9. New Residences The average number of moves a person makes in his or her lifetime is 12. If the standard deviation is 3.2 , find the probability that the mean of a sample of 36 people is a Less than 13

Step 1 ：This problem involves probability and normal distribution. The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger, regardless of the shape of the population distribution. This theorem applies here as we are dealing with a sample size of 36 people.

Step 2 ：We are given the population mean \(\mu = 12\), the standard deviation \(\sigma = 3.2\), and the sample size \(n = 36\). We are asked to find the probability that the sample mean is less than 13.

Step 3 ：To solve this, we need to standardize the score of 13 using the Z-score formula for sample means, which is \(Z = \frac{X - \mu}{\sigma/\sqrt{n}}\), where \(X\) is the sample mean. The Z-score will give us the number of standard deviations the sample mean of 13 is away from the population mean.

Step 4 ：We can then use the Z-score to find the probability that the sample mean is less than 13 by looking up the Z-score in the Z-table (also known as the standard normal distribution table), which gives the area to the left of a given Z-score. This area represents the probability that a value is less than the given value.

Step 5 ：Let's calculate the Z-score and find the probability. Using the given values, we find that \(Z = 1.875\).

Step 6 ：Looking up the Z-score in the Z-table, we find that the probability \(p = 0.9696036382347386\).

Step 7 ：The probability that the sample mean is less than 13 is approximately 0.97. This means that there is a 97% chance that the mean number of moves for a sample of 36 people will be less than 13.

Step 8 ：Final Answer: The probability that the mean of a sample of 36 people is less than 13 is approximately \(\boxed{0.97}\).