Please answer the following questions.

Question

The second dose of the COVID-19 vaccine is more likely to produce side effects than the first. The time between receiving the shot and the onset of side effects is called the lag time. Suppose according to one study, the lag times (among the subjects that reported side effects from a second dose of the vaccine) were normally distributed with a mean of 15 hours, with a standard deviation of 2 hours. Use the empirical rule to answer the following.

(a)

What is the probability that a person selected at random from among those that experienced side effects from their second dose of the vaccine had a lag time of less than 13 hours?

 

(b)

What is the probability that a person selected at random from among those that experienced side effects from their second dose of the vaccine had a lag time between 15 and 19 hours?

Answer 1

Hi Tamera,

The Empirical Rule is another name for the 68-95-99.7 rule. You had a figure in previous problems that depicted this. In short, in normal distributions, 68% of the data lies within one standard deviation of the mean, 95% lies within two, and 99.7% lies within three. To our problem

(a)

Mean = mu = 15

Sigma = standard deviation = 2

You were asked for lag time less than 13 hours. 13 hours is exactly one standard deviation below the mean, so, think back to the figure. You want the percentages for (mu - 2sigma), (mu - 3sigma), and the remaining. Using the 68-95-99.7 rule, we know 32% of data falls outside one standard deviation, but that is for both sides, Since you want only the range below:

32/2 = 16%

(b) 19 hours is exactly two standard deviations above 15 hours, and the 68-95-99.7 rule tells us that 95% of data falls within two standard deviations, but again, you want only one side, the positive side, here. So:

95/2 = 47.5%

I hope this helps.