MATH 3200
PROBABILITY AND STATISTICS
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This examination has twenty problems. All 20 are modifications of the recommended homework problems. Each problem will be
graded on a three-point scale, with 3=fully correct, 2=mostly correct, 1=mostly incorrect, and 0=fully incorrect. Please draw a box or
oval around each final answer so I can locate it easily. Of course, for me to assign partial credit, I will also need to see your steps.
This is especially important when you use the statistical capabilities of your calculator. For example, if you use the normalcdf
function to solve a problem, one step should be to write down the function and its arguments exactly as they appear on your
calculator screen (except you don’t need to wrap the text the way it appears on the small calculator screen).
1. (7.17) Diabetic patients monitor their blood sugar levels with a home glucose monitor which analyzes a drop of
blood from a finger stick. Although the monitor gives precise results in a laboratory, the results are too variable when it
is used by patients. A new monitor is developed to improve the precision of the assay results under home use. Home
testing on the new monitor is done by 9 persons using drops from a sample having a glucose concentration of 118
mg/dl.
If σ < 10 mg/dl, then the precision of the new device under home use is better than the current monitor. The readings
from 9 tests are as follows.
125 123 117 123 115 112 128 118 124
Test H0: σ ≥ 10 vs. H1: σ <10 at the 0.10 level. Also find an upper one-sided 90% confidence interval for σ.
2. (7.18) A bottling company uses a filling machine to fill bottles. A bottle is to contain 475 milliliters ( about 16 oz.)
of beverage. The actual amount is normally distributed with a standard deviation of 1.0 ml. The purchase of a new
machine is contemplated. Based on a sample of 16 bottles filled by the new machine, the sample mean is 476.4 ml. and
the standard deviation is s = 0.6 ml. Does the new machine have a significantly less variation than the current machine?
To answer the question posed, test the hypothesis H0: σ ≥ 1.0 vs. H1: σ < 1.0 at the 0.10 level, where σ is the standard
deviation of the new machine.
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3. (7.21) Refer to the information in Problem 2 for the sample of 16 bottles filled by the new filling machine. Suppose
that the specifications on the contents of a bottle are 476 ± 1 ml of beverage. Using the value 2.903 from n=16,
α=γ=0.05 in Table A12, find a 95% tolerance interval that will include the amounts of beverage in 95% of all bottles.
Does the tolerance interval fall within the specification limits? What does this indicate?
4. (7.22) The durability of upholstery fabric is measured in double rubs (DR), which simulates a person getting in and
out of a chair. The manufacturing label on one fabric gives its durability range as 68,000-82,000 DR. The company’s
quality control department independently evaluated the fabric by testing 16 one-yard samples. The sample mean was
74,283 DR and the sample standard deviation was 4676 DR. Although the mean is acceptable, is the standard deviation
consistent with the labeled range? Assume that the DR measurements are normally distributed, and consider the range
68,000-82,000 to be four standard deviations. Find a 95% prediction interval for the durability of this fabric. If an
office is buying this fabric to cover furniture in a waiting room and requires a fabric with durability of at least 70,000
DR, would this be a good purchase?
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5. (8.1) Tell in each of the following instances whether the study uses an independent samples or a matched design.
a) Two computing algorithms are compared in terms of the CPU times required to do the same six test problems.
b) A survey is conducted of teens from inner city schools and suburban schools to compare the proportion who have
tried drugs.
c) A psychologist measures the response times of subjects under two stimuli; each subject is observed under both
stimuli in a random order.
d) An agronomist compares the yields of three varieties of soybean by planting each variety in 10 separate plots of land,
each one of which has been subdivided into three subplots, with the three varieties randomly allocated to the subplots.
6. (8.6) Production lines in a manufacturing plant are set to make steel ball bearings with a diameter of 1 micron. Six
ball bearings are randomly selected from two production lines. The diameters of the ball bearings measured in microns
were as follows.
First line
1.18 1.42 0.69 0.88 1.62 1.09
Second line
1.72 1.62 1.69 0.79 1.79 0.77
These samples are independent. Therefore make a Q-Q plot of the data. Does one set of ball bearing diameters tend to
be larger than the other?
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7. (8.13) Refer to the data in Problem 6 on the diameters of ball bearings from two production lines.
Calculate 95% confidence intervals for the difference between the mean diameters of the ball bearings from the two
production lines, with and without the assumption of equal variances.
8. (8.13) Using the data from Problem 6, test the null hypothesis that the variances of Lines 1 and 2 are equal to each
other. Use the usual-and-customary two-sided test at the 0.05 significance level.
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9. (9.6) A blood test intended to identify patients at “high risk” of cardiac disease gave positive results on 90 out of 100
known cardiac patients. Find a 90% confidence interval for the sensitivity of the test, which is defined as the
probability that a cardiac patient is correctly identified.
10. (9.16) In a speech class, two persuasive speeches, one pro and the other con, were given by two students on whether
the university should hire a sexual assault coordinator. The opinions of the other 50 students in the class were obtained
on this issue before and after the speeches with the following responses:
After
Pro
Con
Before
Pro
2
12
Con
23
13
Perform a hypothesis test at α = .05 to determine whether or not there is a change in opinion of the students. Use a twosided alternative and calculate the P-value of the test, then state whether or not you reject H0. Note: There are three
variations on this test, which will yield slightly different P-values. Any one of the three is acceptable.
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11. (9.28) Reader’s Digest conducted an experiment to find out how honest people are in different cities. Three cities
of each type were selected: big cities (Atlanta GA, Seattle, WA, and St Louis, MO), suburbs (Boston, MA, Houston,
TX, and Los Angeles, CA), medium cities (Dayton , OH, Greensboro, PA, and Las Vegas, NV) and small cities
(Cheyenne, WY, Concord, NH, and Meadville, PA). In each selected city, 10 wallets were left in public places. Each
wallet contained $50 cash and a telephone number and an address where the owner could be reached. A record was
kept of the number of wallets returned by people who found the wallets. The [altered] data are summarized in the table
below.
City Type
Big Cities
Suburbs
Medium Cities
Small Cities
Returned
10
16
22
23
Kept
20
14
08
07
Set up the hypotheses to check whether there are significant differences between the percentages of people who returned
the wallets in different types of cities. Do a chi-square test of the hypotheses at the .05 significance level.
12. (10.4) The time between eruptions of Old Faithful geyser in Yellowstone National Park is random but is related to
the duration of the last eruption. The table below shows these times for 5 consecutive eruptions.
Duration of Last Eruption
4.0
1.7
1.8
4.9
4.2
Time between Eruptions
70
43
48
70
79
Fit a least squares regression line predicting time between eruptions from duration of the last eruption. Use it to predict
the time to the next eruption if the last eruption lasted 2 minutes. Also calculate the mean square error estimate of σ.
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13. (10.4) Calculate the standard error of the regression slope in Problem 12.
14. (10.11) From the data in Problem 12, calculate the 95% CI for the time to the next eruption if the last eruption
lasted 2 minutes.
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15. (10.11) From the data in Problem 12, calculate the 95% PI for the time to next eruption if the last eruption lasted 3
minutes.
16. (10.8) Derive the least squares estimator for the slope of a regression line that is constrained to pass through the
origin. That is, y = β1 x.
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17. (10.29) Counts of the numbers of finger ridges for 6 pairs of identical twins are given in the following table.
Twin Pair
1
2
3
4
5
6
Twin 1 Ridge Count
71
79
102
115
72
83
Twin 2 Ridge Count
71
82
99
114
70
82
Calculate the correlation coefficient and test the null hypothesis H0: ρ=0 by means of a two-sided Student’s t test with
α = 0.05.
18. Using the data from Problem 17, calculate a 95 % confidence interval for ρ.
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19. Using the Old Faithful data from Problem 12, calculate the sums of squares and degrees of freedom portion of an
analysis of variance table.
20. Complete the analysis of variance table you began in Problem 19, ending with the F ratio and its P-value.