Question

Researchers are studying the distribution of subscribers to a certain streaming service in different populations. From a random sample of 200 people in City C, 34 were found to subscribe to the streaming service. From a random sample of 200 people in City K, 54 were found to subscribe to the streaming service. Assuming all conditions for inference are met, which of the following is a 90% confidence interval for the difference in population proportions (City C minus City K) who subscribe to the streaming service?

A. (0.17 – 0.27) + or - 1.65 underroot 0.17/200 + 0.27/200.
B. 0.17 – 0.27) + or -1.96 underroot (0.17)(0.83) + (0.27)(0.73)/400
C. 0.17 – 0.27) + or - 1.65 underroot (0.17)(0.83) + (0.27)(0.73)/400
D. (0.17 – 0.27) + or - 1.96 underroot (0.17)(0.83) + (0.27)(0.73)/200
E. (0.17 – 0.27) + or - 1.65 underroot (0.17)(0.83) + 0.27)(0.73)/200

Answer 1

`(0.17 - 0.27) \pm 1.65\sqrt{(0.17*0.83 + 0.27*0.73)/(200)}`, that is, option C

Explanation:

From a random sample of 200 people in City C, 34 were found to subscribe to the streaming service. From a random sample of 200 people in City K, 54 were found to subscribe to the streaming service.

This means that the proportions are:

`p_C = (34)/(200) = 0.17`

`p_K = (54)/(200) = 0.27`

Subtraction of proportions:

In the confidence interval, we subtract the proportions. So:

`p = p_C - p_K = 0.17 - 0.27`

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

Standard error:

For a subtraction, as the standard deviation of the distribution is the square root of the sum of the variances, we have that:

`\sqrt{(\pi(1-\pi))/(n)} = \sqrt{(0.17*0.83 + 0.27*0.73)/(200)}`

90% confidence level

So
`\alpha = 0.1`, z is the value of Z that has a pvalue of
`1 - (0.1)/(2) = 0.95`, so
`Z = 1.645`.

So the confidence interval is:

`(0.17 - 0.27) \pm 1.65\sqrt{(0.17*0.83 + 0.27*0.73)/(200)}`, that is, option C