Hypothesis Tests – 1 sample Means 17. TV safety. The manufacturer

Hypothesis Tests – 1 sample Means
17. TV safety. The manufacturer of a metal stand for home
TV sets must be sure that its product will not fail under
the weight of the TV. Since some larger sets weigh
nearly 300 pounds, the company's safety inspectors
have set a standard of ensuring that the stands can
support an average of at least 500 pounds. Their
inspectors regularly subject a random sample of the
stands to increasing weight until they fail. They test the
hypothesis H0:  = 500 against HA:  < 500, using the
level of significance  = .01. If the sample of stands
fail to pass this safety test, the inspectors will not
certify the product for sale to the general public.
a) Is this an upper-tail or lower-tail test? In the
context of the problem, why do you think this is
important?
b) Explain what will happen if the inspectors commit
a Type I error.
c) Explain what will happen if the inspectors commit
a Type II error.
18. Catheters. During an angiogram, heart problems can
be examined via a small tube (a catheter) threaded into
the heart from a vein in the patient's leg. It's important
that the company who manufactures the catheter
maintain a diameter of 2.00 mm. (The standard
deviation is quite small.) Each day, quality control
personnel make several measurements to test H0:  =
2.00 against HA:   2.00 at a significance level of  =
.05. If they discover a problem, they will stop the
manufacturing process until it is corrected.
a) Is this a one-sided or two-sided test? In the context
of the problem, why do you think this is important?
b) Explain in this context what happens if the quality
control people commit a Type I error.
c) Explain in this context what happens if the quality
control people commit a Type II error.
19. TV safety revisited. The manufacturer of the metal TV
stands in Exercise 17 is thinking of revising its safety
test.
a) If the company's lawyers are worried about being
sued for selling an unsafe product, should they
increase or decrease the value of ? Explain.
b) In this context, what is meant by the power of the
test?
c) If the company wants to increase the power of the
test, what options does it have? Explain the
advantages and disadvantages of each option.
20. Catheters again. The catheter company in Exercise 18
is reviewing its testing procedure.
a) Suppose the significance level is changed to  =
.01. Will the probability of Type II error increase,
decrease, or remain the same?
b) What is meant by the power of the test the
company conducts?
c) Suppose the manufacturing process is slipping out
of proper adjustment. As the actual mean diameter
of the catheters produced gets farther and farther
above the desired 2.00 mm, will the power of the
quality control test increase, decrease, or remain
the same?
d) What could they do to improve the power of the
test?
21. Marriage. In 1960, census results indicated that the
age at which American men first married had a mean
of 23.3 years. It is widely suspected that young people
today are waiting longer to get married. We want to
find out if the mean age of first marriage has increased
during the past 40 years.
a) Write appropriate hypotheses.
b) We plan to test our hypothesis by selecting a
random sample of 40 men who married for the first
time last year. Do you think the necessary
assumptions for inference are satisfied? Explain.
c) Describe the approximate sampling distribution
model for the mean age in such samples.
d) The men in our sample married at an average age
of 24.2 years with a standard deviation of 5.3
years. What's the P-value for this result?
e) Explain (in context) what this P-value means.
f) What's your conclusion?
22. Fuel economy. A company with a large fleet of cars
hopes to keep gasoline costs down, and sets a goal of
attaining a fleet average of at least 26 miles per gallon.
To see if the goal is being met, they check the gasoline
usage for 50 company trips chosen at random, finding
a mean of 25.02 mpg and a standard deviation of 4.83
mpg. Is this strong evidence that they have failed to
attain their fuel economy goal?
a) Write appropriate hypotheses.
b) Are the necessary assumptions to perform
inference satisfied?
c) Describe the sampling distribution model of mean
fuel economy for samples like this.
d) Find the P-value.
e) Explain what the P-value means in this context.
f) State an appropriate conclusion.
25. Cars. One of the important factors in auto safety is the
weight of the vehicle. Insurance companies are
interested in knowing the average weight of cars
currently licensed in the United States; they believe it
is 3000 pounds. To see if that estimate is correct, they
checked a random sample of 91 cars. For that group the
mean weight was 2919 pounds, with a standard
deviation of 531.5 pounds. Is this strong evidence that
the mean weight of all cars is not 3000 pounds?
26. Portable phones. A manufacturer claims that a new
design for a portable phone has increased the range to
150 feet, allowing many customers to use the phone
throughout their homes and yards. An independent
testing laboratory found that a random sample of 44 of
these phones worked over an average distance of 142
feet, with a standard deviation of 12 feet. Is there
evidence that the manufacturer's claim is false?
27. Chips ahoy. In 1998, as an advertising campaign, the
Nabisco Company announced a "1000 Chips
Challenge," claiming that every 18-ounce bag of their
Chips Ahoy cookies contained 1000 chocolate chips.
Dedicated Statistics students at the Air Force Academy
(no kidding) purchased some randomly selected bags
of cookies, and counted the chocolate chips. Some of
their data are given below. (Chance, 12, no. 1 [1999])
1219
1244
1214
1258
1087
1356
1200
1132
1419
1191
1121
1270
1325
1295
1345
1135
a) Check the assumptions and conditions for
inference. Comment on any concerns you have.
b) Create a 95% confidence interval for the average
number of chips in bags of Chips Ahoy cookies.
c) What does this evidence say about Nabisco's
claim? Use your confidence interval to test an
appropriate hypothesis and state your conclusion.
28. Yogurt. Consumer Reports tested 14 brands of vanilla
yogurt and found the following numbers of calories per
serving:
160 130 200 170 220 190 230
80 120 120 180 100 140 170
a) Check the assumptions and conditions for
inference.
b) Create a 95% confidence interval for the average
calorie content of vanilla yogurt.
c) A diet guide claims that you will get 120 calories
from a serving of vanilla yogurt. What does this
evidence indicate? Use your confidence interval to
test an appropriate hypothesis and state your
conclusion.
29. Maze. Psychology experiments sometimes involve
testing the ability of rats to navigate mazes. The mazes
are classified according to difficulty, as measured by
the mean length of time it takes rats to find the food at
the end. One researcher needs a maze that will take rats
an aver age of about one minute to solve. He tests one
maze on several rats, collecting the data at the right.
a) Plot the data. Do you think the conditions for
inference are satisfied? Explain.
b) Test the hypothesis that the mean completion time
for this maze is 60 seconds. What is your
conclusion?
c) Eliminate the outlier, and test the hypothesis again.
What is your conclusion?
d) Do you think this maze meets the "one-minute
average" requirement? Explain.
Time
(sec)
38.4
57.6
46.2
55.5
62.5
49.5
38.0
40.9
62.8
44.3
33.9
93.8
50.4
47.9
35.0
69.2
52.8
46.2
60.1
56.3
55.1
30. Braking. A tire manufacturer is considering a newly
designed tread pattern for its all-weather tires. Tests
have indicated that these tires will provide better gas
mileage and longer treadlife. The last remaining test is
for braking effectiveness. The company hopes the tire
will allow a car traveling at 60 mph to come to a
complete stop within an average of 125 feet after the
brakes are applied. They will adopt the new tread
pattern unless there is strong evidence that the tires do
not meet this objective. The distances (in feet) for 10
stops on a test track were
129,128,130,132,135,123,102,125,128, and 130.
Should the company adopt the new tread pattern? Test
an appropriate hypothesis and state your conclusion.
Explain how you dealt with the outlier, and why you
made the recommendation you did.
Hypothesis Tests – 1 sample Means
Answers:
17. a) Lower-tail. We want to show it will not hold 500
pounds (or more) easily.
b) Reject that the stand can hold at least 500 lbs, when
it can. In other words, They will decide the stands are
unsafe, when they are in fact safe
c) Fail to reject that the stand can only hold at least
500 lbs when it can not. In other words, they will
decide the stands are safe, when they're not.
18. a) Two-sided. If they're too big, they won't fit through
the vein. If they're too small, they probably won't
work well.
b) The catheters are rejected when in fact the
diameters are fine, and the manufacturing process is
needlessly stopped.
c) Catheters that do not meet specifications are
allowed to be produced and sold.
19. a) Increase . This means a larger chance of declaring
the stands safe, if they are not.
b) The probability of correctly detecting that the
stands are not capable of holding more than 500
pounds.
c) Decrease the standard deviation—probably costly.
Increase the sample size—takes more time for testing
and is costly. Increase —more Type I errors.
Increase the "design load" to be well above 500
pounds—again, costly.
20. a) Increase
b) The probability of correctly detecting deviations
from 2 mm in diameter.
c) Increase
d) Increase the sample size or increase a.
21. a) H0:  = 23.3; HA:  > 23.3
b) We have a random sample that comprises less than
10% of the population. Population may not be
normally distributed, as it would be easier to have a
few much older men at their first marriage than some
very young men. However, with a sample size of 40, y
should be approximately Normal. We should check
the histogram for severity of skewness and possible
outliers.
c) t39(23.3, s/ 40 )
d) 0.1414
e) If the average age at first marriage is still 23.3
years, there is a 14.5% chance of getting a sample
mean of 24.2 years or older simply from natural
sampling variation.
f) We have not shown that the average age at first
marriage has increased from the mean of 23.3 years.
22. a) H0:  = 26; HA:  < 26
b) We have a representative sample, less than 10% of
all trips, and a large enough sample that skewness
should not be a problem.
c) t49(25.02, 4.83/ 50 )
d) 0.0789
e) If the average fuel economy is 26 mpg, the chance
of obtaining a sample mean of 25.02 or less by natural
sampling variation is 8%.
f) Since 0.05 < P < 0.10, there is some evidence that
the company may not be achieving the fuel economy
goal.
25. No. Based on this large random sample, t = -1.45; Pvalue = 0.1494. Because the P-value is high, we fail to
reject H0. These data do not provide evidence to
support the claim that the average weight of all cars is
not 3000 pounds. We retain the null hypothesis that
the mean is 3000 pounds.
26. Yes. Based on this large random sample, t = -4.42, Pvalue = 3.29 X 10-5. Because the P-value is low, we
reject H0. These data show that the range of these
phones is not 150 feet; in fact, it is less.
27. a) Random sample; less than 10% of all Chips Ahoy
bags; the nearly Normal condition seems reasonable
from a Normal probability plot. The histogram is
roughly unimodal and symmetric with no outliers.
b) (1187.9, 1288.4) chips
c) Based on this sample, the mean number of chips in
an 18-ounce bag is between 1187.9 and 1288.4, with
95% confidence. The mean number of chips is clearly
greater than 1000. However, if the claim is about
individual bags, then it's not necessarily true. If the
mean is 1188 and the SD deviation is near 94, then
2.5% of the bags will have fewer than 1000 chips
using the Normal model. If in fact the mean is 1288,
the proportion below 1000 will be less than 0.3%, but
the claim is still false.
28. a) Random sample; this is less than 10% of all vanilla
yogurt brands. Nearly Normal condition is reasonable
by examining a Normal probability plot. The
histogram is roughly unimodal (although somewhat
uniform) and symmetric with no outliers.
b) (132.0, 183.7) calories
c) Based on this sample, the mean number of calories
in a serving of vanilla yogurt is between 132 and
183.7, with 95% confidence. We conclude that the
diet guide's claim of 120 calories is too low.
29. a) The Normal probability plot is relatively straight,
with one outlier at 93.8 sec. Without the outlier, the
conditions seem to be met. The histogram is roughly
unimodal and symmetric with no other outliers.
b) t = -2.63, P-value = 0.0160. With the outlier
included, we might conclude that the mean
completion time for the maze is not 60 seconds; in
fact, it is less.
c) t = -4.46, P-value = 0.0003. Because the P-value is
so small, we reject H0. Without the outlier, we
conclude that the average completion time for the
maze is not 60 seconds; in fact, it is less. The outlier
here did not change the conclusion.
d) The maze does not meet the "one-minute average"
requirement. Both tests rejected a null hypothesis of a
mean of 60 seconds.
30. The data value of 102 feet is an outlier. When this is
removed, the Normal probability plot is relatively
straight. The Nearly Normal Condition seems
satisfied. With the outlier removed, the histogram is
roughly unimodal and symmetric with no other
outliers. H0:  = 125; HA:  > 125 feet. With the
outlier eliminated, y = 128.89, t = 3.29, P-value =
0.01. With a P-value this low, we reject H0. There is
strong evidence to suggest that the tires will not bring
the car to a complete stop within 125 feet. On the
basis of these data, the company should not adopt the
new tread pattern. Only 2 out of the 10 data values
were less than the desired 125 feet, and 1 of these was
an outlier.