Question

Consider the probability that no less than 94 out of 149 people have not been in a car accident. Assume the probability that a given person has not been in a car accident is 61%.Approximate the probability using the normal distribution. Round your answer to four decimal places.

Answer 1

Solution

Step-by-step explanation:

Binomial probability distribution

Probability of exactly x successes on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

$E(x)\text{ = np}$

The standard deviation of the binomial distribution is:

$\sigma\text{ = }\sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

$z\text{ = }\frac{x-\mu}{\sigma}$

p = 61% = 0.61

n = 149

$\begin{gathered} \mu\text{ = np = 149 }\times0.61\text{ = 90.89} \\ \\ \sigma\text{ = }\sqrt{np(1-p)}\text{ = }\sqrt{90.89\times0.39}\text{ = 5.95} \end{gathered}$$\begin{gathered} z\text{ = }\frac{x-\mu}{\sigma}\text{ = }\frac{94-90.89}{5.95} \\ \\ z\text{ = 0.523} \end{gathered}$