66% of all americans are home owners. if 41 americans are randomly selected, find the following…

Explanation:

Step1: Identify binomial distribution parameters

Let $n = 41$ (number of trials/sample size), $p=0.66$ (probability of success - being a homeowner), $q = 1 - p=0.34$. The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times q^{n - k}$, where $C(n,k)=\frac{n!}{k!(n - k)!}$.

Step2: Calculate part a

For $k = 26$: $C(41,26)=\frac{41!}{26!(41 - 26)!}=\frac{41!}{26!15!}$ $P(X = 26)=C(41,26)\times(0.66)^{26}\times(0.34)^{15}$ $C(41,26)=\frac{41!}{26!15!}=\frac{41\times40\times\cdots\times16}{15!}\approx1.14\times10^{11}$ $P(X = 26)\approx1.14\times10^{11}\times(0.66)^{26}\times(0.34)^{15}\approx0.1370$

Step3: Calculate part b

$P(X\leq29)=\sum_{k = 0}^{29}C(41,k)\times(0.66)^{k}\times(0.34)^{41 - k}$. Using a binomial - probability calculator or software (e.g., Excel's BINOM.DIST function: =BINOM.DIST(29,41,0.66,TRUE)), we get $P(X\leq29)\approx0.7984$.

Step4: Calculate part c

$P(X\geq26)=1 - P(X\leq25)=\ 1-\sum_{k = 0}^{25}C(41,k)\times(0.66)^{k}\times(0.34)^{41 - k}$. Using a binomial - probability calculator or software (e.g., =1 - BINOM.DIST(25,41,0.66,TRUE)), we get $P(X\geq26)\approx0.7540$.

Step5: Calculate part d

$P(25\leq X\leq29)=\sum_{k = 25}^{29}C(41,k)\times(0.66)^{k}\times(0.34)^{41 - k}=BINOM.DIST(29,41,0.66,TRUE)-BINOM.DIST(24,41,0.66,TRUE)\approx0.5564$

Answer:

a. $0.1370$ b. $0.7984$ c. $0.7540$ d. $0.5564$