Test 14B
A P Statistics
The following information is used in questions 1–4.
A random sample of 80 companies from the Forbes 500 list was selected and the relationship between sales
(in hundreds of thousands of dollars) and profits (in hundreds of thousands of dollars) was investigated by
regression. A least-squares regression line was fit to the data using statistical software, with sales as the
explanatory variable and profits as the response variable. Here is the output from the software:
Dependent variable is Profits
R squares = 66.2%
s = 466.2 with 80 – 2 = 78 degrees of freedom
Variable
Constant
Sales
Coefficient s.e. of Coefficient
–176.644
61.16
0.092498
0.0075
P-value
0.0050
≤0.0001
1.
Using the above data, approximately what is the intercept of the least-squares regression line?
(a) 0.0925
(b)
0.0075
(c)
–176.64
(d) 61.16
(e)
None of the above. The answer is ___________________
2.
Using the above data, approximately what is a 90% confidence interval for the slope of the leastsquares regression line?
(a) 0.0925 ± 0.0075
(b)
0.0925 ± 0.012
(c)
–0.0925 ± 0.0075
(d) –0.0925 ± 0.012
(e)
None of the above. The answer is __________________
3.
Using the above data, what is the value of the t statistic for testing whether the slope of the leastsquares regression line is 0?
(a) 0.0075
(b)
0.082
(c)
0.092
(d) 12.73
(e)
None of the above. The answer is __________________
4.
Using the above data, is there strong evidence (and if so, why) of a straight line relationship between
sales and profits?
(a) Yes, because the slope of the least-squares line is positive.
(b) Yes, because the P-value for testing if the slope is 0 is quite small.
(c)
No, because the value of the square of the correlation is relatively small.
(d) It is impossible to say because we are not given the actual value of the correlation.
(e)
None of the above. The answer is
.
Questions 5–7 refer to the following situation:
Growth hormones are often used to increase the weight gain of chickens. In an experiment using 15
chickens, five different doses of growth hormone (0, .2, .4, .8, and 1.0 mg/kg) were injected into chickens
(three for each dose) and the subsequent weight gain was recorded. An experimenter plots the data and
finds that a linear relationship appears to hold. The output from SAS follows:
PARAMETER
INTERCEPT
DOSE
5.
6.
ESTIMATE
3.7816
4.0416
T FOR H0:
PARAMETER=0
3.23
2.85
PR > |T|
0.0066
0.0137
STD ERROR OF
ESTIMATE
1.1705
1.4195
The fitted regression line is:
(a) WtGain = 4.04 + 3.78 Hormone
(c) WtGain = 2.85 + 3.23 Hormone
(e) WtGain = 1.17 + 1.42 Hormone
(b)
(d)
WtGain = 3.23 + 2.85 Hormone
WtGain = 3.78 + 4.04 Hormone
A 95% confidence interval for the slope is:
(a) 4.04 ± 1.96 (1.42)
(c) 4.04 ± 2.16(1.42)
(e) 3.78 ± 2.16 (1.17)
(b)
(d)
4.04 ± 1.77 (1.42)
3.78 ± 1.77 (1.17)
7.
It is suspected that weight gain should increase with dose. An appropriate null and alternate hypothesis
to test the slope, the test statistic, and the p-value are:
(a) H0:  = 0, Ha:  ≠ 0, t = 2.85, p-value = .0069
(b) H0:  = 0, Ha:  ≠ 0, t = 3.23, p-value = .0066
(c) H0:  = 0, Ha:  > 0, t = 2.85, p-value = .0137
(d) H0:  = 0, Ha:  > 0, t = 3.23, p-value = .0033
(e) H0:  = 0, Ha:  > 0, t = 2.85, p-value = .0069
Free Response
Consider the following data on height (in inches) and number of siblings for a simple random sample of 10
female students at a large university.
Height
# of Siblings
64.2
2
65.4
0
64.6
2
66.1
6
65.1
3
65.5
1
1.
Make a scatterplot of the data. Does there appear to be
a linear relationship between height and # of siblings?
2.
a) Use your calculator to obtain the equation of
the least-squares regression line and the correlation.
67.2
2
66.4
2
63.3
0
61.7
1
b) Is the value of your correlation coefficient
consistent with your answer to #1.
3.
Explain in words what the slope  of the true
regression line says about height and # of siblings.
4.
What is the estimate of  from the data? What is your
estimate of the intercept  of the true regression line?
5.
Use your calculator to calculate the residuals. Report the sum of the residuals and the sum of the
squares of the residuals. Then use these results to estimate the standard deviation  in the regression
model and interpret it in the context of this problem.
6.
Calculate SEb and interpret it in the context of the problem.
7.
Carry out a two sided significance test to decide whether height is a good predictor for # of siblings.
a) Write the hypotheses.
b) Determine the test statistic, the degrees of freedom, and the P-value of t against the alternative.
c) Write your conclusions in plain language. Would you use height to predict the number of siblings a
woman has?
8.
Compute a 95% confidence interval for the slope ß of the true regression line.
Answers
Multiple choice:
1) c
2) b
3) d
4) b
5) d
6) c
7) e
Free Response:
1) Not very linear.
2) a) y = -25.934+.4285x, r = .396 b) Yes, since r is not close to 1.
3) On average, all women have .4285 more siblings for each inch of height.
4) the estimate for ß is b = .4285. the estimate for α is a = -25.934
5) sum of residuals = 0, sum of residuals squared = 22.6789, s = 1.6837 which tells us that, on average, the
actual number of siblings will differ from the predicted number of siblings by 1.6837 in.
6) SEb = 1.6837/√22.985 = .3512 which tells us that in repeated samples, the slope will differ from the
slope of the true (population) regression line by, on average, .3512 siblings per inch.
7) a) H0: ß = 0 (no association between height and # of siblings)
H0: ß ≠ 0 (there is an association between height and # of siblings)
b) df = 8, t = 1.22, P = .257
c) No, I would not use height to predict number of siblings. We have
evidence (P=.257) that there is no association between height and number of siblings.
8) .4285±2.306(.3512) = (-.381, 1.238) This means that the mean # of siblings increases by between -.381
and 1.238 for every inch of height for all women. Because this interval contains 0, this is consistent with
our answer to #7 that there is no linear relationship between these two variables.