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- From: Mathematics, Statistics and Probability
- Posted on: Wed 27 Jan, 2016
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1. Assume you have a data set from a normally distributed random variable. Answer the following questions about it.
a. Will the random variable be discrete, continuous, or neither? How do you know?
b. Will the data be qualitative or quantitative? How do you know?
2. 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
3. A university has been tracking the percentage of alumni giving to its annual fund each year for the past 10 years.
The data is given here.
14%, 13%, 15%, 21%, 19%, 24%, 25%, 28%, 25%. 31%
Answer the following questions about this data.
a. what are its mean and median?
b. What is the procedure for using mean and median to determine whether the data is skewed, and if so, in what direction?
c. Apply the procedure you described to the mean and median computed in part a.
4. A nurse is considering going back to graduate school to earn a Ph.D. in biochemistry. One of the schools she visits tells her that the average time to earn the degree she’s considering is 5.5 years. Show that this statement is not sufficiently precise by giving two different explanations of what it might mean.
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. Are the events A and B independent?
b. Are the events A and Bc and 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 10 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 of injury. In the past year, the factory has had a safety infraction or injury on 6 different days. The factory 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 ailment respond 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.
7. The mean time for a racecar driver’s crew to perform a pit stop is 13.2 seconds, with a standard deviation of 0.9 seconds. To maintain his current lead, the driver needs a pit stop in 12.5 seconds or less. Assuming this random variable is normally distributed, what is the probability of the driver getting the pit stop in a short enough time to maintain is lead?
8. From a sample of size 175, the sample mean is x bar=54.37 and sample standard deviation is s= 7.07.
a. Construct a 95% confidence interval for the population mean and show your work.
b. Explain how your work in part a would have been different if the sample size had been only 12 instead.
9. A random sample from the population of registered voters in California is to be taken and then surveyed about an upcoming election. What sample size should be used to guarantee a sampling error of 3% or less when estimating p at the 95%confidence level?
10. a. Explain the two conditions required for a valid large-sample test of hypothesis for a mean.
b. Explain the two different possible outcomes of a test of hypothesis.
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