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Morality In a recent poll, the Gallup Organization found that45% 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 (a) Find and interpret the probability that exactly 15 of those surveyed feel the state of morals is poor. (b) Find and interpret the probability that no more than 10 of those surveyed feel the state of morals is poor.

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(a) The probability of exactly 15 out of 25 surveyed adult Americans believing that the overall state of moral values in the United States is poor can be calculated using a binomial probability formula.

The formula for calculating the probability of exactly x successes in n trials, where the probability of success is p, is:

P(X = x) = (nCx) * p^x * (1-p)^(n-x)

In this case, n = 25, x = 15, and the probability of an adult American believing that the overall state of moral values in the United States is poor is p = 0.45.

Using this formula, we can calculate the probability:

P(X = 15) = (25C15) * 0.45^15 * (1-0.45)^(25-15)

Using a calculator or statistical software to calculate the combination term (25C15), we find:

P(X = 15) ≈ 0.157

The probability is approximately 0.157. This means that there is a 15.7% chance that exactly 15 out of 25 surveyed adult Americans believe that the overall state of moral values in the United States is poor.

(b) The probability of no more than 10 out of 25 surveyed adult Americans feeling that the state of morals is poor can be calculated by finding the cumulative probability from 0 to 10.

P(X ≤ 10) = P(X = 0) + P(X = 1) + … + P(X = 10)

Using the same binomial probability formula as before, we can calculate each individual probability and sum them up:

P(X ≤ 10) = ∑[P(X = i)] (from i = 0 to i = 10)

Using a calculator or statistical software to calculate each individual probability and sum them up, we find:

P(X ≤ 10) ≈ 0.028

The probability is approximately 0.028. This means that there is a 2.8% chance that no more than 10 out of 25 surveyed adult Americans feel that the state of moral values in the United States is poor.
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