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Unformatted text preview: Provide a practical interpretation of this interval.
(0) Calculate a 99% lower conﬁdence bound on the mean Compare this bound with the Inner bound of the twoside conﬁdence interval and discuss why they are diﬁerent. 8—37. The compressive strength of concrete is being teste by a civil engineer. He tests 12 specimens and obtains th
followmg data. 2216 2237 2249 2204
2225 2301 228] 2263 
2318 2255 2275 2295 (a) Check the assumption that compressive strength is normally
I distributed. Include a graphical display in your answer.
(b) Construct a 95% twosided conﬁdence interval on the
mean strength. 
(0) Construct a 95% IoWer conﬁdence bound on the mean
strength. Compare this'bound with the lower bound of the twosided conﬁdence interval and discuss why they are
different. 838. A machine produces metal rods used in an automobile
suspension system. A random sample of 15 rods is selected, and the diameter is measured. The resulting data (in millime
ters) are as follOWS: 8.24 8.25 8.20 8.23 8.24
5 8.21 8.26 8.26 8.20 8.25
:3 8.23 8.23 8.19 8.28 8.24
I: (a) Check the assumption ofnormality for rod diameter. (b) Calculate a 95% twosided conﬁdence interval on mean
rod diameter. (c) Calculate a 95% upper conﬁdence bound on the mean. Compare this bound with the upper bound of the twosided
conﬁdence interval and discuss why they are different. ﬁdence intervals. 266 CHAPTER 8 STATISTICAL mrERVALs FOR A SINGLE s (b) Calculate a 99% conﬁdence interval on the mean it. 8—3 CONFIDENCE INTERVAL ON THE VARIANCE AND A 7
, STANDARD DEVIATION OF A NORMAL DISTRIBUTION Sometimes conﬁdence intervals on the population varianceor standard deviation are needed.
When the population is modeled by a normal distribution, the tests and intervals described in
this section are applicable. The following result provides the basis of constructing these con F‘wm Ml“ have. 839. An article in Computers a Electrical Engineering
[“Parailel Simulation of Cellular Neural Networks” (1996, Vol. 22,
pp. 61—84)] considered the speedup of cellular neural new/0mg
(CNN) for a parallel generalpurpose computing architecture
based on six transputers in different areas. The data follow: 3.775302 3.350679 4.217981 4.030324 4.639692
4.139665 4.395575 4.824257 4.268119 4.584193
4.930027 4.315973 4.600101 (a) Is there evidence to support the assumption that speedup
of CNN is normally distributed? Include a graphical dis—
play in your answer. (b) Construct a 95% twosided conﬁdence interval on the
mean speedup. (c) Construct a 95% lower conﬁdence bound on the mean
seed—up. The wall thickness of 25 glass 2liter bottles was mea
sured by a qualitycontrol engineer. The sample mean was
3 = 4.05 millimeters, and the sample standard deviation was
s = 0.08 millimeter. Find a 95% lower conﬁdence bound for
mean wall thickness. Interpret the interval you have obtained.
841. An article in Nuclear Engineering International
(February 1988, p. 33) describes several characteristics of fuel
rods used in a reactor owned by an electric utility in Norway. Measurements on the percentage of enrichment of 12 rods
were reported as follows: 2.94 3.00 2.90 2.75 3.00 2.95
2.90 2.75 2.95 2.82 2.81 3.05
(21) Use a normal probability plot to check the normality as
sumption. (b) Find a 99% twosided conﬁdence interval on the mean per
centage of enrichment. Are you comfortable with the state ment that the mean percentage of enrichment is 2.95%?
Why? Xz Distribution Let X1, X2, . .. , X" be a random sample from a normal distribution with mean u and
variance oz, and let S2 be the sample variance. Then the random variable has a chisquare (38) distribution with ii — 1 degrees of freedom. _ {n — l).512 ‘
_ —_._02 _ _ . (8—17) X2 298 CHAPTER 9 TESTS OF HYPOTIIESES FOR A SINGLE SAMPLE 9—2. A semiconductor manufacturer collects data from a
new tool and conducts a hypothesis test with the null hypothe
sis that a critical dimension mean width equals 100 nm. The
conclusion is to not reject the null hypothesis. Does this result
provide strong evidence that the critical dimension mean
equals 100 um? Explain. 9—3. The standard deviation of critical dimension thickness in semiconductor manufacturing is cr = 20 nm. (a) State the null and alternative hypotheses used to demon—
strate that the standard deviation is reduced. (b) Assume that thebprevious test does not reject the null
hypothesis. Does this result provide strong evidence that
the standard deviation has not been reduced? Explain. 9—4. The mean pulloff force of a connector depends on cure e. (a) State the null and alternative hypotheses used to demon
strate that the pulloff force is below 25 newtons. (h) Assume that the previous test does not reject the null hypoth
esis. Does this result provide strong evidence that the pulloff
force is greater than or equal to 25 newtons? Explain. 9—5. A textile ﬁber manufacturer is hivestigating a new drap ery yam, which the company claims has a mean thread elonga— tion of 12 kilograms with a standard deviation of 0.5 kilo grams. The company wishes to test the hypothesis H0: p. = 12 against H1: p._< 12, using a random sample of four specimens. (a) What is the type I error probability if the critical region is
deﬁned as i < 11.5 kilograms? (b) Find B for the case where the true mean elongation is
11.25 kilograms. (c) Find B for the case where the true mean is 11.5 kilograms. 9—6. Repeat Exercise 95 using a sample size of n = 16 and
the same critical region. 9—7. In Exercise 9—5, ﬁnd the boundary of the critical region
it" the type I error probability is (a) 0L=0.01andn=4 (c) ot=0.01andn=16 (b) or = 0.05 andn = 4 ((1)0 = 0.05 andn :16 9—8. In Exercise 95, calculate the probability of a type II error if the true mean elongation is l 1.5 kilograms and (5001 = 0.05 andn : 4 (b) or = 0.05 andn =16 (c) Compare the values of B calculated in the previous parts.
What conclusion can you draw? 9—9. In Exercise 95, calculate the P—value if the observed
statistic is
(a) i=11.25 (h)i=11.0 (c) 5:11.75 9—10. The heat evolved in calories per gram of a cement  mixture is approximately normally distributed. The mean is thought to be 100 and the standard deviation is 2. We wish to test H0: p. = 100 versus H1: p. at 100 with a sample ofri! = 9 specimens. (21) If the acceptance region is deﬁned as 98.5 S 2? 5 101.5,
ﬁnd the type I error probability 0:. (b) Find B for the caso where the true mean heat evolved is 103. HWtHO (c) Find B for the case where the true mean heat evolved is
105. This value of B is smaller than the one found in part
(b) above. Why? 9—11. Repeat Exercise 9—10 using a sample size of n = 5
and the same acceptance region. 912.. In Exercise 910, ﬁnd the boundary of the critical region if the type I error probability is (a)a=0.01andn=9 (c)or=0.01andn=5 (b)a=0.05andn=9 (d)o.=0.05andn=5 9—13. In Exercise 910, calculate the probability of a type 11 error if the true mean heat evolved is 103 and (a) ct = 0.05 andn = 9 (b) ot=0.05andn=5 (c) Compare the values of B calculated in the previous parts.
What conclusion can you draw? 914. In Exercise 910, calculate the P—value if the observed statistic is (a) f=98 (b) i=101 (c) f=102 9—15. A consumer products company is formulating a new shampoo and is interested in foam height (in millimeters). Foam height is approximately normally distributed and has a standard deviation of 20 millimeters. The company wishes to test H0: p. = 175 millimeters versus H1: p. > 175 millime ters, using the results of n : 10 samples. (a) Find the type 1 error probabilityo if the critical region is
f > 185. I i (b) What is the probability of type 11 error if the true mean
foam height is 185 millimeters? (c) Find B for the true mean of 195 millimeters. 916. Repeat Exercise 915 assuming that the sample size is r
n = 16 and the boundary of the critical region is the same. 9—17. In Exercise 915, ﬁnd the boundary of the critical
region if the type I error probability is
(a) (1:001 andn= 10 (c) ot=0.01andn= 16
(b or=0.05andn=10 (d)u=0.05andn=16
@ In Exercise 915, calculate the probability of a type II
eor if the true mean foam height is 185 millimeters and
(a) 01': 0.05 andn =10
(b) (I: (1.05de =16
(0) Compare the values of [3 calculated in the previous parts.
What conclusion can you draw?
9—19. In Exercise 915, calculate the Pvalue if the observed
statistic is .
(a) 3: =180 (b) i = 190 (c) i =‘ll70
9—20. A manufacturer is interested in the output voltage of a‘
power supply used in a PC. Output voltage is assumed to be
normally distributed, With standard deviation 0.25 volt, and
the manufacturer wishes to test H0: p. = 5 volts against
H12“. #3 5 volts= using 7: = 8 units.
(a) The acceptance region is 4.85 5 3: S 5.15. Find the value
of or.
(b) Find the power of the test for detecting a true mean output
voltage of 5.1 volts. ' 92 rssrs ON THE MEAN or ANoaMAL DISTRIBUTION. VARIANCE KNOWN 299 9.21. Rework Exercise 920 when the sample size is 16 and
the boundaries of the acceptance region do not change. What
impact does the change in sample size have on the results of
parts (a) and (b)? @ In Exercise 920, ﬁnd the boundary of the critical re gion if the type I error probability is
(a) 0L=0.01811dn=8 (c) or=0.01andn= 16
(b)or=0.05andn=8 (d) o=0.05andn=16
9.23. In Exercise 920, calculate the Pvalue if the observed
statistic is
(a) i=5.2 Ga) 2:43 (C) i=5.l
924. In Exercise 9—20, calculate the probability of a type 11
error if the true mean output is 5.05 volts and
(a) at = 0.05 andn =10
(b) or: 0.05 andn =16 (c) Compare the values of B calculated in the previous parts. What conclusion can you draw? 9—25. The proportion of adults living in Tempe, Arizona, who are college graduates is estimated to hep = 0.4. To test this hypothesis, a random sample of 15 Tempe adults is selected. If the number of college graduates is between 4 and 8, the hypothesis will be accepted; otherwise, we will conclude that p a5 0.4 . (a) Find the type I error probability for this procedure, assum
ing thatp = 0.4. ' (b) Find the probability of committing a type II error if the
true proportion is really p = 0.2. 9—26. The proportion of residents in Phoenix favoring the building of toll roads to complete the freeway system is
believed to be p = 0.3. If a random sample of 10 residents shows that l or fewer favor this proposal, we will conclude that p < 0.3. (2.) Find the probability of type I error if the true proportion is
p = 0.3. . (b) Find the probability of committing a type H error with this
procedure ifp = 0.2. (c) What is the power of this procedure if the true proportion
is p = 0.2? 9—27. A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels yearround to reduce air pollution. If more than 400 voters respond posi tively, we will conclude that more than 60% of the voters favor the use of these fuels. (a) Find the probability of type I error if exactly 60% of the
voters favor the use of these fuels. (b) What is the type II error probability B if 75% of the voters
favor this action? '
Hint: use the normal approximation to the binomial. 9'28° If we plot the probability of accepting Hozu = no versus various values of p. and connect the points with a smooth curve, we obtain the operating characteristic curve (or the 0C curve) of the test procedure. These curves are used extensively in industrial applications of hypothesis testing to display the sensitivity and relative performance of the test. When the true mean is really equal to no, the probability of accepting H0 is 1 i or. (a) Construct an 0C curve for Exercise 9~15, using values
ofthe true mean u of 178, 181, 184, 187, 190, 193, 196,
and 199. (b) Convert the 0C curve into a plot of the power function of
the test. 92 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN In this section, we consider hypothesis testing about the mean u. of a single normal population
where the variance of the population 0*2 is known. We will assume that a random sample X1,
X2, ... , X, has been taken ﬁom the population. Based on our previous discussion, the sample
mean I? is an unbiased point estimator of u with variance 02/11. 9—2.,1 Hypothesis Tests on the Mean Suppose that we wish to test the hypotheses where no is a speciﬁed constant. We have a random sample X 1 ,X2, . ,Xn from a normal pop . (97) ulation. Since f has a normal distribution (i.e., the sampling distribution of f is normal) r’r‘r'— ——. . L. 1: «Sis"= .Qzl!#m‘—.Aba%'r—éwwmHs—zhaswsemnwa;+eeaari‘newinie—.w.n' 308 converted into a large—sample test procedure for unknown 0'2 that is valid regardless ofthe
form of the distribution of the population. This largesample test relies on the central limit the.
orem just as the large—sample conﬁdence interval on p. that was presented in the previous
chapter did. Exact treatment of the case where the population is normal, 0'2 is unknovm, anﬁ n
is small involves use of the 1‘ distribution and will be deferred until Section 93. ' EXERCISES FOR SECTION 92 929. State the null and alternative hypothesis in each case. (a) A hypothesis test will he used to potentially provide evi
dence that the populationmean is greater than 10. (b) A hypothesis test will be used to potentially provide evi—
dence that the population mean is not equal to 7. (c) A hypothesis test will be used to potentially provide evi
dence that the population mean is less than 5. 930. A hypothesis will be used to test that a population mean
equals 7 against the alternative that the population mean does
not equal 7 with known variance 0'. What are the critical Values
for the test statistic 20 for the following signiﬁcance levels? (a) 0.01 (b) 0.05 (c) 0.10 931. A hypothesis will be used to test that a population
mean equals 10 against the alternative that the population mean
is greater than 10 with known variance 0. What is the critical
value for the test statistic 20' for the following sigmﬁcance levels?
(a) 0.01 (b) 0.05 (c) 0.10 932. A hypothesis will be used to test that a population
mean equals 5 against the alternative that the population mean
is less than 5 with known variance 0'. What is the critical value
for the test statistic Zn for the following signiﬁcance levels?
a 0.01 (b) 0.05 (c) 0.10 a For the hypothesis test H0: p. = 7 against H]: n. =15 7
and variance known, calculate the Pvalue for each of the
following test statistics. ' (a) 20 = 2.05 V (b) 20 2 .—1.84 (c) 20 = 0.4 9—34. Forthe hypothesis testHg: u = 10 against H1: is > 10
and variance known, calculate the P—value for each of the
following test statistics. ' .
(a) 20 = 2.05 (b) 20 = —1.84 (c) 20 = 0.4 9—35. For the hypothesis test H0: p. = 5 against H1: pt < 5
and variance known, calculate the Pvalue for each of the
following test statistics. (3 Z : 20 = (C) In : @ Output from a soﬁware package is given below: OneSample Z:
Test ofmu = 35 = vs not = 35
The assumed standard deﬂation = 1.8 Variable. N StDev SE Mean Z P
x _ 25 1.475 ? ? ? Mean
35.710 (a) Fill in the missing items. What conclusions would you draw? (b) Is this a onesided or a twosided test? CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE (0) Use the normal table and the above data to construct a r 95% twosided CI on the mean. (d) What would the Pvalue be if the alternative hypothesis is H]: u, > 35?
937. Output from a software package is given below: OneSample Z: Test of mu = 20 vs > 20
The assumed standard deviation = 0.75 Variable N Mean StDev SE Mean Z P
x 10 19.889 ? (a) Fill in the missing items. What conclgsions would you draw?
(b) Is this a one—sided or a twosided test? (c) Use the normal table andrthe above data to construct a 95% twosided CI 011 the mean. (d) What would the Pvalue be‘if the alternative hypothesis is H}: p. as 20?
9'38. Output from a software package is given below: _ OneSample Z: Test ofmu = 14.5 vs > 14.5 The assumed standard deviation = 1.1 Variable N Mean StDev SE Mean Z P
x 16 15.016 1.015 '1 ‘ :2 ‘? (a) Fill in the missing items. What conclusions would you draw? (b) Is this a one—sided or a twosided test? (c) Use the normal table and the above data to construct a
95% lower bound on the mean. (d) What would the P—valuc be if the alternative hypothesis is
H]: a we 14.5? ' 9'39. Output from a software package is given below: OneSample Z: Test of mu = 99 vs > 99
The assumed standard deviation = 2.5 Variable N Mean StDev SE Mean Z P
x 12 '100.039 2.365 ? 1.44 0.075 _____‘—_—_4—___—— (a) Fill in the missing items. What conclusions would you draw?
(b) Is this a onesided or a twosided test? 0.237 ? '2. mm. _;4al‘.%_.uu.amammmmmm. ‘ ' ' ‘39 31"?" 1. ' 9c2 TESTS ON THE MEAN OF. A NORMAL DISTRlBUTION. VARIANCE KNOWN 309 (9) If the hypothesis had been H0: u. = 98 versus H .1 u. > 98,
would yOu reject the null hypothesis at the 0.05 level of
signiﬁcauCe? Can you answer this without referring to the
normal table? (d) Use the normal table and the above data to construct a
95% lower bound on the mean. (3) What w0uld the Pvalue be if the alternative hypothesis is
H1: u. at 99? . 9,40. The mean water temperature downstream from a power plant cooling tower discharge pipe should be no more than 100°F. Past experience has indicated that the standard
deviation of temperature is 2°F. The water temperature is
measured on nine randomly chosen days, and the average
temperature is found to be 98°F. 
' (a) Is there evidence that the water temperature is acceptable
'at or = 0.05?
(b) What is the P—value for this test? (c) What is the probability of accepting the null hypothesis ' at or _= 0.05 if the water has a true mean temperature of
104°F? a
941. A manufacturer produces crankshafts for an automo
bile engine. The wear of the crankshaft after 100,000 miles
(0.0001 inch) is of interest because it is likely to have an
impact on warranqr claims. A random sample of n = 15 shaﬁs
is tested and i = 2.78. It is known that 0' : 0.9 and that wear
is normally distributed.
(a) Test H0: p. = 3 versus H,: p. at 3 using or = 0.05.
i (b) What is the power of this test if p. = 3.25?
(c) What sample size would be required to detect a true mean
of 3.75 if we wanted the power to be at least 0.9? 942. A melting point test of a = 10 samples of a binder used in manufacturing ‘a rocket propellant resulted in i = 154.2”F. Assume that the melting point is normally dis tributed with 0' = 15°F. (a) Test H0: p. = 155 versus H1: p. at 155 usingoi = 0.01. (b) What is the Pvalue for this test? (0) What is the Berror if the true mean is p. = 150? (d) :What value of u would be required if we want B < 0.1
when p. = 150? Assume that or = 0.01. The life in hours of a battery is known to be approxi
ately normally distributed with standard deviation 0' = 1.25
hours. A random sample of 10 batteries has a mean life of
J? = 40.5 hours. ' (a) Is there evidence to support the claim that battery life
exceeds 40 hours? Use a = 0.05. (b) What is the Pvalue for the test in part (a)? (c) What is the B—error for the test in part (a) if the true mean
life is 42 hours? (d) What sample size would be required to ensure that B does
not exceed 0.10 if the true meanllife is 44 hours? (8) Explain how you could answer the question in part (a)
by calculating an appropriate conﬁdence bound on life. 944. An engineer who is studying the tensile strength of a steel alloy intended for use in golf club shafts knows that tensile strength is approximately normally distributed with o = 60 psi. A random sample of 12 specimens has a mean tensile strength of it = 3450 psi. (a) Test the hypothesis that mean strength is 3500 psi. Use
or = 0.01. . (b) What is the smallest level of signiﬁcance at which you
would be willing to reject the null hypothesis? (c) What is the Berror for the test in part (a) if the true mean
is 3470? (d) Suppose that we wanted to reject the null hypothesis with
probability at least 0.8 if mean strength u. = 3500. What
sample size should be used? ' ...
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