Question

Automobile mechanics conduct diagnostic tests on 150 randomly selected new cars of particular make and model to determine the extent to which they are affected by a recent recall due to faulty catalytic converters. They find that 42 of the new cars tested do have faulty catalytic converters. What is the margin of error for a​ 99% confidence interval based on these sample​ results?

Answer 1

Answer:

$\mathbf{Margin\; of\; error} =$  .0942816

Step-by-step explanation:

Given

Sample size n = 150

Sample proportion ${\widehat{(p)}$ = $\frac{42}{150}$ = 0.28

Confidence interval = $\frac{99}{100}$ = .99

$Margin\; of\; error = z_{\frac{\alpha }{2}}\sqrt{\frac{{\widehat{(p)}}{(1 - \widehat{p})}}{n}}$

$\alpha =$ 1 - confidence interval = 1 - 0.99 = .01

$margin\; of\; error = z_{\frac{.01 }{2}}\sqrt{\frac{(.28) (1 - 0.28)}{150}$    

For 99$\%$ confidence interval

        ( Z = 2.576 )

$margin\; of\; error = z_{.005 }}\sqrt{\frac{(.28) (1 - 0.28)}{150}$                  

$margin\; of\; error = 2.576\times\sqrt{\frac{(.28) (1 - 0.28)}{150}$

                           =  .0942816