Question

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.

Answer 1

Explanation:

a) The experiment is a binomial experiment because of the following reasons

(i) Binomial experiment has a fixed number of trials, from the statement the experiment has a fixed number of trial, the trial is to check whether the overall state of moral value in the United State is poor or not and a fixed sample of 25 is being considered

(ii) In a binomial experiment, each trial is independent of one another

(iii) In a binomial experiment, there are only two possible outcomes, which are success or failure, yes or no, good or bad etc. from the statement the outcomes are poor or rich

(iv) The probability of each outcome remain constant throughout the experiment , from the experiment above, the probability of "poor" is 0.45 which is denoted p and that of "rich is 0.55 , which is denoted with q

(b) Using the binomial distribution formula: N = 25 , p = 0.45 , q = 0.55 P(X=r)=n∁r p^r q^(n-r)

P(X=15)=25∁15 〖(0.45)〗^15 〖(0.55)〗^10

P(X=15)=3268760×0.0000062833×0.00253295162

P(X=15)=0.052

Interpretation: The probability is low, it is not up to average, it means that the significance of the claim is not high

(c) The probability of no more than 10 implies P(X≤10)

P(X≤10)= P(X=0)+P(X=1)+P(X=2)+⋯+P(X=10)

=0.0000003229+0.00000660476+0.00006484674+0.00040676591+0.00183044658+0.00629008008+0.01715476386+0.03809694312+0.07013300892+0.10838737742+0.14188893044

P(=0.38)

Interpretation: The probability is also low, it means the effect is not also high on the claim