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

Explanation:

Step1: Identify binomial parameters

Let $n = 41$ (number of trials, i.e., number of Americans selected), $p=0.66$ (probability of success, i.e., probability of an American being a homeowner), and $q = 1 - p=1 - 0.66 = 0.34$. 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 probability for part a

For $k = 26$, $C(41,26)=\frac{41!}{26!(41 - 26)!}=\frac{41!}{26!15!}$. [ \begin{align*} P(X = 26)&=C(41,26)\times(0.66)^{26}\times(0.34)^{15}\ &=\frac{41!}{26!15!}\times(0.66)^{26}\times(0.34)^{15}\approx0.1379 \end{align*} ]

Step3: Calculate probability for 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 or a TI - 84 Plus calculator's binomialcdf function), $P(X\leq29)\approx0.7984$.

Step4: Calculate probability for 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, $P(X\geq26)\approx0.7540$.

Step5: Calculate probability for part d

$P(25\leq X\leq29)=\sum_{k = 25}^{29}C(41,k)\times(0.66)^{k}\times(0.34)^{41 - k}=P(X\leq29)-P(X\leq24)$. Using a binomial - probability calculator, $P(25\leq X\leq29)\approx0.5564$.

Answer:

a. $0.1379$ b. $0.7984$ c. $0.7540$ d. $0.5564$