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. if a survey 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 (g).
in 100 trials of this experiment, we expect about 6 to result in no more than 5 adults who feel the state of morals is poor.
(round to the nearest whole number as needed.)
(e) using the binomial distribution, find and interpret the probability that more than 14 of those surveyed feel the state of morals is poor.
the probability that more than 14 of those surveyed feel the state of morals is poor is 0.0064.
(round to four decimal places as needed.)
interpret the results.
in 100 trials of this experiment, we expect about 1 to result in more than 14 adults who feel the state of morals is poor.
(round to the nearest whole number as needed.)
(f) using the binomial distribution, find and interpret the probability that 9 or 10 believe the state of morals is poor.
the probability that 9 or 10 believe the state of morals is poor is
(round to four decimal places as needed.)

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n = 20\), \(p=0.45\), \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Calculate \(P(X = 9)\)

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Using a calculator, \(P(X = 9)\approx0.1593\)

Step3: Calculate \(P(X = 10)\)

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Using a calculator, \(P(X = 10)\approx0.1762\)

Step4: Calculate \(P(X=9\ or\ X = 10)\)

By the addition rule for mutually - exclusive events (\(X = 9\) and \(X = 10\) are mutually exclusive in a binomial distribution), \(P(X=9\ or\ X = 10)=P(X = 9)+P(X = 10)\)
\(P(X=9\ or\ X = 10)\approx0.1593 + 0.1762=0.3355\)

Answer:

The probability that 9 or 10 believe the state of morals is poor is \(0.3355\)