Question

A gallup survey indicated that 72% of 18- to 29-year-olds, if given choice, would prefer to start their own business rather than work for someone else. A random sample of 600 18-29 year-olds is obtained today. What is the probability that no more than 70% would prefer to start their own business?

Answer 1

The probability that no more than 70% would prefer to start their own business is 0.1423.

Explanation:

We are given that a Gallup survey indicated that 72% of 18- to 29-year-olds, if given choice, would prefer to start their own business rather than work for someone else.

Let
`\hat p` = sample proportion of people who prefer to start their own business

The z-score probability distribution for the sample proportion is given by;

Z =
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where, p = population proportion who would prefer to start their own business = 72%

n = sample of 18-29 year-olds = 600

Now, the probability that no more than 70% would prefer to start their own business is given by = P(
`\hat p`
`\leq` 70%)

P(
`\hat p`
`\leq` 70%) = P(
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }`
`\leq`
`\frac{0.70-0.72}{\sqrt{(0.70(1-0.70))/(600) } }` ) = P(Z
`\leq` -1.07) = 1 - P(Z < 1.07)

= 1 - 0.8577 = 0.1423

The above probability is calculated by looking at the value of x = 1.07 in the z table which has an area of 0.8577.