0.50 per pound price of sugar is included in the unit profit given earlier.) Its vendor has already made its deliveries for the day. If Mary Custard’s wishes to purchase additional sugar, it must buy it from Donatelli’s Market that sells sugar in one-pound boxes for $2.25 a box. Should Mary Custard’s purchase any boxes of sugar from Donatelli’s Market? Explain

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6. Cox cable is about to expand its cable T offerings in Poway by adding MTV and other stations. The activities as listed below in the table must be completed before the service expansion is completed. Activity

Description

Immediate

Duration (weeks)

Predecessor A B C D E F

Choose station Get town council approval Order converters Install new dish Install converters Change billing system

A B B C, D B

2 4 3 2 10 4

a.

Draw a PERT/CPM network for this project

b.

Prepare a chart showing the earliest/latest start & finish times (ES/EF, LS/LF) and the slack for each activity)

c.

What are the expected completion times of the project?

d.

What are the critical paths for the project?

e.

How long can activity A be delayed without delaying the minimum completion time of the project?

f.

How long can activity B be delayed by without delaying the minimum completion time of the project?

g.

If activity D delayed three days, how long will the landscaping project be delayed by?

Sample Test 361B

Prof. Pandya

Practice Test #3

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Questions are from Lawrence & Pasternack textbook Practice problems are found at the back of the chapters: Chapter 5: 23, 26, 27 Chapter 8: 10, 13, 19, 22 Chapter 9: 12, 16, 17, 19 Chapter 6: 3, 4, 11

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