Class Handout #6 Homework
Name __________________________
Exercise #1
Suppose we are interested in predicting a person's height from the
person's length of stride (distance between footprints). The following
data is recorded for a random sample of 5 people:
Length of Stride (inches) 14 13 21 25 17
Height (inches)
61 54 63 72 59
(a) Identify the dependent (response) variable and the independent
(explanatory) variable for a regression analysis.
The dependent (response) variable is Y = “height”, and the independent
(explanatory) variable is “stride length”.
(b) Does the data appear to be observational or experimental?
Since the ages look random, it appears that the data is observational.
(c) Use the formulas from Exercise #3 on Class Handout #6 to find the
equation of the least squares line.
^
y = 40.2 + 1.2x = 58.2
(d) Use the least squares line to predict the height of a person whose
length of stride is 15 inches.
^
y = 40.2 + 1.2(15) = 58.2
1
Class Handout #6 Homework
Name __________________________
Exercise #2
Suppose we are interested in predicting a male's right-hand grip
strength from age. The following data is recorded for a random
sample of males:
Age (years) 15 17 19 11 16 22 17 25 12 14 25 23
Grip Strength (lbs.) 50 54 66 46 58 54 64 80 46 70 76 80
(a) Identify the dependent (response) variable and the independent
(explanatory) variable for a regression analysis.
The dependent (response) variable is Y = “grip strength”, and the
independent (explanatory) variable is “age”.
(b) Does the data appear to be observational or experimental?
Since the ages look random, it appears that the data is observational.
(c) First, enter the data into an SPSS data file, with the name age for the
variable consisting of the ages, and with the name grip for the variable
consisting of the grip strengths. Save this SPSS data file with the name
grip_strength. Then, in the document titled Using SPSS Version 19.0, use
SPSS with the section titled Performing a simple linear regression with
bivariate data, with checks of linearity, homoscedasticity, and normality
assumptions to do each of the following:
Follow the instructions in the first five steps to graph the least squares line
on a scatter plot, and submit the SPSS output with this assignment; then state
why the linearity assumption appears to be satisfied.
Since the data points appear to be randomly distributed about the line, the
linearity assumption appears to be satisfied.
2
Class Handout #6 Homework
Name __________________________
Exercise #2(c) - continued
Follow the remaining instructions beginning with the 6th step to create
graphs for assessing whether or not the uniform variance (homoscedasticity)
assumption and the normality assumption appear to be satisfied, and to
generate the output for the linear regression; submit the SPSS output with
this assignment. Then, state why the uniform variance (homoscedasticity)
assumption appears to be satisfied, and state why the normality assumption
appears to be satisfied.
The variation in standardized residuals around the horizontal line looks
reasonable uniform.
The histogram of standardized residuals looks somewhat bell-shaped, and
the points on the normal probability plot seem to be randomly distributed
around the diagonal line.
(d) Based on the results in part (c), explain why we feel it is appropriate
to proceed with the regression analysis.
The results in part (c) suggest that the linearity assumption, the uniform
variance (homoscedasticity), and the normality assumption all appear to be
satisfied.
(e) Use the SPSS output to find each of the following:
n = 12
18
62
r = + 0.770
(11)(4.824)2 = 255.98
3
Class Handout #6 Homework
Name __________________________
Exercise #2 - continued
4
Class Handout #6 Homework
Name __________________________
Exercise #2 - continued
5
Class Handout #6 Homework
Name __________________________
Exercise #2 - continued
(f) Use the SPSS output to find the equation of the least squares line.
^
The least squares line can be written y = 26 + 2x .
(g) Write a one-sentence interpretation of the slope in the least squares
line.
Grip strength appears to increase on average by about 2 pounds with each
increase of one year in age.
(h) Find the coefficient of determination, and write a one-sentence
interpretation.
From the SPSS output, we find r2 = 0.593.
About 59.3% of the variation in grip strength is explained by age.
(i) Find the standard error of estimate.
From the SPSS output, we find s = 8.390.
4
Class Handout #6 Homework
Name __________________________
Exercise #2 - continued
(j) A 0.05 significance level is chosen for a hypothesis test to see if there
is any evidence that the linear relationship between age and grip strength is
significant, that is, that the slope in the regression is significantly different
from zero (0). Write the results of this hypothesis test two different ways:
Write the results of the t test about the slope in a format suitable for a
journal article to be submitted for publication.
Since t10 = 3.814 and t10; 0.025 = 2.228, we have sufficient evidence to reject
H0 at the 0.05 level. We conclude that the linear relationship between age
and grip strength is significant (p = 0.003 OR 0.001 < p < 0.01). The data
suggest that the linear relationship is positive.
Write the results of the f test in the ANOVA table in a format suitable
for a journal article to be submitted for publication.
Since f1, 10 = 14.545 and f1, 10; 0.05 = 4.96, we have sufficient evidence to reject
H0. We conclude that the linear relationship between age and grip strength
is significant (p = 0.003 OR 0.001 < p < 0.01). The data suggest that the
linear relationship is positive.
5
Class Handout #6 Homework
Name __________________________
Exercise #2 - continued
(k) Considering the results of the hypothesis test in part (j), decide
whether or not a 95% confidence interval for the slope in the regression
would be of interest. If yes, find and interpret the confidence interval; if not,
explain why.
Since rejecting H0 suggests that the hypothesized zero slope is not correct, a
95% confidence interval will provide us with some information about the
slope, which estimates the average change in grip strength with an increase
of one year in age.
2  (2.228)(8.390/255.98) , 2 + (2.228)(8.390/255.98)
We are 95% confident that the slope in the regression to predict grip strength
from age is between 0.832 and 3.168 lbs.
(l) A 0.05 significance level is chosen for a hypothesis test to see if there
is any evidence that the mean grip strength for 20 year old right-handed
males is different from 80 lbs. Write the results of this hypothesis test in a
format suitable for a journal article to be submitted for publication.
Since t10 = – 5.304 and t10; 0.025 = 2.228, we have sufficient evidence to reject
H0. We conclude that the mean grip strength for 20 year old right-handed
males is different from 80 lbs. (p < 0.001). The data suggest that this mean
is less than 80 lbs.
6
Class Handout #6 Homework
Name __________________________
Exercise #2 - continued
(m) Considering the results of the hypothesis test in part (k), decide
whether or not a 95% confidence interval for the mean grip strength for 20
year old right-handed males would be of interest. If yes, find and interpret
the confidence interval; if not, explain why.
Since rejecting H0 suggests that the hypothesized mean grip strength for 20
year old right-handed males is not correct, a 95% confidence interval will
provide us with some information about this mean.
____________________
66  (2.228)(8.3901/12 + (20 – 18)2/255.98) ,
____________________
66 + (2.228)(8.390/1/12 + (20 – 18)2/255.98)
We are 95% confident that the mean grip strength for 20 year old righthanded males is between 60.12 and 71.88 lbs.
(n) Find and interpret a 95% prediction interval for the grip strength of a
20 year old right-handed male.
_______________________
66  (2.228)(8.3901 + 1/12 + (20 – 18)2/255.98),
_______________________
66 + (2.228)(8.390/1 + 1/12 + (20 – 18)2/255.98)
We are 95% confident that the grip strength for a randomly selected 20-year
old right-handed male will be between 46.40 and 85.60 lbs.
OR
At least 95% of 20-year old right-handed males have a grip strength between
46.40 and 85.60 lbs.
(o) For what age group of right-handed males will the confidence interval
for mean grip strength and the prediction interval for particular grip strength
both have the smallest length?
18 year olds
7
Class Handout #6 Homework
Name __________________________
Exercise #3
Suppose we are interested in predicting a subject’s reaction time to a
particular stimulus from dosage of a certain drug. The following data
is recorded for a random sample of observations on the subject:
Drug Dosage (grams)
4
4
6
6
8
8
10 10
Reaction Time (seconds) 7.5 6.8 4.0 4.4 3.9 3.1 1.4 1.7
(a) Identify the dependent (response) variable and the independent
(explanatory) variable for a regression analysis.
The dependent (response) variable is Y = “reaction time”, and the
independent (explanatory) variable is “drug dosage”.
(b) Does the data appear to be observational or experimental?
Since the drug dosages do not look random, it appears that the data is
experimental.
(c) First, enter the data into an SPSS data file, with the name age for the
variable consisting of the ages, and with the name grip for the variable
consisting of the grip strengths. Save this SPSS data file with the name
reaction. Then, in the document titled Using SPSS Version 19.0, use SPSS
with the section titled Performing a simple linear regression with
bivariate data, with checks of linearity, homoscedasticity, and normality
assumptions to do each of the following:
Follow the instructions in the first five steps to graph the least squares line
on a scatter plot, and submit the SPSS output with this assignment; then state
why the linearity assumption appears to be satisfied.
Since the data points appear to be randomly distributed about the line, the
linearity assumption appears to be satisfied.
8
Class Handout #6 Homework
Name __________________________
Exercise #3(c) - continued
Follow the remaining instructions beginning with the 6th step to create
graphs for assessing whether or not the uniform variance (homoscedasticity)
assumption and the normality assumption appear to be satisfied, and to
generate the output for the linear regression; submit the SPSS output with
this assignment. Then, state why the uniform variance (homoscedasticity)
assumption appears to be satisfied, and state why the normality assumption
appears to be satisfied.
The variation in standardized residuals around the horizontal line looks
reasonable uniform.
The histogram of standardized residuals looks somewhat bell-shaped, and
the points on the normal probability plot seem to be randomly distributed
around the diagonal line.
(d) Based on the results in part (c), explain why we feel it is appropriate
to proceed with the regression analysis.
The results in part (c) suggest that the linearity assumption, the uniform
variance (homoscedasticity), and the normality assumption all appear to be
satisfied.
(e) Use the SPSS output to find each of the following:
n=8
7
4.1
r =  0.963
(7)(2.390)2 = 39.9847
9
Class Handout #6 Homework
Name __________________________
Exercise #3 - continued
10
Class Handout #6 Homework
Name __________________________
Exercise #3 - continued
11
Class Handout #6 Homework
Name __________________________
Exercise #3 - continued
(f) Use the SPSS output to find the equation of the least squares line.
^
The least squares line can be written y = 10.225  0.875x .
(g) Write a one-sentence interpretation of the slope in the least squares
line.
Reaction time appears to decrease on average by about 0.875 seconds with
each increase of one gram in drug dosage.
(h) Find the coefficient of determination, and write a one-sentence
interpretation.
From the SPSS output, we find r2 = 0.927.
About 92.7% of the variation in reaction time is explained by drug dosage.
(i) Find the standard error of estimate.
From the SPSS output, we find s = 0.6344.
10
Class Handout #6 Homework
Name __________________________
Exercise #3 - continued
(j) A 0.05 significance level is chosen for a hypothesis test to see if there
is any evidence that the linear relationship between drug dosage and reaction
time is significant, that is, that the slope in the regression is significantly
different from zero (0). Write the results of this hypothesis test two different
ways:
Write the results of the t test about the slope in a format suitable for a
journal article to be submitted for publication.
Since t6 = – 8.721 and t6; 0.025 = 2. 447, we have sufficient evidence to reject
H0 at the 0.05 level. We conclude that the slope in the linear regression to
predict reaction time from drug dosage is different from zero (p < 0.001).
The data suggest that the linear relationship is negative.
Write the results of the f test in the ANOVA table in a format suitable
for a journal article to be submitted for publication.
Since f1, 6 = 76.087 and f1, 6; 0.05 = 5.99, we have sufficient evidence to reject
H0 at the 0.05 level. We conclude that the linear regression to predict
reaction time from drug dosage is significant (p < 0.001). The data suggest
that the linear relationship is negative.
11
Class Handout #6 Homework
Name __________________________
Exercise #3 - continued
(k) Considering the results of the hypothesis test in part (j), decide
whether or not a 95% confidence interval for the slope in the regression
would be of interest. If yes, find and interpret the confidence interval; if not,
explain why.
Since rejecting H0 suggests that the hypothesized zero slope is not correct, a
95% confidence interval will provide us with some information about the
slope, which estimates the average change in reaction time with an increase
of one gram in drug dosage.
– 0.875  (2.447)(0.6344/39.98) , – 0.875 + (2.447)(0.6344/39.98)
We are 95% confident that the slope in the regression to predict reaction
time from drug dosage is between – 1.1205 and – 0.6295 seconds.
12
Class Handout #6 Homework
Name __________________________
Exercise #3 - continued
(l) Find and interpret each of these intervals: (i) a 95% confidence
interval for the mean reaction time with a drug dosage of 9 grams, and (ii) a
95% prediction interval for the reaction time with a particular drug dosage of
9 grams.
__________________
(i) 2.35  (2.447)(0.6344/1/8 + (9  7)2/39.9847) ,
__________________
2.35 + (2.447)(0.6344/1/8 + (9  7)2/39.9847)
We are 95% confident that the mean reaction time with a drug dosage of 9
grams is between 1.61 and 3.09 seconds.
_____________________
(ii) 2.35  (2.447)(0.6344/1 + 1/8 + (9  7)2/39.9847) ,
_____________________
2.35 + (2.447)(0.63441 + 1/8 + (9  7)2/39.9847)
We are 95% confident that a particular reaction time with a drug dosage of 9
grams will be between 0.63 and 4.07 seconds.
OR
At least 95% of the reaction times with a drug dosage of 9 grams are
between 0.63 and 4.07 seconds.
(m) For what drug dosage will the confidence interval for mean reaction
time and the prediction interval for particular reaction time both have the
smallest length?
7 grams
13
Class Handout #6 Homework
Name __________________________
Exercise #4
Suppose ????????OLDTEXT EX4.40 in OLD POWERPOINT
HW13????? we are interested in predicting a subject’s reaction time to
a particular stimulus from dosage of a certain drug. The following
data is recorded for a random sample of observations on the subject:
Drug Dosage (grams)
4
4
6
6
8
8
10 10
Reaction Time (seconds) 7.5 6.8 4.0 4.4 3.9 3.1 1.4 1.7
(a) Identify the dependent (response) variable and the independent
(explanatory) variable for a regression analysis.
The dependent (response) variable is Y = “reaction time”, and the
independent (explanatory) variable is “drug dosage”.
(b) Does the data appear to be observational or experimental?
Since the drug dosages do not look random, it appears that the data is
experimental.
(c) First, enter the data into an SPSS data file, with the name age for the
variable consisting of the ages, and with the name grip for the variable
consisting of the grip strengths. Save this SPSS data file with the name
reaction. Then, in the document titled Using SPSS Version 19.0, use SPSS
with the section titled Performing a simple linear regression with
bivariate data, with checks of linearity, homoscedasticity, and normality
assumptions to do each of the following:
Follow the instructions in the first five steps to graph the least squares line
on a scatter plot, and submit the SPSS output with this assignment; then state
why the linearity assumption appears to be satisfied.
Since the data points appear to be randomly distributed about the line, the
linearity assumption appears to be satisfied.
14
Class Handout #6 Homework
Name __________________________
Exercise #5
Read the “PRACTICAL EXERCISES” section at the end of Chapter
3. You should find that there are eight answers required. The SPSS
data file Well-Being.sav needed to get the answers is available from
the course web page schedule.
(a) Answer #1 by stating the null and alternative hypotheses
corresponding to the ANOVA table for the simple linear regression.
(b) Answer #2 by ????????????????.
by first using the Analyze> Descriptive Statistics> Frequencies options in
SPSS to obtain the range of values in the data set for each variable; then, for
each variable, write a statement of how the range in the data set compares
with the range stated in the description of this data file in the appendix of the
textbook. Attach the SPSS output to this assignment
15
Class Handout #6 Homework
Name __________________________
Chapter 3 Homework
Read the “PRACTICAL EXERCISES” section at the end
of Chapter 3. Then, answer the 8 questions by doing the
following:
(1) create and submit a Word document which contains
your answers to questions 1, 2, 5, 6, 7, and 8;
(2) submit the SPSS output file(s) corresponding to
questions 3, 4, 6, and 7.
16
Class Handout #6 Homework
Name __________________________
17