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 of a random sample of 15 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). (b) using the binomial distribution, determine the values of n and p. n = 15 (type an integer or a decimal. do not round.) p = 0.45 (type an integer or a decimal. do not round.) (c) using the binomial distribution, find and interpret the probability that exactly 8 of those surveyed feel the state of morals is poor. the probability that exactly 8 of those surveyed feel the state of morals is poor is 0.1647 (round to four decimal places as needed.) interpret the results. in 100 trials of this experiment, we expect about 16 to result in exactly 8 adults who feel the state of morals is poor. (round to the nearest whole number as needed.) (d) using the binomial distribution, find and interpret the probability that no more than 4 of those surveyed feel the state of morals is poor. the probability that no more than 4 of those surveyed feel 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 \(C(n,k)=\frac{n!}{k!(n - k)!}\). For \(P(X\leqslant4)\), we need to calculate \(P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4)\)

Step2: Calculate \(P(X = 0)\)

\(n = 15\), \(p=0.45\), \(k = 0\)
\(C(15,0)=\frac{15!}{0!(15 - 0)!}=1\)
\(P(X = 0)=1\times(0.45)^{0}\times(1 - 0.45)^{15-0}=(0.55)^{15}\approx0.0001\)

Step3: Calculate \(P(X = 1)\)

\(k = 1\)
\(C(15,1)=\frac{15!}{1!(15 - 1)!}=\frac{15!}{1!14!}=15\)
\(P(X = 1)=15\times(0.45)^{1}\times(0.55)^{14}=15\times0.45\times(0.55)^{14}\approx0.0016\)

Step4: Calculate \(P(X = 2)\)

\(k = 2\)
\(C(15,2)=\frac{15!}{2!(15 - 2)!}=\frac{15\times14}{2\times1}=105\)
\(P(X = 2)=105\times(0.45)^{2}\times(0.55)^{13}=105\times0.2025\times(0.55)^{13}\approx0.0093\)

Step5: Calculate \(P(X = 3)\)

\(k = 3\)
\(C(15,3)=\frac{15!}{3!(15 - 3)!}=\frac{15\times14\times13}{3\times2\times1}=455\)
\(P(X = 3)=455\times(0.45)^{3}\times(0.55)^{12}=455\times0.091125\times(0.55)^{12}\approx0.0318\)

Step6: Calculate \(P(X = 4)\)

\(k = 4\)
\(C(15,4)=\frac{15!}{4!(15 - 4)!}=\frac{15\times14\times13\times12}{4\times3\times2\times1}=1365\)
\(P(X = 4)=1365\times(0.45)^{4}\times(0.55)^{11}=1365\times0.04100625\times(0.55)^{11}\approx0.0710\)

Step7: Sum up the probabilities

\(P(X\leqslant4)=P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4)\)
\(P(X\leqslant4)\approx0.0001 + 0.0016+0.0093+0.0318+0.0710=0.1138\)

Answer:

\(0.1138\)