Question

1) In a recent poll, the Gallup organization found that 45% of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. Answer the questions below, showing work. Bare answers are not acceptable. (Showing work means writing the calculator command you used with correct input values in the correct order.)

a) Explain why this is a binomial experiment.

b) Find and interpret the probability that exactly 15 of those surveyed felt the overall state of moral values in the United States is poor.

c) Find and interpret the probability that no more than 10 of those surveyed felt the overall state of moral values in the United States is poor.

d) Find and interpret the probability that more than 16 of those surveyed felt the overall state of moral values in the United States is poor.

e) Find and interpret the probability that 13 or 14 of those surveyed felt the overall state of moral values in the United States is poor.

f) Would it be unusual to find 20 or more adult Americans who believe the overall state of moral values in the United States is poor? Why or why not?

g) Now based on a random sample of 500 adult Americans, compute the mean and standard deviation of the random variable X, the number of adults who believe the overall state of moral values in the United States is poor. Interpret the mean.

h) Would it be unusual to identify 240 adult Americans who believe the overall state of moral values in the United States is poor, based on a random sample of 500 adult Americans? Why? Now using the Normal Approximation to the Binomial, approximate the probability that:

i) Exactly 250 of those surveyed feel the overall state of moral values in the United States is poor.

j) Between 220 and 250, inclusive, of those surveyed feel the overall state of moral values in the United States is poor.

k) At least 260 adult Americans who believe the overall state of moral values in the United States is poor, based on a random sample of 500 adult Americans. Would this be unusual? Why?

2 ) According to www.meretrix.com traffic fatalities occur at a rate of 1.32 fatal accidents per 100 million miles. Find the probability that, during the next 100 million miles, there will be

a) exactly zero accidents. Interpret the results.

b) at least one fatal accident. Interpret the results.

C) more than one fatal accident. Interpret the results.

Answer 1

a)
There is 45% probability of adult americans with 25 sample of americans are given

b)

As per binomial distribution,
P(X=r= nCr * p^r * (1-p)^(n-r)

p = 0.45 , n =25

P(x =15) = 25C15 * 0.45^15 * 0.55^10
= 0.0520

There is 5% probability that exactly 15 of those surveyed felt the overall state of moral values in the United States is poor.

c)
P(x < =10) = P(x =0) + P(x =1) +...... + P(x =10)
= 25C0 * 0.45^0 * 0.55^25 + 25C1 * 0.45^1 * 0.55^124 + ..... + 25C10 * 0.45^10 * 0.55^15

= 0.3843

There is 38.43% probability that no more than 10 of those surveyed felt the overall state of moral values in the United States is poor

d)

P(x > 16) = 1 - P(x < =16)
= 1 - P(x +0) = P(x =1) + ....... + P(x =16)
= 1- (25C0 * 0.45^0 * 0.55^25 + 25C1 * 0.45^1 * 0.55^12 + ..... + 25C16 * 0.45^16 * 0.55^19)

= 1 - 0.9826
= 0.0174

There is 1.74% probability that more than 16 of those surveyed felt the overall state of moral values in the United States is poor

e)

P(x =130 or P(x =14) = P(x =13) + P(x =14)
= 25C13 * 0.45^13 * 0.55^12 + 25C14 * 0.45^14 * 0.55^11
= 0.1236 + 0.0867
= 0.2103
There is 21.03% probability that 13 or 14 of those surveyed felt the overall state of moral values in the United States is poor.