Exercises for Section 6.1
2. The pH of an acid solution used to etch aluminum
varies somewhat from batch to batch. In a sample
of 50 batches, the mean pH was 2.6, with a
standard deviation of 0.3. Let 𝜇 represent the mean
pH for batches of this solution.
a. Find the P-value for testing 𝐻0 : 𝜇 ≤ 2.5 versus
𝐻1 : 𝜇 > 2.5.
b. Either the mean pH is greater than 2.5 mm, or
the sample is in the most extreme _______% of
its distribution.
4. In a process that manufactures tungsten-coated
silicon wafers, the target resistance for a wafer is 85
m. In a simple random sample of 50 wafers, the
sample mean resistance was 84.8 m, and the
standard deviation was 0.5 m. Let 𝜇 represent the
mean resistance of the wafers manufactured by this
process. A quality engineer tests 𝐻0 : 𝜇 = 85 versus
𝐻1 : 𝜇 ≠ 85.
a. Find the P-value.
b. Do you believe it is plausible that the mean is
on target, or are you convinced that the mean is
not on target? Explain your reasoning.
8. Lasers can provide highly accurate measurements
of small movements. To determine the accuracy of
such a laser, it was used to take 100 measurements
of a known quantity. The sample mean error was
25 𝜇m with a standard deviation of 60 𝜇m. The
laser is properly calibrated if the mean error is 𝜇 =
0. A test is made of 𝐻0 : 𝜇 = 0 versus 𝐻1 : 𝜇 ≠ 0.
a. Find the P-value.
b. Do you believe it is plausible that the laser is
properly calibrated, or are you convinced that it
is out of calibration? Explain your reasoning.
test of the hypotheses 𝐻0 : 𝜇 = 12 versus 𝐻1 : 𝜇 ≠
12, the P-va1ue is 0.30.
a. Should 𝐻0 be rejected on the basis of this test?
Explain.
b. Can you conclude that the machine is calibrated
to provide a mean fill weight of 12 oz.? Explain.
18. A shipment of fibers is not acceptable if the mean
breaking strength of the fibers is less than 50 N. A
large sample of fibers from this shipment was
tested, and a 98% lower confidence bound for the
mean breaking strength was computed to be 50.1
N. Someone suggests using these data to test the
hypotheses 𝐻0 : 𝜇 ≤ 50 versus 𝐻1 : 𝜇 > 50.
a. Is it possible to determine from the confidence
bound whether P < 0.01? Explain.
b. Is it possible to determine from the confidence
bound whether P < 0.05? Explain.
Exercises for Section 6.3
6. A random sample of 80 bolts is sampled from a
day’s production, and 4 of them are found to have
diameters below specification. It is claimed that the
proportion of defective bolts among those
manufactured that day is less than 0.10. Is it
appropriate to use the methods of this section to
determine whether we can reject this claim? If so,
state the appropriate null and alternate hypotheses
and compute the P-value. If not, explain why not.
8. A grinding machine will be qualified for a particular
task if it can be shown to produce less than 8%
defective parts. In a random sample of 300 parts,
12 were defective. On the basis of these data, can
the machine be qualified?
Exercises for Section 6.4
Exercises for Section 6.2
6. George performed a hypothesis test. Luis checked
George’s work by redoing the calculations. Both
George and Luis agree that the result was
statistically significant at the 5% level, but they got
different P-values. George got a P-value of 0.20,
and Luis got a P-value of 0.02.
a. Is it possible that George’s work is correct?
Explain.
b. Is it possible that Luis’s work is correct? Explain.
12. A machine that fills cereal boxes is supposed to be
calibrated so that the mean fill weight is 12 oz. Let 𝜇
denote the true mean fill weight. Assume that in a
4. A certain manufactured product is supposed to
contain 23% potassium by weight. A sample of 10
specimens of this product had an average
percentage of 23.2 with a standard deviation of 0.2.
If the mean percentage is found to differ from 23,
the manufacturing process will be recalibrated.
a. State the appropriate null and alternate
hypotheses.
b. Compute the P-value.
c. Should the process be recalibrated? Explain.
6. The thicknesses of six pads designed for use in
aircraft engine mounts were measured. The results,
in mm, were 40.93, 41.11, 41.47, 40.96, 40.80, and
41.32.
a. Can you conclude that the mean thickness is
greater than 41 mm?
b. Can you conclude that the mean thickness is
less than 41.4 mm?
c. The target thickness is 41.2 mm. Can you
conclude that the mean thickness differs from
the target value?
8. The article “Solid-Phase Chemical Fractionation of
Selected Trace Metals in Some Northern Kentucky
Soils” (A. Karathanasis and J. Pils, Soil and Sediment
Contamination, 2005:293-308) reports that in a
sample of 26 soil specimens taken in a region of
northern Kentucky, the average concentration of
chromium (Cr) in mg/kg was 20.75 with a standard
deviation of 3.93.
a. Can you conclude that the mean concentration
of Cr is greater than 20 mg/kg?
b. Can you conclude that the mean concentration
of Cr is less than 25 mg/kg?
Exercises for Section 6.5
6. The article referred to in Exercise 3 categorized
firms by size and percentage of full-operatingcapacity labor force currently employed. The
numbers of firms in each of the categories are
presented in the following table.
Percent of Full OperatingCapacity Labor Force
Currently Employed
>100%
95-100%
90-94%
85-89%
80-84%
75-79%
70-74%
<70%
Small
Large
6
29
12
20
17
15
33
39
8
45
28
21
22
21
29
34
Can you conclude that the distribution of labor
force currently employed differs between small and
large firms? Compute the relevant test statistic and
P-value.
8. For the given table of observed values:
a. Construct the corresponding table of expected
values.
b. If appropriate, perform the chi-square test for
the null hypothesis that the row and column
outcomes are independent. If not appropriate,
explain why.
Observed Values
1
2
3
A
25
4
11
B
3
3
4
C
42
3
5
12 The article “Determination’ of Carboxyhemoglobin
Levels and Health Effects on Officers Working at the
Istanbul Bosphorus Bridge” (G. Kocasoy and H.
Yalin, Journal of Environmental Science and Health,
2004:1129-1139) presents assessments of health
outcomes of people working in an environment
with high levels of carbon monoxide (CO).
Following are the numbers of workers reporting
various symptoms, categorized by work shift. The
numbers were read from a graph.
Shift
Morning Evening
Night
Influenza
16
13
18
Headache
24
33
6
Weakness
11
16
5
Shortness of
7
9
9
Breath
Can you conclude that the proportions of workers
with the various symptoms differ among the shifts?
Exercises for Section 6.6
4. A hypothesis test is to be performed, and the null
hypothesis will be rejected if P 5 0.05. If 𝐻0 is in
fact true, what is the maximum probability that it
will be rejected?
6. A wastewater treatment program is designed to
produce treated water with a pH of 7. Let 𝜇
represent the mean pH of water treated by this
process. The pH of 60 water specimens will be
measured, and a test of the hypotheses 𝐻0 : 𝜇 = 7
versus 𝐻1 : 𝜇 ≠ 7 will be made. Assume it is known
from previous experiments that the standard
deviation of the pH of water specimens is
approximately 0.5.
a. If the test is made at the 5% level, what is the
rejection region?
b. If the sample mean pH is 6.87, will 𝐻0 be
rejected at the 10% level?
c. lf the sample mean pH is 6.87, will 𝐻0 be
rejected at the 1% level?
d. If the value 7.2 is a critical point, what is the
level of the test?
Exercises for Section 6.7
6. A process that manufactures glass sheets is
supposed to be calibrated so that the mean
thickness  of the sheets is more than 4 mm. The
standard deviation of the sheet thicknesses is
known to be well approximated by 𝜎 = 0.20 mm.
Thicknesses of each sheet in a sample of sheets will
be measured, and a test of the hypothesis 𝐻0 : 𝜇 ≤
4 versus 𝐻1 : 𝜇 > 4 will be performed. Assume
that, in fact, the true mean thickness is 4.04 mm.
a. If 100 sheets are sampled, what is the power of
a test made at the 5% level?
b. How many sheets must be sampled so that a 5%
level test has power 0.95?
c. If 100 sheets are sampled, at what level must
the test be made so that the power is 0.90?
d. If 100 sheets are sampled, and the rejection
region is 𝑋̅ ≥ 4.02, what is the power ofthe
test?
8. Water quality in a large estuary is being monitored
in order to measure the PCB concentration (in parts
per billion).
a. If the population mean is 1.6 ppb and the
population standard deviation is 0.33 ppb, what
is the probability that the null
hypothesis𝐻0 : 𝜇 ≤ 1.50 is rejected at the 5%
level, if the sample size is 80?
b. If the population mean is 1.6 ppb and the
population standard deviation is 0.33 ppb, what
sample size is needed so that the probability is
0.99 that 𝐻0 : 𝜇 ≤ 1.50 is rejected at the 5%
level?
Exercises for Section 6.8
2. Five different variations of a bolt-making process
are run to see if any of them can increase the mean
breaking strength of the bolts over that of the
current process. The P-values are 0.13, 0.34, 0.03,
0.28, and 0.38. Of the following choices, which is
the best thing to do next?
i. Implement the process whose P-value was 0.03,
since it performed the best.
ii. Since none of the processes had Bonferroniadjusted P-values less than 0.05, we should
stick with the current process.
iii. Rerun the process whose P-value was 0.03 to
see if it remains small in the absence of multiple
testing.
iv. Rerun all the five variations again, to see if any
of them produce a small P-value the second
time around.
4. Five new paint additives have been tested t0 see if
any of them can reduce the mean drying time from
the current value of 12 minutes. Ten specimens
have been painted with each 0f the new types of
paint, and the drying times (in minutes) have been
measured. The results are as follows:
1
2
3
4
5
6
7
8
9
10
A
14.573
12.012
13.449
13.928
13.123
13.254
12.772
10.948
13.702
11.616
B
10.393
10.435
11.440
9.719
11.045
11.707
11.141
9.852
13.694
9.474
Additive
C
15.497
9.162
11.394
10.766
11.025
10.636
15.066
11.991
13.395
8.276
D
10.350
7.324
10.338
11.600
10.725
12.240
10.249
9.326
10.774
11.803
E
11.263
10.848
11.499
10.493
13.409
10.219
10.997
13.196
12.259
11.056
For each additive, perform a hypothesis test of the
null hypothesis 𝐻0 : 𝜇 ≥ 12 against the alternate
𝐻1 : 𝜇 < 12. You may assume that each population
is approximately normal.
a. What are the P-values for the five tests?
b. On the basis of the results, which of the three
following conclusions seems most appropriate?
Explain your answer.
i. At least one of the new additives results in
an improvement.
ii. None of the new additives result in an
improvement.
iii. Some of the new additives may result in
improvement, but the evidence is
inconclusive.