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Math 275
Quiz #2
Mathematical Statistics
Winter 2005
SOLUTIONS
1. With the standard notation used in class and in the textbook, circle all of the following which
are random variables.

ˆ


p
s2
2nd, 4th, and 6th entries above.
2. A statistical test for Ho:  < 5 versus Ha:   5 has rejection region { T > 3}, where T is the
test statistic. If  = 6 and T = 1 then
the null hypothesis is Accepted and a Type II error occurs.
3. TRUE or FALSE: The significance level of a statistical test is equal to the probability that the
null hypothesis is true
FALSE
4. Consider the problem of estimating the parameter p for a binomial distribution with n = 5
trials. We will use the number of successes X as a test statistic. Here are the binomial probabilities
for several values of p.
p
0.60
0.70
0.75
0
.010
.002
.001
1
.077
.028
.015
2
.230
.132
.088
3
.346
.309
.264
4
.259
.360
.396
5
.078
.168
.237
(a) For testing Ho: p = 0.75 versus Ha: p < 0.75, what is the significance level of a test
whose rejection region is {0,1,2}?
.001 + .015 + .088 = .104
(b) For the test in (a), what is the power of the test if p = 0.60?
.010 + .077 + .230 = .317
5. A study in an economics journal concluded that the proportion of heads of households who
earn the minimum wage has increased from the 10% figure of six years ago. What were the null
and alternative hypotheses for the statistical test that was performed? (Use appropriate symbols.)
Ho: p = .10 versus Ha: p > .10
6. Consider a hypothesis test Ho:  = 3 versus Ha:  > 3. Here is the graph of the likelihood
function for the data.
What is the value of the likelihood ratio test statistic?
(a)
(b)
(c)
(d)
***(e)
0.05
0.2
0.4
0.6
0.8 = lik(3) / lik(MLE) = 20/25
7. In an experiment to determine whether Drug X is effective in reducing hypertension the
following data was collected on 200 subjects.
Reduction in blood pressure
No reduction
Drug X
17
83
100
Placebo
13
30
87
170
100
200
FILL IN THE BLANKS: Using Fisher’s Exact Test, under the null model the number of people
who use Drug X and see a reduction in blood pressure can be modeled by picking balls at random
from an urn. In particular, the probability that that number is equal to k is equal to the probability
of picking 100 balls from an urn made up of 30 red balls and 170 black balls and having k of
them be red.
8. Grades in an elementary statistics class were classified by the students’ majors.
A
B
C
D or F
Psychology
8
14
15
3
Biology
16
19
4
1
Other
13
15
7
4
(a) State the null and alternative hypotheses for the chi-square test to determine if there is
a relationship between students’ majors and their grades.
Ho: No association between major and grade versus Ha: Some association between major and
grade.
(b) Below is the density function for the distribution of the chi-square statistic used in this
test. What are the degrees of freedom for the distribution?
df = (4-1)x(3-1) = 6
(c) The value of the chi-square statistic for this data is  2  13.27 . Guesstimate the
P-value and state an appropriate conclusion in the context of this problem.
P-value is about 5-10%. There is some evidence to reject the null and conclude that there is a
relationship between major and grade.
9. Data is collected from an exponential distribution and a statistical test is performed for
Ho: λ = 2 versus Ha: λ = 3. (Recall the density function for an exponential distribution
f ( x)  e  x , x  0 .) If n = 14 observations X 1 ,
14
, X 14 are made and  X i  5 , then the
value of the likelihood ratio test statistic is
(a)  = .13
(b)  = .33
****(c)  = .51 = (2^14 * exp(-2*5)) / (3^14 * exp(-3*5))
(d)  = .74
(e)  = .96
i 1