Question

A quality control engineer is interested in estimating the proportion of defective items coming off a production line. In a sample of 100 items, 55 are defective. The lower bound of a 99% confidence interval for the proportion of defectives is ________.

Answer 1

The lower bound of a 99% C.I for the proportion of defectives = 0.422

Explanation:

From the given information:

The point estimate = sample proportion
`\hat p`

`\hat p = (x)/(n)`

`\hat p = (55)/(100)`

`\hat p` = 0.55

At Confidence interval of 99%, the level of significance = 1 - 0.99

= 0.01

`Z_(\alpha/2) =Z_(0.01/2) \\ \\ = Z_(0.005) = 2.576`

Then the margin of error
`E = Z_(\alpha/2) * \sqrt{(\hat p(1-\hat p))/(n)}`

`E = 2.576 * \sqrt{(0.55(1-0.55))/(100)}`

`E = 2.576 * \sqrt{(0.2475)/(100)}`

`E = 2.576 *0.04975`

E = 0.128156

E ≅ 0.128

At 99% C.I for the population proportion p is:
`\hat p - E`

= 0.55 - 0.128

= 0.422

Thus, the lower bound of a 99% C.I for the proportion of defectives = 0.422