Question

The creators David Benioff and D. B. Weiss of the TV show "Game of Thrones", want to know how popular Season 8 was among the show’s fans. They hired an analyst who collected a random sample of 40,361 Game of Thrones fans, and found that 8,337 of those fans said that they like Season 8. Using this data, the analyst wants to estimate the actual proportion of fans who like Season 8. What is the lower bound for a 90% confidence interval? Give your answer to 4 decimal places.

Answer 1

The lower bound for a 90% confidence interval is 0.2033

Explanation:

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)`.

For this problem, we have that:

`n = 40361, \pi = (8337)/(40361) = 0.2066`

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`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.2066 - 1.645\sqrt{(0.2066*0.7934)/(40361)} = 0.2033`

The lower bound for a 90% confidence interval is 0.2033