9-7 TESTING FOR GOODNESS OF FIT

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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 Horizons consistent 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 con dence interval on p. 9-55. A manufacturer of interocular lenses is qualifying a new grinding machine and will qualify the machine if the percentage 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 determine if the machine can be quali ed. Use 0.05. (b) Find the P-value for the test in part (a).

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9-56. A researcher claims that at least 10% of all football helmets have manufacturing aws that could potentially cause injury to the wearer. A sample of 200 helmets revealed that 16 helmets contained such defects. (a) Does this nding 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 positively, 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 hypothesis-testing procedure with 0.01.

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

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The table in the end papers of this book (inside front cover) presents a summary of all the single-sample inference procedures from s 8 and 9. The table contains the null hypothesis statement, the test statistic, the various alternative hypotheses and the criteria for rejecting H0, and the formulas for constructing the 100(1 )% two-sided con dence interval.

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TESTING FOR GOODNESS OF FIT

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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 hypothesis 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- t test procedure based on the chi-square distribution.

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CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

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The test procedure requires a random sample of size n from the population whose probability distribution is unknown. These n observations are arranged in a frequency histogram, having k bins or class intervals. Let Oi be the observed frequency in the ith class interval. From the hypothesized probability distribution, we compute the expected frequency in the ith class interval, denoted Ei. The test statistic is

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