Question

Experts have predicted that approximately 1 in 12 tractor-trailer units will be involved in an accident this year. One reason for this is that 1 in 3 tractor-trailer units has an imminently hazardous mechanical condition, probably related to the braking systems on the vehicle.

1. Suppose a sample of 124 tractor-trailers is taken and that 45 of them are found to have a potentially serious braking system problem. Find a 90% confidence interval for the true proportion of all tractor-trailers that have this potentially serious problem.

Answer 1

1.

Solution :

Given that,

Point estimate = sample proportion = $\hat p$ = x / n = 45 / 124 = 0.363

1 - $\hat p$ = 1 - 0.363 = 0.637

Z$\alpha$/2 = Z0.05 = 1.645

Margin of error = E = Z$\alpha$ / 2 * $\sqrt$ (($\hat p$ * (1 - $\hat p$ )) / n)

= 1.645 * ($\sqrt$((0.363 * 0.637 ) / 124 )

Margin of error = E = 0.071

A 90% confidence interval for population proportion p is ,

$\hat p$ - E < p < $\hat p$ + E

0.363 - 0.071 < p < 0.363 + 0.071

0.292 < p < 0.434

The 90% confidence interval for the population proportion p is : 0.292 < p < 0.434