Chapter 12.1 Review
Name: ____________________________
Period: ___________
Other
Can we predict the heights of school-aged children from foot length? Below is computer output
from a regression analysis of this relationship for 15 randomly-selected Canadian children from 8 to
15 years old, along with a residual plot. The explanatory variable is each child’s foot length (in
centimeters), and the response variable is the child’s height (in centimeters).
1. What is the equation of the least-squares regression line based on these data? Define any
parameters used.
2. Interpret the slope of the regression line.
3. If we are trying to determine the relationship between these two variables for all Canadian children
from 8 to 15 years old, is the slope you provided in part A. a statistic or a parameter? Explain.
4. Based on the information given, discuss whether the conditions have been met to use t-procedures to
make inferences about the slope of the regression line. If you do not have enough information to
determine if a condition is met, indicate what other information or analysis is required.
5. Assuming all conditions have been met, construct and interpret a 99% confidence interval for the
slope of the least squares regression of height on foot length.
6. If you were to perform a test of the hypotheses
versus
at the a = 0.01 level, what
would you conclude? Justify your answer by using your result in Question 5.
7. A psychologist counsels people who are getting divorced. He suspects that it is possible to
predict—among the couples he works with—how long a marriage will last from how long the
divorcing couple’s courtship was before they got married. He selects a random sample of 10 of the
hundreds of couples for whom he has records and performs a regression analysis. Below is
numerical and graphical output from his computer software.
(a) Use the computer output to discuss whether the conditions for regression inference have been
met. If you do not have enough information to check a condition, describe what further information
would be required.
For the remaining questions using these data, assume that the conditions for inference have been
satisfied.
(b) Do these data provide convincing evidence that there is a linear relationship between length of
courtship and length of marriage? Perform the appropriate significance test to support your
conclusion.
(c) Construct a 95% confidence interval for the slope of the population regression line for predicting
length of marriage from length of courtship.
(d) Can the psychologist conclude that his data establishes that a longer courtship will cause a longer
marriage? Why or why not?
Chapter 12.1 Review
Answer Section
OTHER
1. ANS:
, where
predicted height of child and x = child’s foot length.
PTS: 1
2. ANS:
A one-centimeter increase in a child’s foot length is associated with a predicted increase of 2.044
centimeter increase, on average, in a child’s height
PTS: 1
3. ANS:
This is a statistic: it is an estimate of the population regression line’s slope based on this particular
random sample of 15 children.
PTS: 1
4. ANS:
Linear: The residual plot shows a random scattering of points around the residual = 0 line,
suggesting that the linear model is a good fit for the data. Independent: observations for one child
should be independent of observations for any other child. The population is finite but very large,
so the 10% rule is satisfied. Normal: to determine whether values of y for a given x are
approximately normally distributed, we need to determine if the residuals are roughly Normally
distributed, using either a histogram or Normal probability plot. This information has not been
provided, Equal standard deviation: the residual plot shows approximately the same spread around
residual = 0 for all values of x. Random: the data came from a random sample of 15 children
PTS: 1
5. ANS:
State: We want to estimate b, the true slope of the population regression line relating foot length to
height, with 99% confidence. Plan: We are told to assume all conditions for inference have been
met, so we will use a t-interval for the slope to estimate b. Do: We use the t distribution with 15
– 2 = 13 degrees of freedom to find the critical value. For a 99% confidence level, the critical value
is t*=3.012, so the 99% confidence interval for b is
or
. Conclude: We are 99% confident that the interval from –1.429 to 5.517 captures the
actual slope of the population regression line relating height to foot length in Canadian children
between 8 and 15 years old
PTS: 1
6. ANS:
Since the 99% confidence interval contains the null value of 0, we cannot reject H0 at the a = 0.01
level. We do not have convincing evidence that the slope of the population regression line relating
foot length and height is different from 0.
PTS: 1
7. ANS:
A. Linear: the scatterplot shows a weak linear relationship between courtship length and marriage
length, and the residual plot shows a random scatter of points about the line residual = 0.
Independent: Courtship and marriage lengths for randomly-selected couples should be
independent. We are sampling without replacement, but we are told that the psychologist has
records for “hundreds” of couples, so there are more than 10x10=100. Normal: The Normal
probability plot of the residuals is roughly linear, which suggests that marriage lengths are roughly
Normally distributed for each length of courtship. Equal standard deviation: the small sample size
makes this difficult to assess, but the residual plot show no obvious differences in variability around
the line residual = 0 for different values of courtship length. Random: The problem states that the
10 couples were randomly selected.
B. We are testing the hypotheses
versus
, where b is the true slope of the
population regression line relating marriage length to courtship length for all couples in the
psychologist’s records. We will use a significance level of a = 0.05. Plan: The procedure is a
t-test for regression slope. Conditions were checked in part A. . Do: From the computer output,
the t-statistic (with 8 degrees of freedom) is 3.68, yielding a P-value of 0.006. Conclude: Since the
P-value is less than a = 0.05, we reject H0. We have enough evidence to conclude that there is a
linear relationship between length of courtship and length of marriage for this population of couples.
C. State: We want to estimate b, the true slope of the population regression line relating marriage
length to courtship length for all couples in the psychologist’s records, with 95% confidence. Plan:
We will use a t-interval for the slope to estimate b. Conditions were checked in part A. . Do:
We use the t distribution with 10 – 2 = 8 degrees of freedom to find the critical value. For a 95%
confidence level, the critical value is , so the 95% confidence interval for b is
or
. Conclude: We are 95% confident that the
interval from 0.918 to 3.994 captures the true slope of the population regression line relating length
of marriage and courtship for couples who work with this psychologist.
D. No. Since this was not a controlled experiment, we cannot establish a causal relationship
between marriage length and length of courtship. There may be other variables that are confounded
with length of courtship.
PTS: 1