Let's analyze and solve the problem step-by-step.

Problem Recap:

1. A population is normally distributed with:
   • Mean ($\mu$) = 151.6
   • Standard deviation ($\sigma$) = 21.5
2. You are tasked with:
   • Finding the probability $P(X > 154.7)$ for a single randomly selected value.
   • Finding the probability $P(M > 154.7)$, where $M$ is the sample mean from a sample of size $n = 188$.

Step 1: Finding $P(X > 154.7)$

For a single value, use the standard normal distribution:

1. Calculate the z-score: $z = \frac{X - \mu}{\sigma} = \frac{154.7 - 151.6}{21.5} = \frac{3.1}{21.5} \approx 0.1442$

2. Find the probability from the z-score:

   • Using a z-table or a statistical calculator, find the area to the left of $z = 0.1442$: $P(Z < 0.1442) \approx 0.5574$
   • The probability to the right is: $P(X > 154.7) = 1 - P(Z < 0.1442) = 1 - 0.5574 = 0.4426$

So, $P(X > 154.7) = 0.4426$. (This matches the answer you entered.)

Step 2: Finding $P(M > 154.7)$

For the sample mean, adjust the standard deviation using the formula for the standard error: $\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}} = \frac{21.5}{\sqrt{188}} \approx \frac{21.5}{13.711} \approx 1.567$

1. Calculate the z-score for the sample mean: $z = \frac{M - \mu}{\text{SE}} = \frac{154.7 - 151.6}{1.567} = \frac{3.1}{1.567} \approx 1.978$

2. Find the probability from the z-score:

   • Using a z-table or a statistical calculator, find the area to the left of $z = 1.978$: $P(Z < 1.978) \approx 0.9762$
   • The probability to the right is: $P(M > 154.7) = 1 - P(Z < 1.978) = 1 - 0.9762 = 0.0238$

So, $P(M > 154.7) = 0.0238$. (This is close to the answer you entered, but your value might differ slightly due to rounding.)

Final Answers:

1. $P(X > 154.7) = 0.4426$ (correctly computed).
2. $P(M > 154.7) = 0.0238$.

Would you like a more detailed breakdown of any step or have further questions?

Related Questions:

1. How do you calculate the standard error for a sample mean?
2. Why do probabilities for sample means decrease as sample size increases?
3. What is the importance of using z-scores in a normal distribution?
4. How do rounding errors impact final probability calculations?
5. What tools or calculators are most accurate for normal distribution probabilities?

Tip:

Always double-check whether you are working with single values or sample means, as the formulas and standard deviations differ.