Investigation 3: COVID-19 Vaccine Summary in Virginia (no data set) The Virginia Department of Health publishes COVID-19 vaccination data on their website. As of 10/5/2021, 60.6% of the population of Virginia is fully vaccinated. This rate varies drastically from county to county. Specifically, Fairfax County's full vaccination rate is 74% and Lee County's full vaccination rate is 40%. A researcher took a random sample of 14 individuals from Fairfax County and another random sample of 14 individuals from Lee County and asked them whether they could verify that they are fully vaccinated. Each sample was selected in a single day.

a) Using four complete sentences, verify that these samples satisfy the conditions of the binomial experiment. provideone sentence to check each requirement in context.

b) Assuming the sample from Fairfax County is a binomial experiment, build the probability distribution in a single table form in StatCrunch. There are two ways to do this. You may use Data Compute Expression and choose the function dbinom. This method relies on you entering all the outcome values of the random variable in the first column of your data table. The other way to do this is to use the binomial calculator and calculate the probability of each of the values of the random variable from X = 0 to X = 14. You may present this table horizontally or vertically and leave the probabilities unrounded.

c) Calculate the probability that at least 12 individuals from Fairfax County in this sample are fully vaccinated using the probability distribution table you created in part (b). Show all of your calculations and use proper probability notation. Round your answer to four decimal places.

d) Verify your answer to part (c) using the StatCrunch binomial calculator. Copy the image with values below to your document. Write a one sentence interpretation of the probability in context of the question.

e) Assuming the sample from Lee County is a binomial experiment, calculate the probability that at least 12 individuals from Lee County in this sample are fully vaccinated using the StatCrunch binomial calculator. Round your answer to four decimal places and compare this probability to the probability you calculated in part (d) in one sentence. 7

f) Calculate the mean and standard deviation of the number of individuals from Fairfax County and Lee County in this sample who are fully vaccinated. Show your work using the binomial mean and binomial standard deviation formulas and provide your two means and two standard deviations in your document. Round your answers to two decimal places. (It is not necessary to use StatCrunch for this part.)

g) Calculate the probability that exactly seven of the 14 individuals from Fairfax County in this sample are fully vaccinated using the StatCrunch binomial calculator. Copy this image with values from StatCrunch into your document. Then, once you obtain your answer, round your answer to four decimal places and write a one sentence interpretation of the probability in context of the question.

h) Imagine you repeated taking a sample of 14 individuals from the Fairfax County population 20,000 times. We can simulate this in StatCrunch. First, go to Data Simulate Binomial. Next, enter 20,000 for Rows, 1 for Columns, 14 for n, 0.74 for p, and click compute. Then, to visualize these data, go to Graph Bar Plot With Data. Produce a properly titled and labeled relative frequency bar chart and paste it into your solutions.

i) Using StatCrunch, state the height of the bar above seven and compare this with your answer to part (g) in context. Identify which type of probability (empirical or theoretical) each value is in your comparison.