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INSTRUCTIONS: I. Assignments are to be handed in prior to beginning of class on the due date. II. For written questions, show all of your work. No credit will be given for answers without justificatio
INSTRUCTIONS: I. Assignments are to be handed in prior to beginning of class on the due date. II. For written questions, show all of your work. No credit will be given for answers without justification. III. Do not use MINITAB for a question unless it specifically says to do so. IV. No late assignments will be accepted. 1. During off hours, cars arrive at a highway service station at an average rate of 5 cars per 10 minutes. The number of cars arriving at the service station is distributed according to a Poisson distribution. (a) What is the probability that during the next minute three cars will arrive? (b) What is the probability that during the next five minutes three cars will arrive? (c) What is the probability that during the next five minutes at least two cars will arrive? 2. Suppose that the times required for a cable company to fix cable problems in its customers’ homes are uniformly distributed between 40 minutes and 65 minutes. (a) What is the probability that a randomly selected cable repair visit will take at least 50 minutes? (b) What is the probability that a randomly selected cable repair visit will take at most 55 minutes? (c) What is the expected length of the repair visit? (d) What is the standard deviation? 3. In a certain study, it is found that the life of a light bulb is exponentially distributed with a mean life of 1,200 hours. (a) What is the probability that a randomly selected bulb will last more than 1,400 hours? (b) What is the probability that the bulb will last less than 1,000 hours? (c) What is the median life of a bulb? 4. A professor grades his students on a normal distribution with a mean score of 70 and a standard deviation of 10. If there are 125 students in his class, about how many scores are between 75 and 85? 5. Suppose that the waiting time for a pizza to be delivered to an individual’s residence has been found to be normally distributed with a mean of 30 minutes and a standard deviation of 8 minutes. What is the probability that a randomly selected individual will have a waiting time: (a) Between 15 and 45 minutes? (b) At least 10 minutes? (c) No more than 22 minutes? 6. In an exam, an instructor plans to give those students in the top 5% an A. If the average score on this exam is 75 with a standard deviation of 8, then what would be the minimum score for an A? Assume normality for the distribution of scores. 7. It has been reported that the average time to download the home page from a government website was 0.9 seconds. Suppose that the download times were normally distributed with a standard deviation of 0.3 seconds. If random samples of 36 download times are selected: (a) Describe the shape of the sampling distribution. How was this determined? (b) Calculate the mean of the sampling distribution of the sample mean. (c) Calculate the standard deviation of the sampling distribution of the sample mean. (d) 80% of the sample means will be between what two values that are symmetrically distributed around the population mean? (e) What is the probability that the sample mean will be less than 0.80 seconds? 8. MINITAB question: Normal approximation to binomial (a) Use MINITAB to find cumulative probabilities P(X ≤ x) for a binomial distribution with parameters n = 30 and p = 0.2. Show your MINITAB output. To find the cumulative probabilities, follow the sequence: In column c1, put values of x; Choose Calc > Probability Distributions > Binomial; Specify the parameters in the box; Put c1 in Input column. (b) Compute the probability P(4 ≤ X ≤ 8) using the MINITAB output. (c) Use the normal approximation to the binomial distribution to calculate the probability P(4 ≤ X ≤ 8) by hand. How does this probability compare to that in part (b)? 9. MINITAB question: Normal distribution. Generate a random sample of 200 observations from a normal distribution with mean 5 and standard deviation 2 and store the data in column c2. To generate the data, follow the sequence: Calc > Random Data > Normal; Input 200 in Generate window; Put c2 in Store In Column(s) window; 5 for mean and 2 for standard deviation. (a) Draw a histogram of these data. Comment on the shape of the distribution. (b) Using MINITAB, calculate the sample mean ¯x and the sample standard deviation s of the 200 observations in c2. Are these values what you expected
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