SAT College Project Evaluation



Normal Approximation to the Binomial Name ___________________________________

1. Are attitudes toward shopping changing? Sample surveys show that fewer people enjoy shopping than in the past. A

survey asked a nationwide random sample of 2500 adults if they agreed or disagreed that “I like buying new clothes,

but shopping is often frustrating and time-consuming.” The population that the poll wants to draw conclusions about

is all U.S. residents aged 18 and over. Suppose that in fact 60% of all adult U.S. residents would say “agree” if

asked the same question.

(a) Verify that a binomial situation applied.

(b) Verify that you can use the normal distribution to approximate the binomial distribution.

(c) Find the mean and standard deviation of the number of people who would say agree.

(d) Find the probability that at least 1520 people agree?

(e) Find the probability at most 1430 people agree?

(f) Find the probability that between 1430 and 1520 people agree?

(g) Find the probability that the sample proportion of people who agree will be more than .65.

(h) Find the probability that the sample proportion of people who agree will lies within .05 of the true

proportion.

2. According to Benford’s Law, the probability that the first digit of a randomly chosen invoice is a 1 or a 2 is 0.477.

You examine 90 invoices from a vendor and find that 29 have first digits 1 or 2. If Benford’s Law holds, the count

of 1’s and 2’s will have the binomial distribution with n = 90 and p = .477. Too few 1’s and 2’s suggests fraud.

(a) Find the mean and standard deviation of the number of 1’s and 2’s.

(b) Find the probability that 29 or fewer of the invoices follow Benford’s Law.

(c) Find the probability that 47 or more of the invoices follow Benford’s Law.

(d) Find the probability that the sample proportion of the number of 1’s and 2’s is less than 0.4.

(e) Find the probability that the sample proportion of the number of 1’s and 2’s is between 0.45 and 0.50.

3. In 1998, Mark McGuire of the St. Louis Cardinals hit 70 home runs, a new major league record. Was this feat as

surprising as most of us thought? In the three seasons before 1998, McGuire hit a home run in 11.6% of his times at

bat. He went to bat 509 times in 1998. McGwire’s home run count in 509 times at bat has approximately the

binomial distribution with n = 509 and p = 0.116.

(a) What is the mean and standard deviation of the number of home runs he will hit in 509 times at bat?

(b) Find the probability that he will hit 70 or more home runs.

(c) Find the probability that he will hit between 50 and 70 home runs.

(d) Find the probability that the sample proportion of home runs will be less than .10.

(e) Find the probability that the sample proportion of home runs will be more than .15.

4. Many local polls of public opinion use samples of size 400 to 800. Consider a poll of 400 adults in Richmond that

the question, “Do you approve of President Bush’s response to the World Trade Center terrorist attacks in September

2001?” Suppose we know that President Bush’s approval rating on this issue nationally is 92% a week after the

incident.

(a) What is the mean and standard deviation of the number of people who approve of President Bush?

(b) Find the probability that at most 358 of the 400 people sampled will answer “yes” to the question.

(c) Find the probability that between 340 and 370 people sampled will answer “yes” to the question.

(d) Find the probability that the sample proportion of people answering “yes” is more than .95.

(e) Find the probability that the sample proportion of people answering “yes” is within .03 of the actual

proportion.

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