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

Answer:

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

Step-by-step explanation:

From the given information:

The point estimate = sample proportion $\hat p$

$\hat p = \dfrac{x}{n}$

$\hat p = \dfrac{55}{100}$

$\hat p$ = 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 $E = Z_{\alpha/2} \times \sqrt{\dfrac{\hat p(1-\hat p)}{n}}$

$E = 2.576 \times \sqrt{\dfrac{0.55(1-0.55)}{100}}$

$E = 2.576 \times \sqrt{\dfrac{0.2475}{100}}$

$E = 2.576 \times0.04975$

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