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Unformatted text preview: 243 Review Exercises which has, approximately, a standard normal distribution. When the pop
ulation is normal, the same statistic has a 2‘ distribution with n — 1 degrees
of freedom. . Understand the interpretation of a level or test. If the null hypothesis is
true, before the data are collected, the probability is a that the experiment
will produce observations that lead to the rejection of the null hypothesis.
Consequently, after many independent experiments, the proportion that lead
to rejection of the null hypothesis will be nearly 07. Don’ts . Don’t routinely apply the statistical procedures above if the sample is not
random but collected from convenient units or the data show a trend in time. Review Exercises 7.76 Specify the null hypothesis and the alternative hypoth 7.77 7.78 7.79 7.80 7.81 esis in each of the following cases. (a) An engineer hopes to establish that an additive will
increase the viscosity of an oil. (b) An electrical engineer hopes to establish that a
modiﬁed circuit board will give a computer a
higher average operating speed. With reference to the example on page 19, ﬁnd a
95% conﬁdence interval for the mean strength of the
aluminum alloy. While performing a certain task under simulated
weightlessness, the pulse rate of 32 astronaut trainees
increased on the average by 26.4 beats per minute with
a standard deviation of 4.28 beats per minute. What can
one assert with 95% conﬁdence about the maximum
error if f = 26.4 is used as a point estimate of the true
average increase in the pulse rate of astronaut trainees
performing the given task? With reference to the preceding exercise, construct a
95% conﬁdence interval for the true average increase
in the pulse rate of astronaut trainees performing the
given task. It is desired to estimate the mean number of days of
continuous use until a certain kind of computer will
ﬁrst require repairs. If it can be assumed that a = 48
hours, how large a sample is needed so that one will
be able to assert with 90% conﬁdence that the sample
mean is off by at most 10 hours? A sample of 12 camshafts intended for use in gasoline
engines has an average eccentricity of 1.02 and a stan
dard deviation of 0.044 inch. Assuming the data may
be treated as a random sample from a normal popula
tion, determine a 95% conﬁdence interval for the actual
mean eccentricity of the camshaft. 7.82 7.83 7.84 7.85 7.86 In order to test the durability of a new paint, a highway
department had test strips painted across heavily trav—
eled roads in 15 different locations. If on the average
the test strips disappeared after they had been crossed
by 146,692 cars with standard deviation of 14,380 cars,
construct a 99% conﬁdence interval for the true average
number of cars it takes to wear off the paint. Assume
a normal population. Referring to Exercise 7.82 and using 14,380 as an es
timate of a, ﬁnd the sample size that would have been
needed to be able to assert with 95% conﬁdence that
the sample mean is off by at most 10,000. [Hint First
estimate n1 by using 2 = 1.96, then use t0.025 for n] ——1
degrees of freedom to obtain a second estimate n2, and
repeat this procedure until the last two values of n thus
obtained are equal] A laboratory technician is timed 20 times in the perfor—
mance of a task, getting f = 7.9 and s = 1.2 minutes.
If the probability of a Type I error is to be at most
0.05, does this constitute evidence against the null hy
pothesis that the average time is less than or equal to
7.5 minutes? Suppose that in the drying time example on page 222,
n is changed from 36 to 50 while the other quantities
remain [1.0 = 20, a = 2.4, and a = 0.03. Find (a) the new dividing line of the test criterion; (b) the probability of Type II errors for the same values
of ,u as shown in the table on page 238. In an airpollution study, ozone measurements were
taken in a large California city at 5.00 RM. The eight
readings (in parts per million) were 7.9 11.3 6.9 12.7 13.2 8.8 9.3 10.6 Assuming the population sampled is normal, construct
a 95% conﬁdence interval for the corresponding true
mean. 244 Chapter 7 Inferences Concerning a Mean 7.87 An industrial engineer concerned with service at a large medical clinic recorded the duration of time from the
time a patient called until a doctor or nurse returned
the call. A sample of size 180 calls had a mean of
1.65 hours and a standard deviation of 0.82. (a) Obtain a 95% conﬁdence interval for the popula—
tion mean of time to return a call. (b) Does ,u lie in your interval obtained in part (a)?
Explain.
(C) In a long series or repeated experiments. with new random samples collected for each experi
ment, what proportion of the resulting conﬁdence
intervals will contain the true population mean?
Explain your reasoning. 7.88 Refer to Exercise 7.87. (a) Perform a test with the intention of establishing
that the mean time to return a call is greater than
1.5 hours. Use or = 0.05. 7.89 (b) In light of your conclusion in part (a), what error
could you have made? Explain in the context of
this problem. (c) In a long series of repeated experiments, with new
random samples collected for each experiment,
what proportion of the resulting tests would reject
the null hypothesis if it prevailed? Explain your reasoning. The compressive strength of parts made from a
composite material are known to be nearly normally
distributed. A scientist, using the testing device for
the ﬁrst time, obtains the tensile strength (psi) of
20 specimens 95 102 105 107 109 110 111 112 114 115
134 135 136 138 139 141 142 144 150 155 shown in Figure 7.12. Should the scientist report the
95% conﬁdence interval based on the t—distribution?
Explain your reasoning. O O O O .000 O. m 00 O. O O O I  I I  L  I
Figure 7. I 2 90 100 110 120 130 140 150 160
Dot diagram of tensile strength Strength (psi)
Key Terms
Alternative hypothesis 225 Large sample Z test 232 Point estimation 204
Classical theory of testing Level of signiﬁcance 225 Power 238
hypotheses 226 Likelihood function 217 P value 231
Composite hypothesis 227 Maximum likelihood estimator 217 Simple hypothesis 227
Conﬁdence 208 Neyman—Pearson theory 226 Tail probability 230
Conﬁdence interval 210 Null hypothesis 227 Twosided alternative 225
Conﬁdence limits 210 One sample I test 233 Two—sided criterion 226 Critical regions 229 Onesided alternative 225
Critical Values 230 Onesided criterion 226
Degree of conﬁdence 210 onesided test 226
Estimated standard error 204 onetaﬂed test 226 Hat notation 205
Interval estimate 209 curve 238 Operating characteristic (0C) Two—sided test 226
Two—tailed test 226
223
223
Unbiased estimator Type I error
Type 11 error
205 266 Chapter 8 Comparing Two Treatments Review Exercises 8.26 8.27 8.28 8.29 8.30 8.31 With reference to Exercise 2.64, test that the mean charge of the electron is the same for both tubes. Use
or = 0.05. With reference to the previous exercise, ﬁnd a 90%
conﬁdence interval for the difference of the two
means. Two chemical additives for drying paint are to be com
pared. Five spray cans are prepared using Additive A
and six are prepared using Additive B. Then 11 differ—
ent boards are sprayed, one can per board. (a) The response is the time in minutes for the surface
to dry, and the summary statistics are Sample Standard size Mean deviation Additive A 5 16.3 2 7
12.1 1 1 Should you pool or not pool the estimates of
variance in order to conduct a test of hypothe
ses that is intended to show that there is a
difference in means? Explain how you would
proceed. (b) Conduct the test for part (a) using 01 = 0.05. (c) Describe how you would randomize the assign
ment of paints when conducting this experiment. With reference to the example on page 14, test that the
mean copper content is the same for both heats. With reference to the previous exercise, ﬁnd a 90%
conﬁdence interval for the difference of the two
means. Random samples are taken from two normal popula—
tions with 01 = 10.8 and 02 = 14.4 to test the null
hypothesis u] — [22 = 53.2 against the alternative
hypothesis m — M2 > 53.2 at the level of signiﬁ—
cance a = 0.01. Determine the common sample size
n = n1 = mg that is required if the probability of
not rejecting the null hypothesis is to be 0.09 when
,LL} — M2 = 66.7. Key Terms 8.32 8.33 8.34 8.35 With reference to the example on page 254, ﬁnd a 90%
conﬁdence interval for the difference of mean strengths
of the alloys (a) using the pooled procedure;
(b) using the large samples procedure. How would you randomize, for a two sample test, in
each of the following cases? (a) Twenty cars are available for a mileage study and
you want to compare a modiﬁed spark plug with
the regular. (b) A new oven will be compared with the old. Fifteen
ceramic specimens are available for baking. With reference to part (a) of Exercise 8.33, how would
you pair and then randomize for a paired test? Two samples in C1 and C2 can be analyzed using the
MINITAB commands Dialog box: Stat > Basic Statistics > 2Sample t
Type C l in First C2 in Second.
Click Samples in different columns. Click Assume equal variances.
Click OK. If you do not click Assume equal variances, the
SmithSatterthwaite test is performed. The output relating to the example on page 254 is TWO SAMPLE T FOR ALLOY 1 VS ALLOY 2 N MEAN STDEV SE MEAN
ALLOY l 58 70.70 1.80 0.24
ALLOY 2 27 76.13 2.42 0.47 95 PCT Cl FOR MU ALLOY l — MU ALLOY 2:
(~6.36, —4.50) T TEST MU ALLOY l = MU ALLOY 2 (VS NE):
T = ——11.58 P = 0.000 DF = 83.0 Perform the test for the data in Exercise 8.10. Matched pairs
Matched pairs t test
Paired t test Pairing 264 259
260 Randomization
260 Treatment 245 Pooled estimator of variance 252 264 Smith—Satterthwaite test 256 25 3
249 Two sample I test
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 Fall '08
 Mendell
 Normal Distribution, Standard Deviation, Null hypothesis

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