Question

If the scientist is accurate, what is the probability that the proportion of airborne viruses in a sample of 477 viruses would differ from the population proportion by more than 3%.

Answer 1

0.01596

Explanation:

A scientist claims that 8% of the viruses are airborne

Given that:

The population proportion p = 8%

The sample size = 477

We can calculate the standard deviation of the population proportion by using the formula:

`\sigma_p = \sqrt{(p(1-p))/(n)}`

`\sigma_p = \sqrt{(0.8(1-0.8))/(477)}`

`\sigma_p = \sqrt{(0.0736)/(477)}`

`\sigma_p = 0.02098`

The required probability can be calculated as:

`P(| \hat p - p| > 0.03) = P(\hat p - p< -0.03 \ or \ \hat p - p > 0.03)`

`= P \bigg ( \frac{\hat p -p }{\sqrt{(p(1-p))/(n)}} < -(0.03)/(0.0124) \bigg ) + P \bigg ( \frac{\hat p -p }{\sqrt{(p(1-p))/(n)}} >(0.03)/(0.0124) \bigg )`

= P(Z < -2.41) + P(Z > 2.41)

= P(Z < -2.41) + P(Z < -2.41)

= 2P( Z< - 2.41)

From the Z-tables;

`P(| \hat p - p| > 0.03)` = 2 ( 0.00798

`P(| \hat p - p| > 0.03)` = 0.01596

Thus, the required probability = 0.01596