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=47​,

pequals=0.5

and

Xequals=33

Answer 1

Answer)

As there are fixed number of trials and probability of each and every trial is same and independent of each other

Here we need to use the binomial formula

P(r) = ncr*(p^r)*(1-p)^n-r

Ncr = n!/(r!*(n-r)!)

N! = N*n-1*n-2*n-3*n-4*n-5........till 1

For example 5! = 5*4*3*2*1

Special case is 0! = 1

P = probability of single trial = 0.5

N = number of trials = 47

R = desired success = 33

P(33) = 0.00242752502

Now by normal approximation

First we need to check the conditions of normality that is if n*p and n*(1-p) both are greater than 5 or not

N*p = 23.5

N*(1-p) = 23.5

Both the conditions are met so we can use standard normal z table to estimate the probability

P(33) = p(32.5<x<33.5) = p(x<33.5) - p(x<32.5)

Z = (x - mean)/s.d

Mean = n*p = 23.5

S.d = √{n*p*(1-p)} = 3.42782730020

P(x<33.5)

Z = (33.5 - 23.5)/3.42782730020 = 2.92

From z table, p(z<2.92) = 0.9982

P(x<32.5)

Z = (32.5-23.5)/3.42782730020 = 2.63

From z table, P(z<2.63) = 0.9957

Required probability is 0.9982 - 0.9957 = 0.0025

Answer 2

n=73, p=0.83, and x=58

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.

Answer 3

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.

n=73, p=0.83, and x=58