A population of values has a normal distribution with μ=183.3 and σ=48.1. You intend to draw a random sample of size n=25

Find the probability that a sample of size is randomly selected with a mean greater than 166.

P(M > 166) = 

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.

To find the probability \( P(M > 166) \), where \( M \) is the sample mean, we will follow these steps:

### 1. **Identify the given values:**

- Population mean \( \mu = 183.3 \)
- Population standard deviation \( \sigma = 48.1 \)
- Sample size \( n = 25 \)
- Sample mean \( M = 166 \)

### 2. **Calculate the standard error of the mean (SEM):**

The standard error of the mean is given by the formula:

\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\]

Substitute the known values:

\[
\text{SEM} = \frac{48.1}{\sqrt{25}} = \frac{48.1}{5} = 9.62
\]

### 3. **Find the z-score:**

To find the probability, we need to calculate the z-score for \( M = 166 \). The z-score is given by:

\[
z = \frac{M - \mu}{\text{SEM}}
\]

Substitute the values:

\[
z = \frac{166 - 183.3}{9.62} = \frac{-17.3}{9.62} \approx -1.798
\]

### 4. **Find the corresponding probability:**

Now, we need to find the probability corresponding to this z-score. From the z-table, or using a standard normal distribution calculator:

- The probability for \( z = -1.798 \) is approximately 0.0361 (this is the area to the left of \( z = -1.798 \)).

However, we are looking for the probability that the sample mean is greater than 166, which is the area to the right of this z-score. Thus:

\[
P(M > 166) = 1 - 0.0361 = 0.9639
\]

### Final Answer:
\[
P(M > 166) \approx 0.9639
\]

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:
1. How does the sample size affect the standard error of the mean?
2. What is the significance of the z-score in normal distribution problems?
3. How would the probability change if the sample size increased?
4. Can you explain why we subtract the z-score probability from 1?
5. What would happen if the population standard deviation was smaller?

**Tip:** The standard error decreases as the sample size increases, making the sample mean more likely to be closer to the population mean.

A population of values has a normal distribution with μ=183.3 and σ=48.1. You intend to draw a random sample of size n=25. Find the probability that a sample of size is randomly selected with a mean greater than 166. P(M > 166) = ?

Find the probability that a sample of size 25, drawn from a population with a normal distribution (μ=183.3, σ=48.1), has a mean greater than 166. This step-by-step solution involves calculating the standard error of the mean, determining the z-score, and finding the probability using normal distribution properties.

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