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 .
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P(121.7
Solution
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 \(\boxed{0.0579}\).
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Solution
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 \(\boxed{0.0579}\).
