Page 367 - Applied Statistics And Probability For Engineers

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9-7 TESTING FOR GOODNESS OF FIT 315

9-53. Consider the defective circuit data in Exercise 8-48. 9-56. A researcher claims that at least 10% of all football
(a) Do the data support the claim that the fraction of defective helmets have manufacturing flaws that could potentially cause
units produced is less than 0.05, using 0.05? injury to the wearer. A sample of 200 helmets revealed that 16
(b) Find the P-value for the test. helmets contained such defects.
9-54. An article in Fortune (September 21, 1992) claimed (a) Does this finding support the researcher’s claim? Use
that nearly one-half of all engineers continue academic studies 0.01.
beyond the B.S. degree, ultimately receiving either an M.S. or (b) Find the P-value for this test.
a Ph.D. degree. Data from an article in Engineering Horizons 9-57. A random sample of 500 registered voters in Phoenix
(Spring 1990) indicated that 117 of 484 new engineering is asked if they favor the use of oxygenated fuels year-round
graduates were planning graduate study. to reduce air pollution. If more than 315 voters respond posi-
(a) Are the data from Engineering Horizons consistent with tively, we will conclude that at least 60% of the voters favor
the claim reported by Fortune? Use 0.05 in reaching the use of these fuels.
your conclusions. (a) Find the probability of type I error if exactly 60% of the
(b) Find the P-value for this test. voters favor the use of these fuels.
(c) Discuss how you could have answered the question in part (b) What is the type II error probability
if 75% of the voters
(a) by constructing a two-sided confidence interval on p. favor this action?
9-55. A manufacturer of interocular lenses is qualifying a 9-58. The advertized claim for batteries for cell phones is set
new grinding machine and will qualify the machine if the per- at 48 operating hours, with proper charging procedures. A study
centage of polished lenses that contain surface defects does of 5000 batteries is carried out and 15 stop operating prior to 48
not exceed 2%. A random sample of 250 lenses contains six hours. Do these experimental results support the claim that less
defective lenses. than 0.2 percent of the company’s batteries will fail during the
(a) Formulate and test an appropriate set of hypotheses to de- advertized time period, with proper charging procedures? Use a
termine if the machine can be qualified. Use 0.05. hypothesis-testing procedure with 0.01.
(b) Find the P-value for the test in part (a).

9-6 SUMMARY TABLE OF INFERENCE PROCEDURES
FOR A SINGLE SAMPLE

The 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 , and the formulas for constructing the 100(1 )% two-sided confidence
0
interval.

9-7 TESTING FOR GOODNESS OF FIT

The 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 plotting and illustrated how it was applied in the case of a normal distribution.
In this section, we describe a formal goodness-of-fit test procedure based on the chi-square
distribution.