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Assume the following data were gathered by a manufacturer of a robotics component, in units of days of continuous use until the component fails. There are 60 measurements in this data set. Show a histogram of this data set with 10 bins of equal size, spanning the range from the data minimum to the data maximum.

142, 147, 127, 161, 145, 137, 122, 123, 141, 139, 139

135, 135, 130, 147, 118, 154, 133, 136, 129, 139, 131

143, 130, 160, 127, 127, 145, 144, 155, 128, 124,144

133, 136, 133, 151, 131, 133, 119, 122, 139, 128, 121

142, 136, 148, 136, 121, 131, 125, 120, 123, 145, 140

150, 136, 135, 133, 134

4. A large department store records the number of returns per day in women's dresses for reasons such as wrong size, didn't like it, color, and so on. The manager of the women's dresses department rexalls from her statistics class in college that the Poisson distribution might describe such events. A random sample of 150 days is taken. The number of returns per day in the sample as well as the observed frequencies are show below.

Numbers of returns per day Observed frequency

0 21

1 37

2 44

3 30

4 14

>4 4

The manager tests the hypothesis that returns per day are Poisson distributed with a population mean equal to 1.90. Her significance level is 0.05. State the null and alternative hypothesis. What are the appropriate degrees of freedom? Identify the critical value. Calculate the expected frequencies.

5. In the diagram below, events A, B and C are shown with numbers in various regions of the graph indicating how many sample points lie in each. For example, the number 3 in the top left of the diagram indicates that there are 3 sample points in B that are not also in either A or C.

A= 9, 0, 10,12 B= 3, 1, 10, 12 C= 5, 1, 0, 10 Out of all A, B and C= 15

a. Are the events A and B independent? b. Are the events for the intersection of A and Bc, intersection of C and Bc mutually exclusive?

6. Under which of the following conditions would it be appropriate to use a binomial random variable? In each case, explain why your answer is correct.

a. A department will interview 10m candidates for a position, and call back for second interviews those who answer the interview questions to the satisfaction of all the interviewers. They hope to call back at least 3, but past experience suggests an average of about 1 call back per 4 interviews.

b. A factory posts on the wall the number of days since its last safety infraction or injury. In the past year, the factory has had a safety infraction or injury on 6 different days. The factor is interested in the number of days that can be expected to elapse without an injury.

c. Fifteen of a doctor’s patients have the same ailment. Studies have shown that about 86.5% of patients with this aliment resond to a certain drug. The doctor prescribes the drug to all 15, but the number who will respond in this case is, of course, not known in advance.

10. An elementary school teacher learned that 40% of school children have at least three cavities. The teacher has 30 students in his class. How many students would he expect in his class to hace at least three cavities? What is the standard deviation? Using the appropriate approximation, determine P(X > 20); that is, the probability that more than 20 students in his class will have 3 cavities.

Assume the following data were gathered by a manufacturer of a robotics component, in units of days of continuous use until the component fails. There are 60 measurements in this data set. Show a histogram of this data set with 10 bins of equal size, spanning the range from the data minimum to the data maximum.

142, 147, 127, 161, 145, 137, 122, 123, 141, 139, 139

135, 135, 130, 147, 118, 154, 133, 136, 129, 139, 131

143, 130, 160, 127, 127, 145, 144, 155, 128, 124,144

133, 136, 133, 151, 131, 133, 119, 122, 139, 128, 121

142, 136, 148, 136, 121, 131, 125, 120, 123, 145, 140

150, 136, 135, 133, 134

4. A large department store records the number of returns per day in women's dresses for reasons such as wrong size, didn't like it, color, and so on. The manager of the women's dresses department rexalls from her statistics class in college that the Poisson distribution might describe such events. A random sample of 150 days is taken. The number of returns per day in the sample as well as the observed frequencies are show below.

Numbers of returns per day Observed frequency

0 21

1 37

2 44

3 30

4 14

>4 4

The manager tests the hypothesis that returns per day are Poisson distributed with a population mean equal to 1.90. Her significance level is 0.05. State the null and alternative hypothesis. What are the appropriate degrees of freedom? Identify the critical value. Calculate the expected frequencies.

5. In the diagram below, events A, B and C are shown with numbers in various regions of the graph indicating how many sample points lie in each. For example, the number 3 in the top left of the diagram indicates that there are 3 sample points in B that are not also in either A or C.

A= 9, 0, 10,12 B= 3, 1, 10, 12 C= 5, 1, 0, 10 Out of all A, B and C= 15

a. Are the events A and B independent? b. Are the events for the intersection of A and Bc, intersection of C and Bc mutually exclusive?

6. Under which of the following conditions would it be appropriate to use a binomial random variable? In each case, explain why your answer is correct.

a. A department will interview 10m candidates for a position, and call back for second interviews those who answer the interview questions to the satisfaction of all the interviewers. They hope to call back at least 3, but past experience suggests an average of about 1 call back per 4 interviews.

b. A factory posts on the wall the number of days since its last safety infraction or injury. In the past year, the factory has had a safety infraction or injury on 6 different days. The factor is interested in the number of days that can be expected to elapse without an injury.

c. Fifteen of a doctor’s patients have the same ailment. Studies have shown that about 86.5% of patients with this aliment resond to a certain drug. The doctor prescribes the drug to all 15, but the number who will respond in this case is, of course, not known in advance.

10. An elementary school teacher learned that 40% of school children have at least three cavities. The teacher has 30 students in his class. How many students would he expect in his class to hace at least three cavities? What is the standard deviation? Using the appropriate approximation, determine P(X > 20); that is, the probability that more than 20 students in his class will have 3 cavities.

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