252x9721
ECO252 QBA II
2nd Hour Exam
March 17, 1997
Class Registered_____________
Class Attended_____________
I. Do all the following (14 Points). Make diagrams!
Show all your work!
x ~ N N(6,11)
1.
P(0  x  7)
2.
P(-2  x  5)
3.
P(7  x  48)
4.
F (5) (The cumulative probability)
5.
The 43rd percentile of the distribution.
6.
x.16
7.
A symmetrical interval about the mean with 64 percent of the probability.
II.
mileage
Do all the following (9 Points-2 Point Penalty for not handing in output)
1.
Hand in your computer output for problem 1.(3) Do not do the questions below unless you hand in your
output.
2.
The following two pages show a comparison of gas mileage before and after use of an additive to
gasoline. x1 is mileage without the additive (MwoA), x2 is mileage with the additive (MwA) and d is x1-x2.
Assume that the significance level is 10%. Each line represents the experience with one automobile
model for example: on the first line 37.2 is mileage for automobile 1 without the additive, 37.7 is
for the same automobile with the additive and -0.5 is the difference.
Most of these tests are the t-test you did as the first computer assignment. The remaining tests use the
instruction TWOSAMPLE t which compares the means of x 1 and x2 assuming that x1 and x2 are independent
samples.
a.
b.
For each of tests a-h do the following (i) write Ho and H1 (ii) draw a diagram showing p-value
(iii) explain whether we accept H0 (use a 10% significance level) (12)
Which test is the correct one to test whether the additive improves mileage? Why? Does the
additive work? Why? (3)
QBA II 2nd Exam
R. Bove, 3/17/96
QBA II 2nd Exam
R. Bove, 3/17/96
QBA II 2nd Exam
R. Bove, 3/17/96
III.
You must do Problem 3a. (2 Point Penalty) Do at least 3 of the following 5 questions. (At least 10 points each).
Show your work! State Ho and H1. Use a significance level of 5% unless you are told otherwise. (You actually
need only 21 more points if you did all of the previous parts.)
1.
In the computer output in Part I. look at test g. Using the values of the means and standard deviations
shown on the output (and assuming that
 12   22
and x1 and x2 are independent).
( x1 = 28.51, Error! Switch argument not specified. s1 = 5.86, Error!
Switch argument not specified.
x2 = 28.68, Error! Switch argument not
specified. s2 = 5.75) Error! Switch argument not specified.
1.
a.
b.
Compute the degrees of freedom in the problem. (5)
Find a critical value for d  x1  x2 Error! Switch
in the one-sided test and do the test at
confidence level. (5)
c.
argument not specified.
the
90%
Redo the test by (i) testing for the equality of variances in x1 and x2. (3) (ii) assuming that the
variances are equal and constructing a 1-sided confidence interval. (4)
QBA II 2nd Exam
R. Bove, 3/17/96
2.
x1
MwoA
37.2
36.7
33.3
30.6
29.1
26.8
24.9
23.0
22.5
21.0
x2
MwA
37.3
36.8
32.9
30.3
29.4
26.7
25.2
23.6
22.9
21.3
d
-5.0
-0.1
0.4
0.3
-0.3
0.1
-0.3
-0.6
-0.4
-0.3
Data from Part II. are repeated above
a.
b.
Assuming that the data come from a nonnormal distribution and is paired, do a 2-sided 10% test
of the equality of the medians. (5)
Repeat the test in a., assuming that the data represent 2 independent samples. (5)
QBA II 2nd Exam
R. Bove, 3/17/96
3.
Data is given for two convenience stores.
Daily Sales ($100 units)
Store 1
Store 2
6
1
8
19
10
2
12
18
14
10
9
10
11
3
7
17
13
4
11
17
x2 = 10.1 Error! Switch argument not specified.
argument not specified.
a.
b.
c.
d.
e.
s2 =7.24 Error! Switch
Compute the sample variance for x1. (2)
Test the hypothesis that the variance for Store 1 is 10. (3)
Test the statement that the variance for Store 2 is larger than the variance for Store 1. (2)
Do a confidence interval for the ratio of the variances. (2)
Records for 200 days for each store give
,
x1 = 9.9, Error! Switch argument not specified.
x2 = 1 0 . 1Error!
Switch argument not specified. s1 = 4.5 Error! Switch argument not specified.
and s2 = 8.1. Error! Switch argument not specified.
Do a confidence interval for the difference between the means. (3)
QBA II 2nd Exam
R. Bove, 3/17/96
4.
A survey asked men and women their choice between three brands of automobiles.
Honda
Men
Women
a.
b.
c.
Nissan
140
Saturn
70
50
Total
30
65
240
125
240
Test whether gender and automobile preference are independent. (6)
Test the hypothesis that the proportion preferring Nissan was the same among men and
women. (4)
For the group as a whole, test the hypothesis that the proportion favoring Nissan was 25%. (3)
QBA II 2nd Exam
R. Bove, 3/17/96
5.
In a training program 300 farmers are broken into 100 groups of 3 farmers each. Farmers that expand
production by 20% or more are considered successes. The record shows that of the 300 farmers, 120
succeed. 22 groups had no successes, 42 had one, 30 had 2 and 6 had 3.
Success
0
1
2
3
Frequency
22
42
30
6
100
Success
0
1
2
3
Frequency
21
45
28
6
100
a.
Use a chi-squared test to determine if a Binomial Distribution fits the data, getting your value of
 from the data. (4)
b.
Use a Kolmogorov-Smirnov test to see if a binomial distribution with  =.5 fits the data. (4)
c.
(i) in a., could you use a Kolmogorov-Smirnov or Lilliefors test? Why or why not? (2) (ii) in b.,
could you use a chi-squared or Lilliefors test? Why or why not? (2)
d.
100 data points were collected and their sample mean and standard deviation computed as 3.92
and 0.468 respectively. They were then grouped by intervals of one standard deviation on either
side of the sample mean.
Interval
2.516-2.984
2.984-3.452
3.452-3.920
3.920-4.388
4.388-4.856
4.856-5.324
Count
3
18
36
23
18
2
Test to see if these come from the normal distribution using the most powerful possible
method. (8)
QBA II 2nd Exam
R. Bove, 3/17/96