Unit 5 – Sampling Distributions Name: _______________________
HW 1 – The Normal Distribution and Combining Normal Random Variables Period: _____
1) Consider a set of 9000 scores on a national test that is known to be approximately normally distributed with a mean of 500 and a standard deviation of 90.
(a) What is the probability that a randomly selected student has a score greater than 600?
(b) How many scores are there between 450 and 600?
(c) Megan needs to be in the top 1% of the scores on this test to qualify for a scholarship. What is the minimum score Megan needs?
2) A normal distribution has mean 700 and standard deviation 50. The probability is 0.6 that a randomly selected term from this distribution is above x. What is x?
3) The normal random variable X has a standard deviation of 12. We also know that . Find the mean of the distribution.
4) Gus and Cal go bowling every week. Gus's scores are normally distributed with a mean of 175 pins and a standard deviation of 30 pins. Cal's scores are normally distributed with a mean of 150 pins and a standard deviation of 40 pins. Assume that their scores in any given game are independent. Let G be Gus's score in a random game, C be Cal's score in a random game, and D be the difference between Gus's and Cal's scores where D = G – C. What is the probability that Gus will knock down more pins than Cal?
Unit 5 – Sampling Distributions Name: _______________________
HW 2 – Sampling Distribution of Sample Proportions Period: _____
1) Suppose a large candy machine has 15% orange candies. Imagine taking an SRS of 25 candies from the machine and observing the sample proportion of orange candies.
(a) What is the mean of the sampling distribution of ? Why?
(b) Find the standard deviation of the sampling distribution of . Check to see if the 10% condition is met.
(c) Is the sampling distribution of approximately Normal? Check to see if the Large Counts condition is met.
(d) If the sample size were 225 rather than 25, how would this change the sampling distribution of ?
2) A mail-order company advertises that it ships 90% of its orders within three working days. You select an SRS of 100 of the 5000 orders received in the past week for an audit. The audit reveal that 86 of these orders were shipped on time.
(a) If the company really ship 90% of its orders on time, what is the probability that the proportion in an SRS of 100 orders is 0.86 or less?
(b) A critic seats, "Aha! You claim 90% that in your sample the on time percentage is lower than that. So the 90% claim is wrong." Explain in simple language why your probability calculation in (a) shows that the result of the sample does not refuse the 90% claim.
3) The Harvard College Alcohol Study finds that 67% of college students support efforts to "crack down on underage drinking". Does this result hold at a large local college? To find out, college administrators surveyed an SRS of 100 students and find that 62 suppose a crackdown on underage drinking. What is the probability that the proportion in an SRS of 100 students is 0.62 or less? Does this refute the claim made by Harvard? Explain using the 4 step process.
Unit 5 – Sampling Distributions Name: ________________________
HW 3 – Sampling Distribution of Sample Means Period: ___________
1) The Wechsler Adult Intelligence Scale (WAIS) is a common “IQ test” for adults. The distribution of WAIS scores for people over 16 years of age is approximately normal with mean 100 and standard deviation 15. What is the probability that the average WAIS score of an SRS of 60 people is 105 or higher?
2) The Harvard College Alcohol Study interviewed a SRS of 14,941 college students about their drinking habits. Suppose that half of all college students “drink to get drunk” at least once in a while. That is, p = 0.5. Find the probability of getting a sample proportion between 0.49 and 0.51 in your sample of size 14,941.
3) A company that owns and services a fleet of cars for its sales force has found that the service lifetime of disc brake pads varies from car to car according to a Normal distribution with mean μ = 55,000 miles and standard deviation σ = 4500 miles. The company installs a new brand of brake pads on 8 cars. The average life on the pads on these 8 cars turns out to be 51,800 miles. What is the probability that the sample mean lifetime is 51,800 miles or less if the lifetime distribution is unchanged? (The company takes this probability as evidence that the average lifetime of the new brand of pads is less than 55,000 miles.)
4) Voter registration records show that 68% of all voters in Indianapolis are registered as Republicans. To test whether the numbers dialed by a random digit dialing service device really are random, you use the device to call 150 randomly chosen residential telephones in Indianapolis. Of the registered voters contacted, 73% are registered Republicans. Find the probability of obtaining an SRS of size 150 from the population of Indianapolis voters in which 73% or more are registered Republicans. How well is your random digit device working?
Unit 5 – Sampling Distributions Name: _________________
HW 4 – Sampling Distribution of a Difference in Sample Means & Proportions Period: _____
1) Suppose that there are two large high schools, each with more than 2000 students, in a certain town. At School 1, 70% of students did their homework last night. Only 50% of the students at School 2 did their homework last night. The counselor at School 1 takes an SRS of 100 students and records the proportion that did homework. School 2’s counselor takes an SRS of 200 students and records the proportion that did homework. School 1’s counselor and School 2’s counselor meet to discuss the results of their homework surveys. After the meeting, they both report to their principals that .
(a) Describe the shape, center, and spread of the sampling distribution of
(b) Find the probability of getting a difference of 0.10 or less between the two surveys.
(c) Does the result in part (b) give us reason to doubt the counselor’s reported value?
2) For boys, the average number of absences in the first grade is 15 with a standard deviation of 7; for girls, the average number of absences is 10 with a standard deviation of 6. In a nationwide survey, suppose 100 boys and 50 girls are sampled. What is the probability that the male sample will have at most three more days of absences than the female sample?
3) In a study of heart surgery, one issue was the effect of drugs called beta-blockers on the pulse rate of patients during surgery. The available subjects were divided at random into two groups of 30 patients each. One group received a beta-blocker; the other group received a placebo. The pulse rate of each patient at a critical point during the operation was recorded. The treatment group had a mean pulse rate of 65.2 and standard deviation 7.8. For the control group, the mean pulse rate was 70.3 and the standard deviation was 8.3. What is the probability that the difference in the samples (treatment group – placebo group) is greater than 0?