Keller MATH 533 Final Exam -
1. A random sample of 20 cars driving down I-294 is selected and their speed is monitored. The results are as follows (in mph).
68 65 50 79 77 60 55 61 78 75
75 67 72 58 70 62 67 72 70 74
a. Compute the mean, median, mode, and standard deviation, Q1, Q3, Min, and Max for the above sample data on speed per car.
b. In the context of this situation, interpret the Median, Q1, and Q3.
2. Consider the following data on newly hired employees in relation to which part of the country they were born and their highest degree attained.
|
|
HS
|
BS
|
MS
|
PHD
|
Total
|
|
East
|
3
|
5
|
2
|
1
|
11
|
|
Midwest
|
7
|
9
|
2
|
0
|
18
|
|
South
|
5
|
8
|
6
|
2
|
21
|
|
West
|
1
|
7
|
8
|
6
|
22
|
|
Total
|
16
|
29
|
18
|
9
|
72
|
If you choose one person at random, then find the probability that the person
a. has a PHD.
b. is from the East and has a BS as the highest degree attained.
c. has only a HS degree, given that person is from the West.
3. A source in the Internal Revenue Service has stated that historically 90% of federal tax returns filed are free of arithmetic errors. A random sample of 25 returns are selected and checked carefully for arithmetic errors. Assuming independence, find the probability that
a. all 25 returns are free of arithmetic errors.
b. at most 23 returns are free of arithmetic errors.
c. more than 17 are free of arithmetic errors.
4. CJ Computer Disks stocks and sells recordable CDs. The monthly demand for these CDs is closely approximated by a normal distribution with a mean of 20,000 disks and standard deviation of 4,000 disks. CJ receives shipments from the supplier once per month (at the beginning of each month).
a. Find the probability that the demand for recordable CDs exceeds 30,000 for a particular month.
b. Find the probability that the demand for recordable CDs is between 12,000 and 18,000.
c. How large an inventory must CJ have available at the beginning of the month so that the probability of running out of recordable CDs (a stock out) during the month is no more than .05?
5. The Ford Motor Company wishes to estimate the mean dollar amount of damage done to a Ford Explorer as a result of a 10 mph crash into the rear bumper of a parked car. The sample results are as follows.
Sample Size = 36
Sample Mean = $638
Sample Standard Deviation = $115
a. Construct a 95% confidence interval for the average dollar amount of damage.
b. Interpret this interval.
c. How large a sample size will need to be selected if we wish to have a 95% confidence interval for the average dollar amount of damage with a margin for error of $10?
6. A clock company is concerned about errors in assembly of their custom made clocks. A random sample of 120 clocks from today's production yields nine clocks with assembly errors.
a. Compute the 95% confidence interval for the percentage of clocks with assembly errors in today's production.
b. Interpret this confidence interval.
c. How many clocks should be selected in order to be 95% confident of being within 2% of the population percentage of clocks with assembly errors in today's production?
7. 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 (with a = 0.05) that the percentage of households in the population that prefer the new dishwashing liquid is more than 25%?
8. A manufacturer of athletic footwear claims that the mean life of his product will exceed 50 hours. A random sample of 36 shoes leads to the following results in terms of useful life.
Sample Size = 36 shoes
Sample Mean = 52.3 hours
Sample Standard Deviation = 9.6 hours
Does the sample data provide evidence to conclude that the manufacturer's claim is correct (using a = .10)? 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 non-rejection 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 the sample data provide evidence to conclude that the manufacturer's claim is correct (using a = .10)?
9. McCave Development Enterprises is considering whether to build a shopping mall in Statesville. The manager wants you to analyze the relationship between mall size and the rate of return on invested capital. You select a random sample of 16 cities similar to Statesville in demographic and economic characteristics and collect the following data on FOOTAGE (in 10,000 square feet) and RETURN (rate of return as a %).
|
RETURN
|
FOOTAGE
|
PREDICT
|
|
18.3
|
12.8
|
15.0
|
|
11.7
|
18.6
|
7.5
|
|
19.5
|
10.3
|
|
|
17.5
|
14.3
|
|
|
15.4
|
14.2
|
|
|
9.8
|
21.4
|
|
|
11.4
|
18.6
|
|
|
14.5
|
16.7
|
|
|
16.3
|
15.5
|
|
|
19.0
|
9.8
|
|
|
17.0
|
14.2
|
|
|
15.1
|
16.2
|
|
|
19.5
|
12.8
|
|
|
10.9
|
19.4
|
|
|
16.3
|
15.0
|
|
|
16.3
|
15.4
|
|
Regression Analysis: RETURN versus FOOTAGE
The regression equation is
RETURN = 30.0 - 0.943 FOOTAGE
Predictor Coef SE Coef T P
Constant 29.976 1.238 24.22 0.000
FOOTAGE -0.94257 0.07921 -11.90 0.000
S = 0.969721 R-Sq = 91.0% R-Sq(adj) = 90.4%
Analysis of Variance
Source DF SS MS F P
Regression 1 133.15 133.15 141.59 0.000
Residual Error 14 13.17 0.94
Total 15 146.31
Predicted Values for New Observations
New Obs Fit SE Fit 95% CI 95% PI
1 15.838 0.244 (15.315, 16.360) (13.693, 17.982)
2 22.907 0.666 (21.479, 24.334) (20.384, 25.429)X
X denotes a point that is an outlier in the predictors.
Values of Predictors for New Observations
New Obs FOOTAGE
1 15.0
2 7.5
Correlations: RETURN, FOOTAGE
Pearson correlation of RETURN and FOOTAGE = -0.954
P-Value = 0.000
a. Analyze the above output to determine the regression equation.
b. Find and interpret in 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 Footage can be used to predict Return? 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 the mean rate of return on capital investment for malls that have square footage of 150,000. Interpret this interval.
g. Find the 95% prediction interval for the rate of return on capital investment for a mall that has square footage of 150,000. Interpret this interval.
h. What can we say about the rate of return on capital investment for a mall that has square footage of 75,000?
10. At an auction, a national car rental agency sold 12 comparably equipped 3-year-old Chevrolet Corsicas. The data on mileage (X1), type of car (X2), and selling price (Y) are found below.
Y = PRICE ($)
X1= MILEAGE (miles)
X2= TYPE (dummy variable 0=sedan, 1=coupe)
The data is given below (in MINITAB).
|
PRICE
|
Mileage
|
Type
|
PredMile
|
PredType
|
|
7000
|
60000
|
0
|
54000
|
0
|
|
8500
|
52000
|
0
|
54000
|
1
|
|
7000
|
62000
|
0
|
|
|
|
8900
|
48000
|
0
|
|
|
|
7600
|
55000
|
0
|
|
|
|
7200
|
60000
|
1
|
|
|
|
8500
|
50000
|
1
|
|
|
|
7800
|
53000
|
1
|
|
|
|
7200
|
58000
|
1
|
|
|
|
9000
|
48000
|
1
|
|
|
|
7200
|
60000
|
1
|
|
|
|
7700
|
55000
|
0
|
|
|
Correlations: PRICE, Mileage, Type
PRICE Mileage
Mileage -0.970
0.000
Type 0.023 -0.053
0.942 0.871
Cell Contents: Pearson correlation
P-Value
Regression Analysis: PRICE versus Mileage, Type
The regression equation is
PRICE = 15841 - 0.146 Mileage - 39 Type.
Predictor Coef SE Coef T P
Constant 15840.9 671.4 23.59 0.000
Mileage -0.14562 0.01205 -12.09 0.000
Type -39.5 114.1 -0.35 0.737
S = 197.278 R-Sq = 94.2% R-Sq(adj) = 92.9%
Analysis of Variance
Source DF SS MS F P
Regression 2 5689732 2844866 73.10 0.000
Residual Error 9 350268 38919
Total 11 6040000
Predicted Values for New Observations
New Obs Fit SE Fit 95% CI 95% PI
1 7977.5 82.1 (7791.7, 8163.3) (7494.1, 8460.9)
2 7938.0 81.2 (7754.4, 8121.6) (7455.4, 8420.6)
Values of Predictors for New Observations
New Obs Mileage Type
1 54000 0.00
2 54000 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 t-tests on,(use two tailed test with (a= .05). Interpret your results.
d. Predict the price for an individual Chevy Corsica with mileage of 54,000 miles and with type being sedan. Use both a point estimate and the appropriate interval estimate.