Ch.15 Review: Questions on the t Test for the Slope of the Regression Line
Questions 1-5 refer to the following data that you collect for explanatory variable A and resp. variable B:

A(x) / B(y)
1 / 14.2
2 / 14.9
3 / 15.5
4 / 16.8
5 / 17.8
6 / 18.9
7 / 20.1
8 / 20.9
9 / 21.5
10 / 22

1. Construct the least-squares regression line that best fits these data.
A. = .941 + 13.08x
B. = 13.08 + .941x
C. = .9359 + 13.11x

D. = 13.11 + .9358x

E. = 13.11
2. What's the critical t value you'd use for a 95% confidence interval for the slope of this regression line?

A. / 1.812
B. / 2.228
C. / 1.833
D. / 2.262
E. / 2.306


3. Calculate SEb and construct a 95% confidence interval for the slope of the regression line.

A. / (.8599, 1.012)
B. / (.8584, 1.013)
C. / (.9023, .9695)
D. / (-1.3701, 3.2419)
E. / (13.03,13.187)

4. Which of these are proper null and alternative hypotheses for a two-sided significance test about the slope of the regression line?

A. / Ho: = 0 Ha: 0
B. / Ho: = .9359 Ha: .9359
C. / Ho: = .9359 Ha: .9359
D. / Ho: = 13.11 Ha: 13.11
E. / None of the above


5. Calculate the test statistic and p-value for the significance test for Ho: = 0 (knowing x is not helpful in predicting y), and Ha: 0 (knowing x is helpful in predicting y).

A. / t = .0359, p = .486
B. / t = .3089, p = .383
C. / t = 17.5, p < .0005
D. / t = 27.88, p < .0005
E. / t = 32, p < .0005

Free Response Questions

1)  You want to determine whether knowing a student's final grade helps to predict how that student will evaluate their teacher. You ask 12 students to assign their math teacher a numeric grade between 0 and 100. You also record each student's final grade. You collect the following data:

Student Grade / Teacher Evaluation
70 / 69
72 / 68
74 / 72
77 / 73
79 / 74
80 / 82
83 / 80
86 / 85
87 / 90
90 / 84
92 / 89
95 / 93

a) Construct a scatterplot for these data. Does there seem to be a linear relationship between the two variables?
b) Use your TI-83 calculator to construct a least-squares regression line.

c) Create a scatterplot of the residuals against the explanatory variable. Does the scatterplot support the assumption that the residuals are normally distributed about the regression line? Can you continue to use regression analysis to analyze these data?
d) What are the null and alternative hypotheses for a two-sided significance test for this regression line?
e) Using your calculator, find your test statistic and p-value for this two-sided test.
f) What's the standard error of the slope of this regression line?

g) Construct a 95% confidence interval for the slope of the regression line.

h) Interpret the 95% confidence interval (0.776, 1.2384) for the slope of the regression line.

Chapter 15 Review Worksheet Solutions

1.  D. = 13.11 + .9359x. The formula = a + bx, where a = - b = 13.11 and
b = r = .9359, gives you = 13.11 + .9359x.

2.  E. 2.306. To find the critical t value for a confidence interval using a table, locate your degrees of freedom (n - 2) and look across the table to find the t value for a two-sided confidence interval (01/2 = .05/2 = .025).

3.  B. (.8584, 1.013). To construct a confidence interval for the slope of a regression line, use the formula: b1 t* = .9359 2.306(.0336) = .9359 .0774816 = (.8584, 1.013)

4.  B. Ho: = 0, Ha: 0. The null and alternative hypotheses for a two-sided significance test about the slope of the regression line are Ho: the slope is zero (knowing x is not helpful in predicting y) and Ha: the slope is not zero (knowing x is helpful in predicting y).

5.  D. t = 27.88, p < .0005. To find your test statistic, use t = = = 27.85, where b = r. To find your p-value, find 8 degrees of freedom on your table. The t value of 27.85 is off the table, so p < .0005.

Free Response
a) Yes, there is a linear relationship between the two variables. To see the relationship construct a scatterplot.

b) y-hat = -2.75494 + 1.00717x

c) The residual plot does support the assumption that the residuals are normally distributed about the regression line. To support this assumption, the residuals have to be randomly scattered above and below the x-axis. This plot shows that the residuals are scattered randomly. If you want to be even more confident about this assumption, create a normal probability plot of residuals. A normal probability plot will be a line if the residuals are normally distributed, and this plot is very close to a line. Because this assumption is met, you can continue to use regression analysis with these data.

d) Ho :B = 0 (Knowing a student's grade does not help to predict how the student will evaluate the teacher.) Ha:B (does not equal) 0 (Knowing a student's grade does help to predict how the student will evaluate the teacher.)
e) t = 9.706, p = .00000209
f) SEb = .10377
The standard error is not given by the t test, so calculate it using the slope and t in the
formula : t = b/SEb

g) (.776,1.2384)
To find a confidence interval, use the following formula: b+-t*SEb
Remember that your value for degrees of freedom for a linear regression t test is n minus 2.

h) We are 95% confident that, for every one-number change in a student's grade, the change in teacher evaluation is between .776 and 1.2384.