Question

Automobile mechanics conduct diagnostic tests on 100 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 25 of the new cars tested do have faulty catalytic converters. What is the margin of error for a 95% confidence interval based on these sample results? 0.077 0.112 0.066 0.085

Answer 1

Solution :

1) Given that,

n = 100

x = 25

Point estimate = sample proportion = $\hat p$ = x / n = 25 / 100 = 0.25

1 - $\hat p$ = 1 - 0.25 = 0.75

At 95% confidence level

$\alpha$ = 1 - 95%

$\alpha$ =1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z0.025 = 1.960

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

E = 1.96 ($\sqrt$((0.25 * 0.75) /100 )

E = 0.085

2) Given that,

n = 600

x = 358

Point estimate = sample proportion = $\hat p$ = x / n = 358 / 600 = 0.5967

1 - $\hat p$ = 1 - 0.5967 = 0.4033

At 95% confidence level

$\alpha$ = 1 - 95%

$\alpha$ =1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z0.025 = 1.960

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

= 1.96 ($\sqrt$((0.5967 * 0.4033) /600 )

= 0.03925

A 95% confidence interval for population proportion p is ,

$\hat p$ ± E   

= 0.5967 ± 0.03925

=( 0.5574, 0.6359 )

3) False

np $\geq$ 10 and nq $\geq$ 10