Question

I. Elena wants to estimate what proportion of computers produced at a factory have a certain defect. A random sample of 200 computers shows that 12 computers have the defect. She is willing to assume independence between computers in the sample. Based on this sample, construct a 95% confidence interval for the proportion of computers that have the defect.

a
(0.4529, 0.5871)
b
(0.02709, 0.09291)
c
(0.02131, 0.09869)
d
(0.0309, 0.0691)

Answer 1

To construct the 95% confidence interval for the proportion of defective computers, we use the sample proportion and standard error, apply the z*-value for a 95% confidence level, and calculate the interval, which is (option b) (0.02709, 0.09291).

Step-by-step explanation:

To construct a 95% confidence interval for the true proportion of computers that have a certain defect, we can use the formula for the confidence interval of a population proportion. The sample proportion (p) is found by dividing the number of defective computers by the total number of computers sampled. In this case, p = 12 / 200 = 0.06. The standard error (SE) for the proportion is calculated using the formula SE = √[p(1 - p) / n], where n is the sample size. Therefore, SE = √[0.06(1 - 0.06) / 200].

To find the critical value (z*), we refer to the standard normal distribution table, considering that we want a 95% confidence level and the tails on both sides of the curve contain 2.5% (0.025) since it's a two-tailed test. The z* value for 0.025 in each tail is approximately 1.96.

Using the formula for the confidence interval: p ± z*SE, we get 0.06 ± 1.96√[0.06(1 - 0.06) / 200]. Calculating this, we find the interval to be approximately (0.02709, 0.09291).

Therefore, the correct answer is (option b): (0.02709, 0.09291).

Answer 2

The answer is (A) 0.4529, 0.5871