(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.