Question

The probability that a person likes ice cream is .80. Assume each person is independent. Suppose 1000 people are selected. Use the normal approximation to the binomial to compute the probability that 822 or fewer people selected like ice cream. You will need to compute the mean and standard deviation first.

Answer 1

Answer: 0.9591

Step-by-step explanation:

Given : The probability that a person likes ice cream is p=0.80.

Sample size : n= 1000

Using the normal approximation to the binomial ,

$\mu=np=1000(0.80)=800\\\\\sigma=\sqrt{np(1-p)}\\\\=\sqrt{1000(0.80)(0.20)}\approx12.65$

Let x be the random variable that represents the number of  people like ice cream.

Now, the probability that 822 or fewer people selected like ice cream will be :-

$P(x\leq882)=P(z\leq\dfrac{822-800}{12.65})\\\\=P(z\leq1.74)$

 [∵ $z=\dfrac{x-\mu}{\sigma}$]

$=0.9591$  [using p-value table for z]

Hence, the required probability = 0.9591