6 - 3
Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). (Hint: Draw a graph in each case.)
Find the probability that a randomly selected adult has an IQ between 110 and 120 (referred to as bright normal).
Use this information (based on data from the National Health Survey):
* Men's heights are normally distributed with mean 69.0 in. and standard deviation 2.8 in.
* Women's heights are normally distributed with mean 63.6 in. and standard deviation 2.5 in.
Marine Corps Height Requirement for Men - The U.S. Marine Corpos requires that men have heights between 64 in. and 80 in.
a. Find the percentage of men who meet the height requirements. Are many men denied the opportunity to become a Marine because they do not satisfy the height requirements?
b. If the height requirements are changed so that all men are eligible except the shortest 3% and the tallest 4%, what are the new height requirements?
Aircraft Seat Width - Engineers want to design seats in commercial aircraft so that they are wide enough to fit 99% of all males. (Accommodating 100% of males would require very wide seats that jwould be much too expensive.) Men have hip breadths that are normally distributed with a mean of 14.4 in. and a standard deviation of 1.0 in. (based on anthropometric survey data from Gordon, Clauser, et al.). Find P99. That is, find the hip breadth for men that separates the smallest 99% from the largest 1%.
6 - 4
Statistical Literacy and Critical Thinking
Sampling Distribution of the Proportion - Samples of size n = 1000 are randomly selected from the population of the last digits of telephone numbers, and the proportion of even numbers is found for each sample. What is the distribution of the sample proportions?
Births: Sampling Distributio of Proportion - When 3 births are randomly selected, the sample space is bbb, bbg, bgb, bgg, gbb, gbg, ggb, and ggg. Assume that those 8 outcomes are equally likely. Describe the sampling distribution of the proportion of girls from 3 births as a probability distribution table. Does the mean of the sample proportions equal the proportion of girls in 3 births? (Hine: See Example 5).
Quality Control: Sampling Distribution of Proportion - After constructing a new manufacturing machine, 5 prototype integrated circuit chips are produced and it is found that 2 are defective (D) and 3 are acceptable (A). Assume that two of the chips are randomly selected with replacement from the population.
a. After identifying the 25 different possible samples, find the proportion of defects in each of them, then use a table to describe the sampling distribution of the proportions of defects.
b. Find the mean of the sampling distribution.
c. Is the mean of the sampling distribution (from part (b)) equal to the population proportion of defects? Does the mean of the sampling distribution of proportions always equal the population proportion?
6 - 5
Mensa - Membership in Mensa requires an IQ score above 131.5. Nine candidates take IQ tests, and their summary results indicated that their mean IQ score is 133. (IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.)
a. If 1 person is randomly selected from the general population, find the probability of getting someone with an IQ score of atleast 133.
b. If 9 people are randomly selected, find the probability that their mean IQ score is at least 133.
c. Although the summary results are available, the individual IQ test scores have been lost. Can it be concluded that all 9 candidates have IQ scores above 131.5 so that they are all eligible for Mensa membership?
Designing Motorcycle Helmets - Engineers must consider the breadths of male heads when designing motorcycle helmets. Men have head breadths that are normally distributed with a mean of 6.0 in. and a standard deviation of 1.0 in. (based on anthropometric survey data from Gordon, Churchill, et al.).
a. If one male is randomly selected, find the probability that his head breadth is less than 6.2 inc.
b. The Safeguard Helmet company plans an initial production run of 100 helmets. Find the probability that 100 randomly selected men have a mean head breadth less than 6.2 in.
c. The production manager sees the result from part (b) and reasons that all helmets should be made for men with head breadths less than 6.2 in., because they would fit all but a few men. What is wrong with that reasoning?
Correcting for a Finite Population - In a study of Rey's syndrome, 160 children had a mean age of 8.5 years, a standard deviation of 3.96 years, and ages that approximated a normal distribution (based on data from Holtzhauer and others, American Journal of Diseases of Children, Vol. 140). Assume that 36 of those children are to be randomly selected for a follow-up study.
a. When considering the distribution of the mean ages for groups of 36 children, should central limit theorem (standard error) be corrected by using the finite population correction factor? Explain.
b. Find the probability that the mean age of the follow-up sample group is greater than 10.0 years.
These problems demonstrate the use of z-scores and standard normal tables. Other basic concepts include sample mean calculations and the normal approximation to the binomial distribution.