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) =
`(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 =
`√(n.p.(1-p))` = 3.34

Find the z-score:

z =
`(x - mean)/(sd)` =
`(20-15.9)/(3.34) = 1.23`

z-score = 0.8907

Comparing values:

0.8907 - 0.0552 = 0.8355