Question

Mr. Starnes wondered what proportion of Katy Perry's Twitter followers are

female, so he selected a random sample of 500 of Katy Perry's followers. In the

sample, 308 are female. Construct and interpret a 90% confidence interval for the

true proportion of Katy Perry followers that are female.

Answer 1

To construct a 90% confidence interval for the proportion of female followers of Katy Perry, we calculate the sample proportion, standard error, and margin of error before deriving the interval (0.58027, 0.65173), indicating that we are 90% confident that the true proportion lies within this range.

Step-by-step explanation:

To construct a 90% confidence interval for the true proportion of Katy Perry followers that are female, we first calculate the sample proportion. In the sample of 500 followers, 308 are female, so the sample proportion (π) is 308/500 = 0.616. Next, we use the z-score for a 90% confidence interval, which is approximately 1.645.

Now we can calculate the standard error (SE) of the sample proportion: SE = √[π(1 - π) / n], where n is the sample size. Plugging in the values, we get SE = √[0.616(1 - 0.616) / 500] = √[0.616 × 0.384 / 500] = √[0.236544 / 500] ≈ 0.02172.

Then we calculate the margin of error (ME): ME = z × SE, which is 1.645 × 0.02172 ≈ 0.03573. The 90% confidence interval is thus π ± ME, which turns out to be 0.616 ± 0.03573, or (0.58027, 0.65173).

Interpreting this confidence interval, we are 90% confident that the true proportion of Katy Perry followers that are female falls between 58.027% and 65.173%. It is important to note that while we can be confident in the method of estimation, this range does not mean that exactly 90% of all of Katy Perry's followers are female within this interval. Rather, it means that if we were to take many samples and construct a similar interval, 90% of those intervals would contain the true population proportion.

Answer 2

Answer and Step-by-step explanation:

To calculate a confidence interval for a proportion:

1) Determine the proportion

`p=(308)/(500)`

p = 0.616

2) A 90% confidence interval has z-score that is 1.645.

3) Calculate margin of error

`ME=\sqrt{(p(1-p))/(n) }`

where

n is the sample size

So

`ME=\sqrt{(0.616(1-0.616))/(500) }`

`ME=\sqrt{(0.616(0.384))/(500) }`

ME = 0.02175

Then the interval is given by

p ± z(ME)

0.62 ± 1.645(0.02175)

0.62 ± 0.035

The interval will be

0.585 < p < 0.655

What this means is that we are 90% confident that the true proportion of Katy Perry's Twitter female followers is between 0.585 and 0.655.