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.
n=58
p=0.3
x=12
ANSWER A, B, AND C BELOW.
a) For n=58, p= 0.3, and x=12, use the binomial probability formula to find P(X).
b) Approxiamte P(x) using the normal distribution. Use a standard distribution table.
c)By how much to the probabilities differ?

Answer 1

Answer:

a) 0 .0355

b) 0 .0345

c) The probabilities differ by 0.0010

Step-by-step explanation:

We are given the following information:

a) P(Success) = 0.3

n = 58

x = 12

$P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}$

where n is the total number of observations, x is the number of success, p is the probability of success.

Now, we are given n = 58 and x = 12

We have to evaluate:

$P(x = 12) = \binom{58}{12}(0.3)^{12}(1-0.3)^{46} = 0.0355$      

b) Using the normal distribution

n = 12

p = 0.3

q = 1 - p = 1 - 0.3 = 0.7

b) $\text{Mean}=np = 12\times 0.3 = 3.6\\ \text{Standard Deviation} = \sqrt{npq} = \sqrt{12\times 0.3\times 0.7} = 1.58$

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

P(x = 12)

$P( x \leq 12)-P(x < 12$

Calculation the value from standard normal z table, we have,  

$P(x = 12) = 0.0345$    

c) Difference

0.0355 - 0.0345 = 0.0010

Thus, the probabilities differ by 0.0010