A population of values has a normal distribution with $u=123.5$ and $o=61.9$. You intend to draw a random sample of size $n=105$
Find the probability that a sample of size $n=105$ is randomly selected with a mean between 121.7 and 130.7.

Step 1 ：The problem is asking for the probability that the mean of a sample of size 105 is between 121.7 and 130.7. This is a problem of normal distribution. The mean of the population is given as 123.5 and the standard deviation is given as 61.9.

Step 2 ：We can use the Central Limit Theorem which states that if you have a population with mean \(\mu\) and standard deviation \(\sigma\) and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

Step 3 ：This will allow us to standardize our variables and use the Z-table to find the probabilities. The formula to calculate the z-score is: \(Z = \frac{X - \mu}{\sigma / \sqrt{n}}\) where: \(X\) is the value we are looking to find the probability for, \(\mu\) is the mean, \(\sigma\) is the standard deviation, and \(n\) is the size of the sample.

Step 4 ：We will calculate the z-scores for 121.7 and 130.7 and then use the Z-table to find the probabilities.

Step 5 ：Given mean = 123.5, standard deviation = 61.9, and sample size = 105, we calculate the z-scores as follows: \(z1 = -0.2979727201732996\) and \(z2 = 1.1918908806931985\).

Step 6 ：We then find the probabilities corresponding to these z-scores from the Z-table: \(p1 = 0.3828619922710792\) and \(p2 = 0.8833479828124222\).

Step 7 ：The probability that the sample mean is between 121.7 and 130.7 is the difference between these two probabilities: \(probability = p2 - p1 = 0.500485990541343\).

Step 8 ：Final Answer: The probability that a sample of size \(n=105\) is randomly selected with a mean between 121.7 and 130.7 is approximately \(\boxed{0.5005}\).