A recent poll by the Gallup organization found that the...

A recent poll by the Gallup organization found that the proportion of Americans stating that they were not going to get the Covid-19 vaccine was 20%. You, being an aspiring statistician are surprised and have a feeling this statement is too low due to the chatter on all of the social media platforms. You decide to construct a confidence interval to estimate the true population proportion. In your survey of 350 persons, 98 of them stated they would not be getting vaccinated. a. Construct a 99% confidence interval for the proportion of Americans who will not get vaccinated for Covid-19. b. If you lowered your confidence level to 95%, how would the result compare to the answer you calculated in part a? c. Interpret this confidence interval verbally. d. Does the Gallup's claim that the percentage is 22% seem reasonable? Why or why not?

Transcript

00:01 We have some sample data.

00:02 So we took a sample of 350 people, so there's our sample size, out of these 98 of them said they would not be getting a vaccine.

00:12 So x would be 98.

00:13 We want to take this information and make a confidence interval for the proportion of americans who won't get vaccinated.

00:20 So the formula we need for that is p -hat, sample proportion, point estimate, plus and minus the margin of error, z root p -hat, 1 minus p -hat, over n.

00:33 So what is p -hat, first of all? well, it is the proportion of our sample who won't get vaccinated.

00:40 98 out of 350.

00:43 Let me see if this is a nice number or not.

00:46 Yep, 0 .28.

00:50 Z we get from a level of confidence.

00:53 So right now this is a binomial experiment.

00:56 We have n independent trials, two outcomes, they won't get vaccinated, or they will.

01:02 Same probability p, whatever it is, of not getting vaccinated.

01:07 The binomial variable is x, the number of people in the sample who say they won't.

01:12 From this we take a normal approximation.

01:16 So this is now a normal distribution, still concerning variable x.

01:20 I want it to concern p -hat.

01:23 So all i do is take the distribution, take its parameters, divide everything by n, sample size.

01:30 Now it's a sampling distribution.

01:33 And i put my point estimate in the middle here, form the interval around it, containing 99 % of the area of this curve.

01:40 99 % confidence.

01:42 It's a bit wide.

01:45 1 % is in the tails, so each tail is half a percent.

01:50 Z is the z -score to exclude those tails.

01:53 You might have a table of these.

01:55 If not, you can use invnorm on excel or your calculator.

02:00 Just put in the area to the left, 0 .005 or 0 .995, and it gives you the critical value, 2 .576.

02:09 So the interval is 0 .28, plus and minus, i'm going to get my margin of error now, 0 .28 times 0 .72, divided by 350, square root it, multiplied by 2 .576, is 0 .06.

02:29 Actually, i'll add an extra decimal place here, 62, to three decimal places.

02:34 If i take that away from my confidence interval, i get the lower bound, 0 .218...

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