Chapter 10
1. The
following MINITAB output is from a data set relating x = average hourly wage
and y = quit rate for a sample of
industries
Predictor Coef Stdev t-ratio P
Constant 4.8615 0.5201 9.35 0.000
quit rate -0.34655 0.05866
-5.91 0.000
s = 0.4862 R-sq = 72.9% R-sq(adj) = 70.8%
Analysis of Variance
SOURCE DF SS MS F P
Regression 1 8.2507 8.2507 34.90 0.000
Error 13 3.0733 0.2364
Total 14 11.32540
A. What is the equation of the least squares regression line? (Use words to
identify your variables in the
equation instead of just letters.)
B. Give point estimates of the slope and y-intercept.
C. What is the magnitude of the typical deviation from the estimated regression
line?
D. Give the point estimate .
E. What is the sample correlation between x and y?
F. Determine and interpret the value of r2.
G. Test the following hypotheses:
2.
The article “Effects of Enhances UV-B Radiation on Peas and
Soybeans” (Environ. And Exper. Botany (1984): 131-143) included
the accompanying data on pea plants, with y = sunburn index and x = distance
(cm) from an ultraviolet light source.
|
Sunburn index (y) |
18 |
21 |
25 |
26 |
30 |
32 |
36 |
40 |
40 |
50 |
51 |
54 |
61 |
62 |
63 |
|
Distance from light source(x) |
4.0 |
3.7 |
3.0 |
2.9 |
2.6 |
2.5 |
2.2 |
2.0 |
2.1 |
1.5 |
1.5 |
1.5 |
1.3 |
1.2 |
1.1 |
A. Estimate the mean change in the sunburn index associated with an increase of
1 centimeter in distance.
B. Find the estimated mean sunburn index on peas if the distance from the light
source is 3.5 cm.
C. Calculate an estimate of .
D. Compute the standard deviation of the statistic b, the estimated slope of
the regression line.
E. Compute a 95% confidence interval for the slope of the population regression
line.
F. Compute and interpret the value of r2.
3. Physical characteristics of
sharks are of interest to surfers and scuba divers, as well as marine
researchers. The length in feet of a
shark (x) and its corresponding jaw width (y) for 44 sharks appeared in the
following magazines: Skin Diver and Scuba News. Here is the Minitab output of this data.
The regression equation is
jaw width = 0.69 + 0.963 length (ft)
Predictor Coef StDev T P
Constant 0.688 1.299 0.53 0.599
length ( 0.96345 0.08228 11.71 0.000
S = 1.376 R-Sq = 76.6% R-Sq(adj) = 76.0%
Analysis of Variance
Source DF SS MS F P
Regression 1 259.53 259.53 137.12 0.000
Residual Error 42 79.49 1.89
Total 43 339.02
A. Give point estimates of the slope and intercept of the population regression line.
B. Calculate an estimate of the mean jaw width for a shark which is 13.75 feet long.
C. Estimate with a 95% confidence interval.
D. Calculate an estimate .
E. Determine the proportion of the observed variation in jaw width that can be attributed to the
simple linear regression model.
4. No tortilla chip lover likes soggy chips. A study looking at the relationship between the amount of frying time in seconds and the amount of moisture content (%) was done. The data are displayed in the table below:
|
|
5 |
10 |
15 |
20 |
25 |
30 |
45 |
60 |
|
|
16.3 |
9.7 |
8.1 |
4.2 |
3.4 |
2.9 |
1.9 |
1.3 |
A. Construct a scatter plot for this data including the least squares
regression.
B. First estimate what a least squares regression line might be, then use your
calculator to find it and compare your results.
C. Is the simple linear regression model useful for predicting the % of
moisture using the amount of frying time?
Explain your answer. If your answer is no, explain how to adjust the
data to find an appropriate model.
D. Use the appropriate model (simple linear regression or the other one) to
predict the percent of moisture remaining in tortilla chips if the frying time
is 35 minutes.
5. Let x maximum outdoor temperature and y = hours of chiller operation per day for a 3-ton residential gas air-conditioning units. The data are as follows.
|
|
72 |
78 |
80 |
86 |
88 |
92 |
|
|
4.8 |
7.2 |
9.5 |
14.5 |
15.7 |
17.9 |
A. Construct a scatter plot for this data
including your estimation of the least squares regression line.
B. What is the equation of the estimated regression line?
C. What is the predicted time required for an maximum outdoor temperature of 90
degrees?
D. What percentage of observed
variation hours can be explained by the simple linear regression
model?
E. What is the magnitude of a typical
deviation from the estimated regression line?
F. Estimate with a 95% confidence interval.
6. Is cardiovascular fitness (as measured by time to exhaustion running on a treadmill) related to an athlete’s performance in a 20-km ski race?
Given the following information
regarding the data calculate as indicated:
A. What is the equation of the value of the correlation coefficient, the
coefficient of determination, and the estimated slope of the regression line?
B. Interpret the slope.
C. Find the estimated equation of the regression line.
D. Find the standard deviation of the
residuals and the standard deviation of the slope.
E. Test the following hypotheses:
7. The articles “Effect of Temperature on the pH of Skim Milk” (Journal of Dairy Research (1988): 277-280) reported on a study involving x, the temperature in degrees Celsius under specified experimental conditions and y, the pH of the milk. A Minitab printout of a regression analysis that was done on this data is shown on the next page. Answer the following questions using this printout.
Regression Analysis
The regression equation is
pH of Milk = 6.84 - 0.00731 temp
Predictor Coef StDev T P
Constant 6.84335 0.01974 346.66 0.000
temp -0.0073061 0.0004158 -17.57 0.000
S = 0.03559
Analysis of Variance
Source DF SS MS F P
Regression 1 0.39104 0.39104 308.68 0.000
Residual Error 14 0.01774 0.00127
Total 15 0.40878
Unusual Observations
Obs temp pH of Mi Fit StDev Fit Residual St Resid
16 78.0 6.34000 6.27347 0.01728 0.06653 2.14R
R denotes an observation with a large standardized residual
Predicted Values
obs Fit StDev Fit 95.0% CI 95.0% PI
24 6.668 0.01173 ( 6.64284, 6.69315) ( 6.58762, 6.74838)
45 6.51457 0.00896 ( 6.49534, 6.53380) ( 6.43585, 6.59329)
67 6.35384 0.01357 ( 6.32474, 6.38293) ( 6.27214, 6.43553)
a.) The 95% confidence
interval for the true average milk pH when the milk temperature is 45 degrees
is:
b) The 95% prediction interval for a
single pH observation to be made when the milk temperature is 45 degrees is:
c) Calculate a 99% confidence interval
for the true average milk pH when the milk temperature is 24 degrees. Compare this to the 95% confidence interval.
d.) What is the standard deviation of
the regression model?
e.) What is the standard deviation of
the slope?
f.) Calculate the coefficient of
determination and the correlation coefficient.