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ConfidenceIntervals
1. A sample of 35 textbooks was taken from the TCU bookstore. The average price of the books was $52.50 with a standard deviation (S) of $5.10. Forty textbooks were sampled from the SMU bookstore. The average price of those 40 books was $65.00 with a variance (S2) of $18.06. Calculate a 90% confidence interval for the difference in the average prices for the two stores. Explain your findings.
2. Two car models are tested for a difference in average gas mileage. Ten cars of Model A averaged 32.1 miles per gallon with a standard deviation of 5.2 mpg. Twelve cars of Model B averaged 35.5 mpg with a standard deviation of 4.4 mpg. Calculate a 95% con- fidence interval for the difference between the two cars’ mileage. There is no evidence to suggest the variances of the populations are equal. Explain your findings.
3. Bendix Fertilizer Company has two plants—one in LA and the other in Atlanta. Recent customer complaints suggest that the Atlanta shipments are underweight as compared to the LA shipments. Ten (10) boxes from LA average 96.3 pounds with a standard deviation (S) of 12.5 pounds and fifteen (15) boxes are selected form Atlanta with an aver- age weight of 101.1 pounds with a standard deviation (S) of 10.3 pounds. Does a 99% confidence interval support those complaining? Assume equal population variances. Explain your findings.
4. The monthly starting salaries in thousands of dollars of 12 maintenance personnel at TCU are compared to those from SMU using the data in the table shown below. Develop and interpret a 95% confidence interval for the difference in mean starting salaries at TCU and SMU.
TCU
SMU
15.6 13.7
16.8 13.6
18.5 15.2
16.5 11.2
15.5 11.6
14.8 15.2
18.8 12.5
19.5 13.5
17.5 13.9
16.5 18.2
14.5 14.5
18.7 11.2
Hypothesis Testing
5. Each patient in Farmer’s Hospital is asked to evaluate the service at the time of dis- charge. Recently there have been complaints that the resident physicians and nurses on the surgical wing responded too slowly to calls from the senior citizens. Other patients seem to be receiving faster service. Mr. Bob Anderson, Hospital Administrator, asked for a study by the quality assurance team. After studying the problem, the QA team, headed by Sharon Schwartz, decided to collect the following sample information and to test the claim at the alpha 0.01 level of significance. The question to be answered was: Are the other patients receiving faster service than the senior citizens? As a side calculation, Mrs. Schwartz decided to determine the p-value even though she knew she would have to explain its meaning to Mr. Anderson.
Conduct the study by choosing the appropriate test statistic and explain your results. The data sets are as follows:
Patient Type
Sample Mean
Sample Standard Deviation
Sample Size
Senior Citizen
5.50 minutes
0.40 minutes
50
Other
5.30 minutes
0.30 minutes
100
6. Owens Lawn Care, Inc. manufactures and assembles lawnmowers, which are shipped to dealers throughout the United States and Canada. Two different procedures have been proposed for mounting the engine on the frame of the lawnmower. The question is: Is there a difference in the mean time to mount the engines on the frames of the lawnmowers? The first procedure was developed by Welles (#1) and the second was developed by Atkins (#2). To evaluate the two methods, it was decided to conduct a time
and motion study. A sample of five employees was timed using procedure #1 and six different employees were timed using procedure #2. The results in minutes are shown below. Is there a difference in the mean mounting times? There is reason to believe the population variances are equal. Use the 0.10 level of significance.
Procedure #1 in Minutes
Procedure #2 in Minutes
2
3
4
7
9
5
3
8
2
4
Average = 4.0 minutes
3 Average= 5.0 minutes
Problems: Large-Size Samples—Confidence Intervals
1. Testing the Mean: American Hallmark Insurance sells policies to residents throughout the DFW Metroplex area. Claims are certain to be filed. The company wants to estimate the difference in the mean claim costs (in dollars) between people living in urban areas and those residing in the suburbs. Of the 180 urban policies selected as a sample, the mean insurance claim of $2,025 was reported with s = $918. The sample of 200 suburban policies revealed a mean insurance claim of $1,802 and s = $512. What does a 95 percent confidence interval tell the company about the claims filed by these two groups? Interpret your results.
2. Testing the Mean: Two production designs are used to manufacture a certain product. The mean time required to produce the product using design A was 3.51 days with s = 0.79 days. Design B required a mean time of 3.32 days with s = 0.73 days. Equal size samples of 150 were used for both designs. What would a 99 percent confidence interval reveal about which design should be used? Interpret your results.
3. Testing the Mean: An accountant for a large corporation in the Midwest must decide whether to select MCI or Sprint to handle the firm’s long-distance telephone service. Data collected for calls using both services are reported here. See Data Table 9.1.
Data Table 9.1 MCI Compared to Sprint.
MCI
Sprint
Number of calls
145
102
Mean cost
$4.07
$3.89
Standard deviation
$0.97
$0.85
What does a 95 percent confidence interval reveal about the difference in the population means? Interpret the results.
Large-Size Sample—Hypothesis Testing
4. Testing the Mean: Samples of size 50 and 60 reveal means of 512 and 587 and standard deviations of 125 and 145, respectively. At the 2 percent alpha level, test the hypothesis that m1 =m2or m1 -m2 =0.
Small-Size Sample—Equal Variances (Confidence Intervals)
5. Testing the Mean: Seventeen cans of Energy Aid report a mean fill level of 17.2 ounces with a standard deviation of 3.2 ounces, and 13 cans of Charged produce a mean fill level of 18.1 ounces and s = 2.7 ounces. Assuming equal variances and normal distributions in population weights, what conclusion can you draw regarding the difference in mean weights based on a 98 percent confidence interval? Interpret your results.
6. Testing the Mean: Grant requests must be submitted to either the National Science Foundation (NSF) or Health and Human Services (HHS), but the TCU Foundation for Health Studies prefers to have an answer as soon as possible. A study of both organizations has yielded the following results. Fourteen grant requests to NSF took an average of 45.7 weeks with a standard deviation of 12.6 weeks. Twelve grant requests to HHS took an average of 32.9 weeks with a standard deviation of 16.8 weeks. If the NSF takes more than five weeks longer than HHS, the TCU foundation will submit the grant requests to HHS. At the 90% confidence level, test the difference in the process time for the two organizations. Assume the population variances are equal. Should the foundation bother submitting to HHS?
Small-Size Sample—Equal Variances (Hypothesis Testing)
7. Testing the Mean: Using the information in problem #6 mentioned above, test the hypothesis that the two foundations process the grants with the same efficiency and speed. Use the alpha level of 0.10. Assume equal population variances. How does the confidence interval approach used in #6 differ from this approach or does it?
8. Testing the Mean: Twenty-six mutual funds, each with $5,000 invested in them, are selected for comparison. Of the 26 funds, 12 are income oriented and yield a mean return of $1,098.60 with a standard deviation of $43.20. The remaining 14 funds are growth oriented and yield a mean return of $987.60 with a standard deviation of $53.40.
a. Calculate and interpret the 80 percent confidence interval for the difference between the population mean returns. There is no reason to believe that the population variances are equal.
b. Sample Size for the Mean: What sample size is necessary to be 95 percent certain that the error does not exceed $10.00?
Small-Size Sample—Unequal Variances (Hypothesis Testing)
9. Testing the Mean: At the 1 percent alpha level, test for the equality of means if samples of 10 and 8 yield means of 36 and 49 and standard deviations of 12 and 18, respectively. Assume unequal population variances.
Matched Pairs—Confidence Interval
10. Testing the Mean: Rankin Associates will accept bids from two construction companies on a remodeling job at the home office. The decision about which offers to accept depends in part on the mean completion times of similar jobs by each company. Data are collected and paired from several previous remodeling jobs done by each company. Based on a 99 percent level of confidence, which company would receive the contract from Rankin Associates? See Data Table 9.2.
CHAPTER 9: HOMEWORK PROBLEMS-HYPOTHESIS TESTING AND CONFIDENCE INTERVALS FOR TWO POPULATIONS 323 Data Table 9.2 Contract Selection Process for Ranking Associates.
Pair Number
Company 1
Company 2
1
10.0
9.2
2
12.2
10.0
3
15.3
9.2
4
9.6
10.5
5
8.6
9.5
6
9.4
8.4
7
12.5
7.2
8
7.3
8.4
9
9.4
10.5
10
8.7
6.2
11
9.1
8.1
11. Costs of services at two local car dealerships tend to confuse many consumers. In an attempt to understand which might be the most costly, the statistics department at TCU conducts a study. The services are matched as closely as possible and the results are shown in Data Table 9.3.
Data Table 9.3 Service Department Cost Study.
Service
Dealer 1–$
Dealer 2–$
1
54
36
2
56
35
3
59
34
4
65
39
5
62
37
6
43
32
7
38
31
8
48
30
9
46
29
10
59
45
Calculate and interpret the 90% confidence interval for the difference between the population means. Assuming the quality of service is the same, which dealer should be used or does it make a difference?
Matched Pairs—Hypothesis Testing
12. Testing the Mean: Two major automobile manufacturers have produced compact cars with the same size engines. Is there a statistically-significant difference in the miles per gallon (MPG) of the two brands? A random sample of 8 cars is selected from each manufacturer. A specified driving distance is established and 8 drivers are selected. The number of drivers is not 16, but 8 who will drive car A and then drive car B. The results of the test are tabulated below in Data Table 9.4. Set a = 0.05.
Data Table 9.4 MPG Study for Two Manufacturers.
Driver
MPG MFG A
MPG MFG B
1
29
27
2
24
23
3
26
28
4
24
23
5
25
24
6
27
26
7
30
28
8
25
27
13. Testing the Mean: Snow White buys her seven dwarfs (bet you can’t name them) new shovels for Christmas. The tons dug with the old and new shovels are shown In Data Table 9.5. At alpha 0.10, did Snow White’s gift to her seven buddies change the output or did it remain the same?
Data Table 9.5 Output in Tons of Old Shovels and New Shovels.
Daily Output in Tons
Dwarf
Old Shovels
New Shovels
Doc
1.7
1.9
Happy
1.4
1.5
Grumpy
2.1
2.2
Bashful
1.9
2.0
Sleepy
2.2
2.2
Dopey
1.4
1.5
Sneezy
1.9
1.8
Proportions—Confidence Intervals and Hypothesis Testing
14. Confidence Interval: A study is conducted to determine if there is a difference in the use of credit cards between men and women when Christmas gifts are purchased. The test included a sample of 150 men with 27% using credit cards and a sample of 130 women with 35% using credit cards. At the confidence level of 99%, develop and interpret the confidence interval for the proportion of men and women who rely on credit cards.
15. Confidence Interval: Of the 40 Dallas accounting firms surveyed, six indicated they were advertising. A similar survey was conducted in Phoenix where 8 out of 50 firms reported they were advertising. Determine a 90% confidence interval (alpha 0.10) to estimate the difference between the proportions of the firms who are advertising in the two cities. Interpret your results.
16. Hypothesis Testing: Using the information given in problem #15, test the hypothesis that the proportions of those advertising in the two cities are the same. Use the alpha level of 0.10. Interpret your results. Explain the difference between the results determined with the confidence interval solution in #15 and this hypothesis testing approach.
1. Large Sample, Z-Solution: Texas Health Care operates 33 medical clinics in the Dallas- Fort Worth Metroplex area. A study was initiated to determine the difference in the mean time spent per visit for men and women patients. Previous studies indicated the standard deviation is 11 minutes for men and 16 minutes for women. A random sample of 100 males and 100 females was selected and the sample means were 34.5 minutes for the males and 42.4 minutes for the females. Determine a 95% confidence interval and interpret the result. If there is a difference, how much is the difference?
2. Small Sample, t-solution, Equal Variances: Baylor Hospital in Grapevine, TX is work- ing with a medication they believe will help patients recover from certain illnesses more quickly. They have noticed that there seems to be a difference in response time in patients 50 and under and those patients over 50. To test this theory, they want to know if there is a difference in time when the medication actually reaches the blood stream. A random sample of six (6) people, ages 50 or under, yields a sample mean of 13.6 minutes with a standard deviation of 3.1 minutes. A random sample of eight (8) people over 50 yields a sample mean of 11.2 minutes with a standard deviation of 5.0 minutes. Determine the confidence interval at an alpha level of 0.05. Can the claim that the older patients process the medication more rapidly than the younger patients be supported?
3. Small Sample, t-solution, Equal Variances: Apex Industries has two manufacturing plants which build sub-assembly parts for shipment to Lockheed-Martin. Plant A is reasonably well automated and Plant B still has a number of manual operations in place. In order to determine if the assembly times are different between the automated plant and the manual plant, a random sample of 15 parts for each plant is tracked. The results were that Plant A (automated) processed the parts in 56.7 hours with a standard deviation of 7.1 hours. Plant B (manual) processed the parts in 70.4 hours with a standard deviation of 8.3 hours. Determine and interpret a 95% confidence interval to see if the automated plant is more efficient than the manual plant.
4. Hypothesis Test, Large Sample, Z-Solution: Bates Manufacturing makes parts for equipment used in the oil and gas business. One part is a coupling, useful on natural gas drilling equipment. Two different machines are used in manufacturing the part. The standard deviation for Machine 1 has been established at 0.025 inches and the standard deviation for Machine 2 has been established at 0.034 inches. The question is does Machine 2 produce a part with higher average diameters than Machine 1? The alternate would be, “is the Machine 1 diameter less than the Machine 2 diameter?” A random sample of 100 parts from each machine is selected. The results yielded a mean of 0.501 inches for Machine 1 and a mean of 0.509 for Machine 2. The claim is that Machine 1 produces a part that is less in diameter than Machine 2. Set up the null and alternate hypothesis and test the parts diameter at an alpha level of 0.05.
5. Confidence Interval: Large Sample, Z-Solution. Use the information from the problem just above (Bates Manufacturing) and calculate a 95% Confidence Interval. Did you find the same answer? How much is the difference, if one exists?
6. P-Value: Using the data from the problem just above (Bates Manufacturing), determine the p-value and interpret the results? Did you get the same answer? Why or Why Not?
7. Hypothesis Testing: Small Sample, t-Solution, Equal Variances: To determine if one retirement plan is more popular than another, Wilson Investments decided to evaluate the plans based on average contribution. Wilson’s management wants to test the theory that the plan yielding the better result is the one into which more people will make larger contributions. A random sample of 15 people for Plan A and 15 people for Plan B is taken. The mean contribution of Plan A is $2,119.70 with a standard deviation of $709.70. The mean contribution of Plan B is $1,777.70 with a standard deviation of $593.90. Wilson’s management wants to know at the alpha 0.05 level if the means can be considered to be equal. Assume equal population variances. Set up the null and alternate hypothesis and interpret the results.
8. Hypothesis Testing: Small Sample, t-Solution, Equal Variances: Budget Car Rental wants to see if there is a difference in the mean mileage per gallon for their most popu- lar SUV. Budget believes that the mean mileage for driving on the highway will exceed that of driving in town. Budget randomly selects 15 SUVs to be driven on the highway and another 15 SUVs to be driven in the city. The vehicles are filled with exactly 14 gal- lons of gasoline. The test driver is asked to drive the vehicle until the car runs out of gas. The sample yielded the following results: Highway average miles per gallon was 19.6468 and City average miles per gallon was 16.1460. The variance for the Highway study was 18.4637 miles squared and for the City study was 29.6064 miles squared. At the alpha 0.05 level, do the Highway miles per gallon exceed the City miles per gallon?
9. Matched Pairs: Students at TCU are curious to see if the cost of books purchased through the bookstore are the same as the cost of those same books purchased online. Nineteen courses are selected and the prices of the textbooks are compared. The results of the study revealed that the mean cost of the textbooks through the bookstore was $139.40 and online was $126.70 and the standard deviation of the difference (Sd) is 30.4488. Using Hypothesis testing and the t-test, determine if there is a difference in the prices through the bookstore and online at the alpha level of 0.05.
10. Confidence Interval: Work the problem just above using a 95% confidence interval. Do the solutions yield the same result? Why or Why Not?
11. Hypothesis Testing: Large Sample, Z-Solution. A think-tank in Washington, DC wants to find out if there is a difference between the salaries of federal workers and those in the private sector. A random sample of 35 federal jobs and 32 private sector jobs is selected. The average salary for the federal jobs was $66,700 and for the private sector jobs was $60,400 with a standard deviation of $12,000 and $11,000, respectively. At the alpha level of 0.05, test to see if there is any difference.
12. P-value: Work the problem just above using the p-value approach (Federal and Private Sector Jobs).
