9-7 TESTING FOR GOODNESS OF FIT 3159-6 SUMMARY TABLE OF INFERENCE PROCEDURES
FOR A SINGLE SAMPLEThe table in the end papers of this book (inside front cover) presents a summary of all the
single-sample inference procedures from Chapters 8 and 9. The table contains the null
hypothesis statement, the test statistic, the various alternative hypotheses and the criteria
for rejecting H 0 , and the formulas for constructing the 100(1 )% two-sided confidence
interval.9-7 TESTING FOR GOODNESS OF FITThe hypothesis-testing procedures that we have discussed in previous sections are designed
for problems in which the population or probability distribution is known and the hypotheses
involve the parameters of the distribution. Another kind of hypothesis is often encountered:
we do not know the underlying distribution of the population, and we wish to test the hypoth-
esis that a particular distribution will be satisfactory as a population model. For example, we
might wish to test the hypothesis that the population is normal.
We have previously discussed a very useful graphical technique for this problem called
probability plottingand illustrated how it was applied in the case of a normal distribution.
In this section, we describe a formal goodness-of-fit testprocedure based on the chi-square
distribution.9-53. Consider the defective circuit data in Exercise 8-48.
(a) Do the data support the claim that the fraction of defective
units produced is less than 0.05, using 0.05?
(b) Find the P-value for the test.
9-54. An article in Fortune(September 21, 1992) claimed
that nearly one-half of all engineers continue academic studies
beyond the B.S. degree, ultimately receiving either an M.S. or
a Ph.D. degree. Data from an article in Engineering Horizons
(Spring 1990) indicated that 117 of 484 new engineering
graduates were planning graduate study.
(a) Are the data from Engineering Horizonsconsistent with
the claim reported by Fortune? Use 0.05 in reaching
your conclusions.
(b) Find the P-value for this test.
(c) Discuss how you could have answered the question in part
(a) by constructing a two-sided confidence interval on p.
9-55. A manufacturer of interocular lenses is qualifying a
new grinding machine and will qualify the machine if the per-
centage of polished lenses that contain surface defects does
not exceed 2%. A random sample of 250 lenses contains six
defective lenses.
(a) Formulate and test an appropriate set of hypotheses to de-
termine if the machine can be qualified. Use 0.05.
(b) Find the P-value for the test in part (a).9-56. A researcher claims that at least 10% of all football
helmets have manufacturing flaws that could potentially cause
injury to the wearer. A sample of 200 helmets revealed that 16
helmets contained such defects.
(a) Does this finding support the researcher’s claim? Use
0.01.
(b) Find the P-value for this test.
9-57. A random sample of 500 registered voters in Phoenix
is asked if they favor the use of oxygenated fuels year-round
to reduce air pollution. If more than 315 voters respond posi-
tively, we will conclude that at least 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 if 75% of the voters
favor this action?
9-58. The advertized claim for batteries for cell phones is set
at 48 operating hours, with proper charging procedures. A study
of 5000 batteries is carried out and 15 stop operating prior to 48
hours. Do these experimental results support the claim that less
than 0.2 percent of the company’s batteries will fail during the
advertized time period, with proper charging procedures? Use a
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