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are equal. Suppose we decide to use k 8 cells. For the standard normal distribution, the intervals that divide the scale into eight equally likely segments are [0, 0.32), [0.32, 0.675) [0.675, 1.15), [1.15, ) and their four mirror image intervals on the other side of zero. For each interval pi 1 8 0.125, so the expected cell frequencies are Ei npi 100(0.125) 12.5. The complete table of observed and expected frequencies is as follows:
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Class Interval 4.948 4.986 5.014 5.040 5.066 5.094 5.132 Totals x x x x x x x x 4.948 4.986 5.014 5.040 5.066 5.094 5.132
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Observed Frequency oi 12 14 12 13 12 11 12 14 100
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The boundary of the rst class interval is x 1.15s 4.948. The second class interval is 3x 1.15s, x 0.675s2 and so forth. We may apply the eight-step hypothesis-testing procedure to this problem. 1. 2. The variable of interest is the form of the distribution of power supply voltage. H0: The form of the distribution is normal.
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9-7 TESTING FOR GOODNESS OF FIT
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3. H1: The form of the distribution is nonnormal. 4. 0.05 5. The test statistic is
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Since two parameters in the normal distribution have been estimated, the chi-square statistic above will have k p 1 8 2 1 5 degrees of freedom. 2 11.07. Therefore, we will reject H0 if 2 0 0.05,5 Computations:
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Ei 2 2 114 12.52 2 12.5 p 114 12.52 2 12.5
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2 Conclusions: Since 2 0.64 11.07, we are unable to reject H0, and there 0 0.05,5 is no strong evidence to indicate that output voltage is not normally distributed. The P-value for the chi-square statistic 2 0.64 is P 0.9861. 0
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EXERCISES FOR SECTION 9-7
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9-59. Consider the following frequency table of observations on the random variable X. Values Observed Frequency 0 24 1 30 2 31 3 11 4 4 de ned as the number of calls during that one-hour period. The relative frequency of calls was recorded and reported as Value Relative Frequency Value Relative Frequency 5 0.067 11 0.133 6 0.067 12 0.133 8 0.100 13 0.067 9 0.133 14 0.033 10 0.200 15 0.067
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(a) Based on these 100 observations, is a Poisson distribution with a mean of 1.2 an appropriate model Perform a goodness-of- t procedure with 0.05. (b) Calculate the P-value for this test. 9-60. Let X denote the number of flaws observed on a large coil of galvanized steel. Seventy-five coils are inspected and the following data were observed for the values of X: Values Observed Frequency 1 1 2 11 3 8 4 13 5 11 6 12 7 10 8 9
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(a) Does the assumption of a Poisson distribution seem appropriate as a probability model for this data Use 0.05. (b) Calculate the P-value for this test. 9-62. Consider the following frequency table of observations on the random variable X: Values Frequency 0 4 1 21 2 10 3 13 4 2
(a) Does the assumption of the Poisson distribution seem appropriate as a probability model for this data Use 0.01. (b) Calculate the P-value for this test. 9-61. The number of calls arriving at a switchboard from noon to 1 PM during the business days Monday through Friday is monitored for six weeks (i.e., 30 days). Let X be
(a) Based on these 50 observations, is a binomial distribution with n 6 and p 0.25 an appropriate model Perform a goodness-of- t procedure with 0.05. (b) Calculate the P-value for this test.
CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE
9-63. De ne X as the number of under lled bottles from a lling operation in a carton of 24 bottles. Sixty cartons are inspected and the following observations on X are recorded: Values Frequency 0 39 1 23 2 12 3 1
Vehicles per Minute 40 41 42 43 44 45 46 47 48 49 50 51 52
Observed Frequency 14 24 57 111 194 256 296 378 250 185 171 150 110
Vehicles per Minute 53 54 55 56 57 58 59 60 61 62 63 64 65
Observed Frequency 102 96 90 81 73 64 61 59 50 42 29 18 15
(a) Based on these 75 observations, is a binomial distribution an appropriate model Perform a goodness-of- t procedure with 0.05. (b) Calculate the P-value for this test. 9-64. The number of cars passing eastbound through the intersection of Mill and University Avenues has been tabulated by a group of civil engineering students. They have obtained the data in the adjacent table: (a) Does the assumption of a Poisson distribution seem appropriate as a probability model for this process Use 0.05. (b) Calculate the P-value for this test.