Question

An environmental agency worries that many cars may be violating clean air emissions standards. The agency hopes to check a sample of vehicles in order to estimate that percentage with a margin of error of 3​% and 90​% confidence. To gauge the size of the​ problem, the agency first picks 40 cars and finds 8 with faulty emissions systems. How many should be sampled for a full​ investigation?

Answer 1

Answer:   482 cars

Step-by-step explanation:

The formula to find the sample size is given by :-

$n= p(1-p)(\dfrac{z_{\alpha/2}}{E})^2$

, where p is the prior estimate of population proportion.

$z_{\alpha/2}$ = Two-tailed z-value for $\alpha$ (significance level).

E= Margin of error .

Given : Margin of error = 3% = 0.03

Confidence level = 90%=0.90

$\alpha=1-0.90=0.10$

By z-value table : $z_{\alpha/2}=z_{0.05}=1.645$

To gauge the size of the​ problem, the agency first picks 40 cars and finds 8 with faulty emissions systems.

Then , the prior estimate of population proportion of  faulty emissions systems =$p=\dfrac{8}{40}=0.2$

Then, the required sample size would be :

$n= 0.2(1-0.2)(\dfrac{1.645}{0.03})^2$

$n=0.16(3006.6944)$

$n=481.071104\approx482$

Hence, 482 cars should be sampled for a full​ investigation.