A population of values has a normal distribution with $\mu=123.5$ and $\sigma=61.9$. You intend to draw a random sample of size $n=105$.
Find the probability that a single randomly selected value is between 121.7 and 130.7 .
\[
P(121.7< X< 130.7)=
\]
Find the probability that a sample of size $n=105$ is randomly selected with a mean between 121.7 and 130.7.
\[
P(121.7< M< 130.7)=
\]
Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact $z$-scores or $z$ scores rounded to 3 decimal places are accepted.

Step 1 ：Given a population of values with a normal distribution, where the mean \(\mu\) is 123.5 and the standard deviation \(\sigma\) is 61.9.

Step 2 ：We are asked to find the probability that a single randomly selected value is between 121.7 and 130.7.

Step 3 ：This can be solved by calculating the z-scores for 121.7 and 130.7, and then finding the area under the normal distribution curve between these two z-scores.

Step 4 ：The z-score is calculated using the formula: \[z = \frac{x - \mu}{\sigma}\] where \(x\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 5 ：Substituting the given values into the formula, we get the z-scores for 121.7 and 130.7 as -0.0290791599353796 and 0.1163166397415184 respectively.

Step 6 ：We then find the area under the curve between these two z-scores, which represents the probability.

Step 7 ：The probability that a single randomly selected value is between 121.7 and 130.7 is approximately 0.0579.

Step 8 ：Final Answer: The probability that a single randomly selected value is between 121.7 and 130.7 is \(\boxed{0.0579}\).