Statistics 101 - Laboratory 5
Modeling linear relationships
1. Forearms and Feet
On last week’s lab you collected data on the length of forearms and feet. In the table below are
the Forearm Length and the Foot Length for a sample of 10 Stat 101 students.
Student
Forearm, x
Foot, y
1
2
3
23.5 23.5 28.5
23.5 23
29
4
26
22
5
26
26
6
25
25
7
8
9
10
25.5 26.5 26 25.5
23
24 24.5 24
a. Why does it make sense to have the Forearm Length as the explanatory variable and Foot
Length for the response variable? Hint: What did you have to do to measure each?
b. Plot the data. Decribe the general relationship between forearm length and foot length.
c. Calculate the sample mean and sample standard deviation for the Forearm Length.
Round final answers to 3 decimal places.
d. Calculate the sample mean and sample standard deviation for the Foot Length. Round
final answers to 3 decimal places.
e. Given that
∑ ( x − x )( y − y ) = +17.35 , verify that the correlation between Forearm
Length and Foot Length is +0.676. What does this indicate about the linear relationship
between Forearm Length and Foot Length?
f. Calculate the least squares regression slope estimate. Round your final answer to 4
decimal places.
g. Calculate the least squares regression y-intercept estimate. Round your final answer to 3
decimal places.
h. Use your prediction equation to predict the Foot Length for student number 6, whose
Forearm Length is 25 cm. What is the residual for this prediction?
i. Put the least squares regression line on the plot of the data. It should be clear that you
have used the regression line to do the plot. Hint: You need two points to draw a straight
line.
2. Forearms and Feet: Using JMP with a larger sample
On the next page is JMP output that summarizes the distribution of Foot Lengths and the
relationship between Forearm Lengths and Foot Lengths for a sample of 100 Stat 101 students.
a. Describe the shape of the distribution of Foot Lengths.
b. Do the sample mean and sample median values agree with your description of the shape
of the distribution? Explain briefly.
c. Report the values of appropriate summary statistic for the center and spread. Why did
you choose these summaries?
d. Give the prediction equation for the line relating Foot Length to the Forearm Length.
e. Give an interpretation of the estimated slope coefficient within the context of the
problem.
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f.
Use the prediction equation to predict the Foot Length of a student whose Forearm
Length is 25 cm. How does this prediction compare to the actual Foot Length of student
6, 25 cm?
g. How much of the variation in Foot Length can be explained by the linear relationship
with Forearm Length?
h. Describe the plot of residuals. What does this indicate about the predictions made using
the prediction equation in d.?
Distribution of Foot Length
100.0%
75.0%
50.0%
25.0%
0.0%
Sample Statistics
maximum
30.5
quartile
27
median
25
quartile
23.7
minimum
20.5
Mean
25.39
Std Dev
2.373
N
100
Bivariate Fit of Foot By Forearm
Linear Fit
Foot = 2.8698302 + 0.8526225*Forearm
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
Parameter Estimates
Term
Estimate
Intercept
2.8698302
Forearm
0.8526225
0.520307
0.515412
1.651813
25.391
100
Std Error
2.190626
0.082698
t Ratio
1.31
10.31
Prob>|t|
0.1932
<.0001*
2
Statistics 101 - Laboratory 5
Group Answer Sheet
Names of Group Members:
____________________, ____________________
____________________, ____________________
1. Forearms and Feet
a. Why does it make sense to have Forearm Length as the explanatory variable and
Foot Length as the response?
b. Plot the data.
3
c. Sample mean and sample standard deviation of Forearm Length, x. Round final
answers to 3 decimal places.
d. Sample mean and sample standard deviation of Foot Length, y. Round final
answers to 3 decimal places.
e. Verify that the correlation between Forearm Length and Foot Length is +0.676. What
does this indicate about the linear relationship between Forearm Length and Foot Length?
f.
Calculate the least squares regression slope estimate. Round final answer to 4 decimal
places.
g. Calculate the least squares regression y-intercept estimate. Round final answer to 3
decimal places.
h. Use your prediction equation to predict the Foot Length for student number 6, whose
Forearm Length is 25 cm. What is the residual for this prediction?
j.
Put the least squares regression line on the plot of the data. It should be clear that you
have used the regression line to do the plot. Hint: You need two points to draw a straight
line.
4
2. Forearms and Feet: Using JMP with a larger sample
a. Describe the shape of the distribution of Foot Length.
b. Does the sample mean and sample median values agree with your description of the
shape of the distribution? Explain briefly.
c. Report the values of appropriate summary statistic for the center and spread. Why did
you choose these summaries?
d. Give the prediction equation for the line relating Foot Length to Forearm Length.
e. Give an interpretation of the estimated slope coefficient within the context of the
problem.
f.
Use the prediction equation to predict the Foot Length of a student whose Forearm
Length is 25 cm. How does this prediction compare to the actual Foot Length of student
6, 25 cm?
g. How much of the variation in Foot Length can be explained by the linear relationship
with Forearm Length?
h. Describe the plot of residuals. What does this indicate about the predictions made using
the prediction equation in d.?
5