Applications using Sample Proportions Preview CT-STEM

Practice with 2 proportions

Suppose we want to investigate the effectiveness of two potential COVID-19 vaccines. We will call these "Vaccine 1" and "Vaccine 2".

During Phase 3 trials of the vaccine development process, thousands of volunteers are randomly assigned one of the two vaccines. After 30 days, researchers take blood samples to detect if antibodies are present. The presence of antibodies would indicate that the vaccine has been effective at preventing the individual from experiencing moderate to severe COVID-19 symptoms. The table below shows the results of some randomly selected volunteers from each vaccine trial.

	Antibodies present	No antibodies present
Vaccine 1	53	22
Vaccine 2	59	16

Questions

Please answer the questions below.

Question 9.1

Identify each of the following, using the table above : \(n_1, n_2, \hat p_1, \hat p_2\)

Question 9.2

Based on the sample results, which vaccine seems to be more effective? Provide evidence to support your reasoning.

Question 9.3

Let's analyze the difference between these two sample proportions. Calculate \(\hat p_1 - \hat p_2\)

Question 9.4

For the standard deviation formula, we need to think back to our work with combining random variables in Chapter 6. Remember: you have a formula for the standard deviation of \(\hat p_1 - \hat p_2\) on your formula sheet!

Calculate this value for our scenario using your answers from question 8.1

Question 9.5

Later on, we found out that the two vaccines are equally effective. What is the probability of observing a difference in sample proportions greater than the one shown in the table? Be sure to check the Normal condition for each population.

Notes

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