1. A sample of 15 computers reveals the following data in years of service (X1, YEARS), whether the computer is a MAC or not (X2, 1=Mac computer, 0=not a Mac computer), and the total number of breakdowns (Y, BREAKDOWNS). The results are found below.
|
YEARS
|
MAC
|
BREAKDOWNS
|
|
1
|
1
|
0
|
|
1
|
0
|
1
|
|
2
|
1
|
0
|
|
2
|
0
|
2
|
|
2
|
1
|
1
|
|
2
|
0
|
3
|
|
3
|
1
|
1
|
|
3
|
0
|
4
|
|
4
|
1
|
2
|
|
4
|
0
|
5
|
|
4
|
0
|
6
|
|
5
|
1
|
3
|
|
5
|
0
|
7
|
|
6
|
0
|
8
|
Correlations: YEARS, MAC, BREAKDOWNS
YEARS MAC
MAC -0.168
0.549
BREAKDOWNS 0.810 -0.664
0.000 0.007
Cell Contents: Pearson correlation
P-Value
Regression Analysis: BREAKDOWNS versus YEARS, MAC
The regression equation is
BREAKDOWNS = 0.462 + 1.19 YEARS - 2.68 MAC.
Predictor Coef SE Coef T P
Constant 0.4625 0.4456 1.04 0.320
YEARS 1.1946 0.1169 10.22 0.000
MAC -2.6805 0.3470 -7.72 0.000
S = 0.649016 R-Sq = 94.2% R-Sq(adj) = 93.3%
Analysis of Variance
Source DF SS MS F P
Regression 2 82.679 41.339 98.14 0.000
Residual Error 12 5.055 0.421
Total 14 87.733
Predicted Values for New Observations
New Obs Fit SE Fit 95% CI 95% PI
1 3.755 0.367 (2.956, 4.554) (2.131, 5.379)
Values of Predictors for New Observations
New Obs YEARS MAC
1 5.00 1.00
a. Analyze the above output to determine the multiple regression equation.
b. Find and interpret the multiple index of determination (R-Sq).
c. Perform the multiple regression t-tests on?ˆ1,?ˆ2 (use two tailed test with (a = .10). Interpret your results.
d. Predict the total number of breakdowns for a single computer that is a 5-year-old MAC. Use both a point estimate and the appropriate interval estimate.
2. A researcher is studying the relationship between earnings for major electronic firms and research and development expenditures in the previous year. A random sample of 15 major electronic firms is selected and gives the following data on R&D (X, research, and development expenditures in the previous year in millions of $) and EARNINGS (Y, earnings in millions of $).
|
R&D
|
EARNINGS
|
PREDICT
|
|
15
|
221
|
9
|
|
8.5
|
83
|
25
|
|
12
|
147
|
|
|
6.5
|
69
|
|
|
4.5
|
41
|
|
|
2
|
26
|
|
|
0.5
|
35
|
|
|
1.5
|
40
|
|
|
14
|
125
|
|
|
9
|
97
|
|
|
7.5
|
53
|
|
|
0.5
|
12
|
|
|
2.5
|
34
|
|
|
3
|
48
|
|
|
6
|
64
|
|
Correlations: R&D, EARNINGS
Pearson correlation of R&D and EARNINGS = 0.919
P-Value = 0.000
Regression Analysis: EARNINGS versus R&D
The regression equation is
EARNINGS = 6.84 + 10.7 R&D
Predictor Coef SE Coef T P
Constant 6.843 9.781 0.70 0.497
R&D 10.671 1.266 8.43 0.000
S = 22.5948 R-Sq = 84.5% R-Sq(adj) = 83.3%
Analysis of Variance
Source DF SS MS F P
Regression 1 36253 36253 71.01 0.000
Residual Error 13 6637 511
Total 14 42890
Predicted Values for New Observations
New Obs Fit SE Fit 95% CI 95% PI
1 102.88 6.83 ( 88.13, 117.63) ( 51.88, 153.87)
2 273.61 24.51 (220.66, 326.56) (201.59, 345.62)XX
XX denotes a point that is an extreme outlier in the predictors.
Values of Predictors for New Observations
New Obs R&D
1 9.0
2 25.0
a. Analyze the above output to determine the regression equation.
b. Find and interpret?ˆ1in the context of this problem.
c. Find and interpret the coefficient of determination (r-squared).
d. Find and interpret coefficient of correlation.
e. Does the data provide significant evidence (a = .05) that money spent on research and development can be used to predict earnings in the following year? Test the utility of this model using a two-tailed test. Find the observed p-value and interpret.
f. Find the 95% confidence interval for mean earnings for all firms that had research and development expenditures was $9 million in the previous year. Interpret this interval.
g. Find the 95% prediction interval for the earnings for one firm that had research and development expenditures of $9 million. Interpret this interval.
h. What can we say about the earnings when research and development expenditures were $25 million?
3. Consider the following age data, which is the result of selecting a random sample of 25 Boeing 747 airplanes that are owned by United Airlines. The age of each airplane is given in years.
17 5 5 12 8 5 8 16 14
12 5 5 18 7 5 4 12 22
13 5 2 24 7 7 1
a. Compute the mean, median, mode, and standard deviation, Q1, Q3, Min, and Max for the above sample data on age of Boeing 747 airplanes owned by United Airlines.
b. In the context of this situation, interpret the Median, Q1, and Q3.
4. East Side Bank has three loan officers, Jim, Sally, and Dave, who process all loan applications. Each application results in an approval, rejection, or delay. The results of the last 100 loan applications are shown below.
| |
Jim
|
Sally
|
Dave
|
Total
|
|
Approved
|
20
|
16
|
18
|
54
|
|
Rejected
|
1
|
7
|
14
|
33
|
|
Delayed
|
5
|
2
|
6
|
13
|
|
Total
|
37
|
25
|
38
|
100
|
If you choose a loan application at random, then find the probability that the loan application
a. was processed by Jim.
b. was approved and processed by Sally.
c. was rejected, given that the application was processed by Dave.
5. Records of a health insurance company show that 40% of policyholders under age 30 submitted a claim during the past year. A random sample of 75 policyholders under age 30 is selected. Assuming the records are correct, then find the probability that
a. exactly 30 submitted a claim during the past year.
b. more than 32 submitted a claim during the past year.
c. at most 29 submitted a claim during the past year.
6. The end-of-month cash flow for a company can be modeled by a normal distribution with a mean of $10,000 and a standard deviation of $7,000. Note that a positive cash flow figure means that there is a surplus of cash, but a negative cash flow means that we have a cash shortage.
a. Find the probability that the end-of-month cash flow will be between $15,000 and $25,000.
b. Find the probability that the end-of-month cash flow will be at least $5,000.
c. The treasurer is concerned about the possibility of a cash shortage and wishes to develop strategies to manage cash flow. The treasurer wants to establish a reserve fund that can be used in case of negative cash flow. Suppose that the treasurer wants to be able to have a fund large enough to cover all but 1% of the months. How much money should be in the reserve fund?
7. A survey is taken of U.S. companies that do business with firms in India. One of the questions in the survey was: "Approximately how many years has your company been trading with firms in India?" A random sample of 44 responses to this question yields the following results.
Sample Size = 44
Sample Mean = 10.46 years
Sample Standard Deviation = 7.7 years
a. Compute the 95% confidence interval for the population mean number of years of trading with firms in India.
b. Interpret this interval.
c. How many firms should be sampled if we wish to generate a 95% confidence interval for the mean number of years of trading with firms in India that is accurate to within 1 year?
8. You are in charge of selling advertising for radio station WQAA. The fee you can set for airtime is directly related to the share of the listening market your station reaches. From time to time you conduct surveys to determine WQAA's share of the market. This month, when you contacted 200 randomly selected residents, 12 respondents said they listen to WQAA.
a. Compute the 99% confidence interval for the percent of the market that are listeners of WQAA.
b. Interpret this confidence interval.
c. How many individuals should be sampled in order to be 99% confident of being within 1% of the actual population percent of the market that are WQAA listeners?
9. The Cottrell Soap Company has produced a new dish washing liquid that it believes is superior to competitive products on the market. The board of directors has indicated that the product will be marketed if more than 25% of the population of households prefer the product over its competitors. The company's market research department has distributed the new product to 400 randomly selected households. The results of this sampling are that 120 of the 400 households prefer the new dish washing liquid. Does the sample data provide evidence to conclude that the percentage of households in the population that prefer the new dishwashing liquid is more than 25% (witha = .05)? Use the hypothesis testing procedure outlined below.
a. Formulate the null and alternative hypotheses.
b. State the level of significance.
c. Find the critical value (or values), and clearly show the rejection and nonrejection regions.
d. Compute the test statistic.
e. Decide whether you can reject Ho and accept Ha or not.
f. Explain and interpret your conclusion in part e. What does this mean?
g. Determine the observed p-value for the hypothesis test and interpret this value. What does this mean?
h. Does this sample data provide evidence (witha = 0.05) that the percentage of households in the population that prefer the new dishwashing liquid is more than 25%?
10. At a supermarket, the average number of register mistakes per day per clerk was 18. The owner of the supermarket purchased new cash registers in an effort to decrease the number of errors. After extensive training on the new registers, the manager took a random sample of 100 clerks on randomly selected days using the new registers and found the following results.
Sample Size = 100 clerks
Sample Mean = 17.25 mistakes per day per clerk
Sample Standard Deviation = 4.35 mistakes per day per clerk
Does the sample data provide evidence to conclude that the population mean number of register mistakes per day per clerk was less than 18 (usinga = .01)? Use the hypothesis testing procedure outlined below.
a. Formulate the null and alternative hypotheses.
b. State the level of significance.
c. Find the critical value (or values), and clearly show the rejection and nonrejection regions.
d. Compute the test statistic.
e. Decide whether you can reject Ho and accept Ha or not.
f. Explain and interpret your conclusion in part e. What does this mean?
g. Determine the observed p-value for the hypothesis test and interpret this value. What does this mean?
h. Does this sample data provide evidence (witha = .01) that the population mean number of register mistakes per day per clerk was less than 18?