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. 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 13 hours and a standard deviation of 2 hours.

Use the empirical rule to answer the following:

What is the probability that a person selected at random from among those who experienced side effects from their second dose of the vaccine had a lag time of less than 11 hours?

To find the probability of a lag time of less than 11 hours, we use the empirical rule, which suggests that approximately 16% of the normal distribution lies more than one standard deviation below the mean when the standard deviation is 2 hours.

The question deals with the concept of the empirical rule, which is a statistical rule stating that for a normal distribution, nearly all data will fall within three standard deviations of the mean.

According to the empirical rule, 68% of the data falls within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations.

Given that the mean lag time is 13 hours and the standard deviation is 2 hours, we want to find the probability of a lag time of less than 11 hours, which is one standard deviation below the mean.

To apply the empirical rule, we note that approximately 34% of the data lies between the mean and one standard deviation below the mean.

Therefore, we can assume that roughly 34% of the data lies between 11 hours and 13 hours (one standard deviation below and the mean).

Since a normal distribution is symmetrical, about 68% of the data is within one standard deviation of the mean, so this means that 34% of the data lies between 13 hours and 15 hours, and another 34% between 11 hours and 13 hours.

The remaining 16% (half of 32%) lies beyond one standard deviation on both tails of the distribution.

Hence, the probability of a lag time of less than 11 hours is approximately 16%.