We can use the empirical rule (also called 68-95-99.7 rule) to evaluate the percentage of people.
99.7% of people should be sen by the doctor between 11 and 23 minutes for this to be considered a normal distribution.
Thus, Option D: 99.7% is correct.
<h2>Given that:</h2>
- The typical time interval of wait time is 17 minutes.
- The standard deviation for wait time is 2 minutes.
<h3>To find:</h3>
Percentage of people that are to be seen by the Doctor for the distribution of wait time to be considered normal distribution.
<h3>What is empirical rule for normal
distribution?</h3>
If
(that means, X pertains normal distribution with mean
and standard deviation
), then we have:
![P(\mu - \sigma < X](https://tex.z-dn.net/?f=P%28%5Cmu%20-%20%5Csigma%20%3C%20X%20%3C%5Cmu%20%2B%20%5Csigma%29%20%5Capprox%2068%5C%25%20%3D%200.68%5C%5C%5C%5CP%28%5Cmu%20-%202%5Csigma%20%3C%20X%20%3C%5Cmu%20%2B%202%5Csigma%29%20%5Capprox%2095%5C%25%20%3D%200.95%5C%5C%5C%5CP%28%5Cmu%20-%203%5Csigma%20%3C%20X%20%3C%5Cmu%20%2B%203%5Csigma%29%20%5Capprox%2099.7%5C%25%20%3D%200.997)
Let X be the random variable tracking the wait time (in minutes) by patients for checkup.
Then we have by given data:
![\mu = 17 \: \rm minutes](https://tex.z-dn.net/?f=%5Cmu%20%3D%2017%20%5C%3A%20%5Crm%20minutes)
![\sigma = 2 \: \rm minutes](https://tex.z-dn.net/?f=%5Csigma%20%3D%202%20%5C%3A%20%5Crm%20minutes)
Thus,
![X \sim N(17, 2)](https://tex.z-dn.net/?f=X%20%5Csim%20N%2817%2C%202%29)
The given limits are 11 and 23 minutes or
![11 < X < 23 = 17 - 6 < X < 17 + 6 = \mu - 3\sigma < X < \mu + 3\sigma](https://tex.z-dn.net/?f=11%20%3C%20X%20%3C%2023%20%3D%2017%20-%206%20%3C%20X%20%3C%2017%20%2B%206%20%3D%20%5Cmu%20-%203%5Csigma%20%3C%20X%20%3C%20%5Cmu%20%2B%203%5Csigma)
Thus, the percentage can be calculated by empirical rule as follows:
![P(11 < X < 23 ) = P(\mu - 3\sigma < X < \mu + 3\sigma) = 0.997 = 99.7\%](https://tex.z-dn.net/?f=P%2811%20%3C%20X%20%3C%2023%20%29%20%3D%20P%28%5Cmu%20-%203%5Csigma%20%3C%20X%20%3C%20%5Cmu%20%2B%203%5Csigma%29%20%3D%200.997%20%3D%2099.7%5C%25)
where
.
Thus, 99.7% of people should be sen by the doctor between 11 and 23 minutes for this to be considered a normal distribution.
Thus, Option D: 99.7% is correct.
Learn more about empirical rule here:
brainly.com/question/16645539