Exam 1 Part 1
Part I.
(5 pts each)
1. Define Type I error in
words and in terms of Ho.
2. Define Type II error in
words and terms of Ho.
3. Define the power of a
statistical test 1) in words and
in terms of Ho; 2) in terms of
Beta.
4. Define the relationship
between alpha and Type I
error?
5. Precisely, how does raising
or lowering alpha affect Type
II error for a given statistical
test?
6. Precisely, how does raising
or lowering alpha affect the
power of a statistical test.
7. Given a choice between two
statistical tests for the same
comparison of means, a
statistician chooses the test
with the smaller p-value. Did
she choose the more powerful
or the less powerful test?
Support your answer.
8. Why should an ANOVA
test, rather than a series of ttests, be used to compare more
than 3 means?
9. Calculate the upper bound
for Type I error, if 5 means
were compared using t-tests.
Math 410b
Name_________________________
Answers
Type I (Alpha) error is the probability of rejecting the Ho in error.
Type II (Beta) error is the probability of not rejecting Ho in error.
1)The power of a test is the probability of rejecting Ho when it should be
rejected, that is, when the statistics being compared differ because of
something other than sampling error!
2) Power = 1 – (Beta)
Alpha is the probability of committing a Type I error.
Alpha and Beta are inversely related. Raising alpha decreases the probability
Type II error; lowering alpha increases the probability of Type II error.
Raising alpha increases the power of a test, while lowering alpha decreases its
power. (Direct proportion.)
She chose the more powerful test, because a smaller p-value means the test
will show significant separation of means “more confidently” than a larger
p-value, thereby allowing rejection of Ho easier.
With more than 3 means, the upper bound for Type I error grows according to
the formula given in problem 9, if they are compared two at a time with ttests.
Upper Bound for Type I error = 1-(1-Alpha)c where c is the number of
comparisons of means. If alpha = .05, then there are 10 comparisons of 5
means possible, so Upper bound for Type I Error = 1-.9510
=.40
9. Complete the ANOVA table
to the right.
Source
Corrected Model
Intercept
VAR00002
Error
Total
10. Knowing that F-crit = 4.96
for alpha =.05, what can be
said about the means compared
in the ANOVA in problem 9?
Type III Sum of
df Mean Square
Squares
4000.000 1
4000
3249000.000 1
3249000
4000.000 1
4000
18000.000 8
2250
3271000.000 10
F
1.77777778
1444
1.77777778
Since the corrected model has an F = 1.8, one cannot reject Ho in this case.
There is not enough evidence to state that any of the means differ from each
other!
Exam 1 Part 1
11. In any ANOVA, the total
(within) variability is divided
up among the Intercepts,
Variables (Factors), and the
“Error,” or unexplained
variability. The ratios of these
Sources of Variability to the
Total Variability measure their
strengths. Calculate the
percents for the Intercept,
VAR00002, and the Error in
the table.
Math 410b
Source
Corrected Model
Intercept
VAR00002
Error
Total
Name_________________________
Type III Sum of
Squares
4000.000
3249000.000
4000.000
18000.000
3271000.000
%variability
99.32742
0.122287
0.55029
100
12. What does it mean for a
statistical test to be robust with
respect to its assumptions?
A statistical test is robust with respect to its assumptions if contradicting one
the assumptions DOES NOT invalidate the test results.
13. In particular, for which
assumption is ANOVA not
robust? Conjecture about why
this might be.
14. In words, BRIEFLY
explain what it means for two
factors to interact in an
ANOVA.
ANOVA is particularly sensitive to violating the homogeneity of variance
assumption. That is, if the levels of a factor differ in their variances, the
outcome of the ANOVA are suspect.
Two factors interact if the mean dependent measure changes differentially with
the factor levels. That is, averages significantly decline across one factor but
rise across another.