Question

16. Let w be a random variable modeled as a binomial with p = 0.42 and n = 35. a. Find the exact value of P(W = 15) by using the binomial probability formula. b. Find the approximate value of P(14 < W< 16) by using a normal curve approximation. C. Round the probabilities in parts a. and b. to two decimal places and compare.

Answer 1

Solution:

Given that:

$W\sim bin(n,p)$

(a)

P(X) = ncx xpx (1 - p)"-*

Here X=W

$P(W=15)=35_{C_{15}}\times 0.42^{15}\times (1-0.42)^{20}$

$\mathbf{{\color{Green} P(W=15)=0.1346}}$

(b)

Using Normal Approximation to Binomial

$Mean=n\times p$

  

= 35 x 0.42

  $=14.7$

$Variance=n\times p\times Q$

$=35\times 0.42\times 0.58$

$=8.526$

$Standard\ deviation=\sqrt{variance}$

  $=\sqrt{8.526}$

$=2.9199$

$P(14.5<W<15.5)=?$

$Z=\frac{W-\mu}{\sigma}$

$Z=\frac{14.5-14.7}{2.9199}=-0.0685$

$Z=\frac{15.5-14.7}{2.9199}=0.2739$

$P(14.5<Z<15.5)=P(Z<0.2739)-P(Z<-0.0685)$

$P(14.5<Z<15.5)=0.60795-0.4727$

$\mathbf{{\color{Green} P(14.5<Z<15.5)=0.1353}}$

(c)

${\color{Green} both\ probabilities\ are\ not\ approximately\ same. (\because difference\ between\ probabilities\ are\ greater\ than\ 0.05)}$