If the mean serum cholesterol level is 225 and the standard deviation is 22 , determine the probability that a randomly selected person would have a cholesterol value between 150 and 250 ?
$12.8 \%$
$87.2 \%$
$2 \%$
$59 \%$

Step 1 ：Given that the mean serum cholesterol level is 225 and the standard deviation is 22, we are asked to determine the probability that a randomly selected person would have a cholesterol value between 150 and 250.

Step 2 ：This is a question of statistics, specifically involving the normal distribution.

Step 3 ：The first step is to convert the cholesterol levels of 150 and 250 into z-scores. The z-score is a measure of how many standard deviations an element is from the mean. The formula for calculating the z-score is: \(z = \frac{X - \mu}{\sigma}\) where X is the value we are interested in, \(\mu\) is the mean and \(\sigma\) is the standard deviation.

Step 4 ：Calculating the z-scores for 150 and 250, we get \(z_{lower} = -3.409090909090909\) and \(z_{upper} = 1.1363636363636365\).

Step 5 ：We can use the standard normal distribution table (also known as the z-table) to find the probabilities corresponding to these z-scores. The probability that a randomly selected person would have a cholesterol value between 150 and 250 is the difference between the probabilities corresponding to the z-scores of 250 and 150.

Step 6 ：From the z-table, we find that \(prob_{lower} = 0.0003258987757201417\) and \(prob_{upper} = 0.8720977960116918\).

Step 7 ：The probability that a randomly selected person would have a cholesterol value between 150 and 250 is \(prob_{between} = prob_{upper} - prob_{lower} = 0.8717718972359717\).

Step 8 ：Converting this to percentage, we get approximately 87.2%.

Step 9 ：Final Answer: The probability that a randomly selected person would have a cholesterol value between 150 and 250 is \(\boxed{87.2\%}\).