Question

Suppose that a recent poll found that 40% of adults in a certain country believe that the overall state of moral values is poor. If a survey of a random sample of 20 adults in this country is conducted in which they are asked to disclose their feelings on the overall state of moral values.

complete parts (a) through (e) below.
Click here to view the table for the binomial probability distribution.
Click here to view the table for the cumulative binomial probability distribution.
(a) Find and interpret the probability that exactly 12 of those surveyed feel the state of morals is poor.
The probability that exactly 12 of those surveyed feel the state of morals is poor is
(Round to four decimal places as needed

Answer 1

0.0355 = 3.55% probability that exactly 12 of those surveyed feel the state of morals is poor. This means that there is a low probability of finding this exact value.

Explanation:

For each adult surveyed, there are only two possible outcomes. Either they think that the state of morals is poor, or they do not. One adult's opinion is independent of other adults. This means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

40% of adults in a certain country believe that the overall state of moral values is poor.

This means that
`p = 0.4`

Sample of 20 adults

This means that
`n = 20`

(a) Find and interpret the probability that exactly 12 of those surveyed feel the state of morals is poor.

This is P(X = 12).

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 12) = C_(20,12).(0.4)^(12).(0.6)^(8) = 0.0355`

0.0355 = 3.55% probability that exactly 12 of those surveyed feel the state of morals is poor. This means that there is a low probability of finding this exact value.