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CHAPTER 7 Tests of Hypotheses and Significance
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Tests involving the chi-square distribution
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7.84. The standard deviation of the breaking strengths of certain cables produced by a company is given as 240 lb. After a change was introduced in the process of manufacture of these cables, the breaking strengths of a sample of 8 cables showed a standard deviation of 300 lb. Investigate the significance of the apparent increase in variability, using a significance level of (a) 0.05, (b) 0.01. 7.85. The annual temperature of a city is obtained by finding the mean of the mean temperatures on the 15th day of each month. The standard deviation of the annual temperatures of the city over a period of 100 years was 16 Fahrenheit. During the last 15 years a standard deviation of annual temperatures was computed as 10 Fahrenheit. Test the hypothesis that the temperatures in the city have become less variable than in the past, using a significance level of (a) 0.05, (b) 0.01. 7.86. In Problem 7.77 a sample of 20 electric light bulbs revealed a standard deviation in the lifetimes of 150 hours. Would you conclude that this is unusual Explain.
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7.87. Two samples consisting of 21 and 9 observations have variances given by s2 16 and s2 8, respectively. Test the 1 2 hypothesis that the first population variance is greater than the second at a (a) 0.05, (b) 0.01 level of significance. 7.88. Work Problem 7.87 if the two samples consist of 60 and 120 observations, respectively. 7.89. In Problem 7.82 can we conclude that there is a significant difference in the variability of the pH values for the two solutions at a 0.10 level of significance
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7.90. Referring to Problem 7.63, determine the probability of accepting the hypothesis that there are equal proportions of red and blue marbles when the actual proportion p of red marbles is (a) 0.6, (b) 0.7, (c) 0.8, (d) 0.9, (e) 0.3. 7.91. Represent the results of Problem 7.90 by constructing a graph of (a) b vs. p, (b) (1 b) vs. p. Compare these graphs with those of Problem 7.25 by considering the analogy of red and blue marbles to heads and tails, respectively.
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7.92. In the past a certain type of thread produced by a manufacturer has had a mean breaking strength of 8.64 oz and a standard deviation of 1.28 oz. To determine whether the product is conforming to standards, a sample of 16 pieces of thread is taken every 3 hours and the mean breaking strength is determined. Find the (a) 99.73% or 3s (b) 99% and (c) 95% control limits on a quality control chart and explain their applications. 7.93. On the average about 3% of the bolts produced by a company are defective. To maintain this quality of performance, a sample of 200 bolts produced is examined every 4 hours. Determine (a) 99%, (b) 95% control limits for the number of defective bolts in each sample. Note that only upper control limits are needed in this case.
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7.94. Fit a binomial distribution to the data of Table 7-24. Table 7-24 x f 0 30 1 62 2 46 3 10 4 2
CHAPTER 7 Tests of Hypotheses and Significance
7.95. Fit a normal distribution to the data of Problem 5.98. 7.96. Fit a normal distribution to the data of Problem 5.100. 7.97. Fit a Poisson distribution to the data of Problem 7.44, and compare with the fit obtained by using the binomial distribution. 7.98. In 10 Prussian army corps over a period of 20 years from 1875 throughout 1894, the number of deaths per army corps per year resulting from the kick of a horse are given in Table 7-25. Fit a Poisson distribution to the data. Table 7-25 x f 0 109 1 65 2 22 3 3 4 1
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7.99. In 60 tosses of a coin, 37 heads and 23 tails were observed. Test the hypothesis that the coin is fair using a significance level of (a) 0.05, (b) 0.01. 7.100. Work Problem 7.99 using Yates correction. 7.101. Over a long period of time the grades given by a group of instructors in a particular course have averaged 12% A s, 18% B s, 40% C s, 18% D s, and 12% F s. A new instructor gives 22 A s, 34 B s, 66 C s, 16 D s, and 12 F s during two semesters. Determine at a 0.05 significance level whether the new instructor is following the grade pattern set by the others. 7.102. Three coins were tossed together a total of 240 times, and each time the number of heads turning up was observed. The results are shown in Table 7-26 together with results expected under the hypothesis that the coins are fair. Test this hypothesis at a significance level of 0.05. Table 7-26 0 heads Observed Frequency Expected Frequency 24 1 head 108 2 heads 95 3 heads 23
7.103. The number of books borrowed from a public library during a particular week is given in Table 7-27. Test the hypothesis that the number of books borrowed does not depend on the day of the week, using a significance level of (a) 0.05, (b) 0.01. Table 7-27 Mon. Number of Books Borrowed 135 Tues. 108 Wed. 120 Thurs. 114 Fri. 146
7.104. An urn consists of 6 red marbles and 3 white ones. Two marbles are selected at random from the urn, their colors are noted, and then the marbles are replaced in the urn. This process is performed a total of 120 times, and the results obtained are shown in Table 7-28. (a) Determine the expected frequencies. (b) Determine at a level of significance of 0.05 whether the results obtained are consistent with those expected.
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