Question

A courier service company wishes to estimate the proportion of people in various states that will use its services. Suppose the true proportion is 0.04. If 349 are sampled, what is the probability that the sample proportion will be less than 0.05

Answer 1

The probability that the sample proportion will be less than 0.05 is 0.80511.

Explanation:

We are given that a courier service company wishes to estimate the proportion of people in various states that will use its services.

Suppose the true proportion is 0.04 and 349 are sampled.

Let
`\hat p` = sample proportion

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

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

where,
`\hat p` = sample proportion

n = sample size = 349

p = population proportion = 0.04

Now, the probability that the sample proportion will be less than 0.05 is given by = P(
`\hat p` < 0.05)

P(
`\hat p` < 0.05) = P(
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` <
`\frac{0.05-0.04}{\sqrt{(0.05(1-0.05))/(349) } }` ) = P(Z < 0.86) = 0.80511

The above probability is calculated by looking at the value of x = 0.86 from the z table which has an area of 0.80511.

Hence, the probability that the sample proportion will be less than 0.05 is 0.80511.