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Automobile mechanics conduct diagnostic tests on 150 new cars of a 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. The​ 99% confidence interval for the true proportion of new cars with faulty catalytic converters is​ _______.

Respuesta :

Answer:

99% Confidence interval: (0.185,0.375)                                                  

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 150

Number of cars that have faulty catalytic converters, x = 42

[tex]\hat{p} = \dfrac{x}{n} = \dfrac{42}{150} = 0.28[/tex]

99% Confidence interval:

[tex]\hat{p}\pm z_{stat}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

[tex]z_{critical}\text{ at}~\alpha_{0.01} = \pm 2.58[/tex]

Putting the values, we get:

[tex]0.28\pm 2.58(\sqrt{\frac{0.28(1-0.28)}{150}}) = 0.28\pm 0.095\\\\=(0.185,0.375)[/tex]

The​ 99% confidence interval for the true proportion of new cars with faulty catalytic converters is​ (0.185,0.375)

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