Question

Compute​ P(X) using the binomial probability formula. Then determine whether the normal distribution can be used to estimate this probability. If​ so, approximate​ P(X) using the normal distribution and compare the result with the exact probability. nequals=5353​, pequals=0.30.3​, and Xequals=20

Answer 1

Answer and Step-by-step explanation: P(X) calculated by the binomial probability formula is:

P(X) = $\left[\begin{array}{ccc}n\\X\end{array}\right]$.$p^{x}.(1-p)^{n-x}$

P(20) = $\left[\begin{array}{ccc}53\\20\end{array}\right] .(0.3)^{20}.(1-0.3)^{33}$

P(20) = $\frac{53!}{33!.20!}.3.5.10^{-11}.7.7.10^{-6}$

P(20) = 0.0552

To determine whether the normal distribution can be used to estimate this probability, both n.p and n.(1-p) must be greater than 5:

n . p = 53*0.3 = 15.9

n.(1-p) = 53(1-0.3) = 37.1

Since both ARE greater than 5, normal distribution can be used.

To approximate:

mean = n . p = 15.9

standard deviation = $\sqrt{n.p.(1-p)}$ = 3.34

Find the z-score:

z = $\frac{x - mean}{sd}$ = $\frac{20-15.9}{3.34} = 1.23$

z-score = 0.8907

Comparing values:

0.8907 - 0.0552 = 0.8355