For a population with a μ = 72 and σ = 20. What is the probability of obtaining a sample mean greater than 75 a. for a random sample of N = 100 scores? 

To solve this problem, we need to determine the probability of obtaining a sample mean greater than 75, given that the population mean (μ) is 72, the population standard deviation (σ) is 20, and the sample size (N) is 100.

### Step 1: Standard Error of the Mean (SE)
The standard error of the mean (SE) can be calculated using the formula:
\[
SE = \frac{\sigma}{\sqrt{N}}
\]
where:
- \( \sigma = 20 \) (population standard deviation),
- \( N = 100 \) (sample size).

\[
SE = \frac{20}{\sqrt{100}} = \frac{20}{10} = 2
\]

### Step 2: Z-score Calculation
Now we can calculate the Z-score, which is a measure of how many standard errors the sample mean is away from the population mean. The Z-score formula is:

\[
Z = \frac{\overline{X} - \mu}{SE}
\]
where:
- \( \overline{X} = 75 \) (sample mean),
- \( \mu = 72 \) (population mean),
- \( SE = 2 \).

\[
Z = \frac{75 - 72}{2} = \frac{3}{2} = 1.5
\]

### Step 3: Finding the Probability
Using the Z-score of 1.5, we can find the probability corresponding to this Z-score from the standard normal distribution table.

The probability for \( Z = 1.5 \) is approximately 0.9332, which means that the probability of obtaining a sample mean less than 75 is 93.32%.

However, we want the probability of obtaining a sample mean **greater** than 75. Since the total area under the curve is 1, the probability of a sample mean greater than 75 is:

\[
P(X > 75) = 1 - 0.9332 = 0.0668
\]

### Final Answer:
The probability of obtaining a sample mean greater than 75 for a sample of \( N = 100 \) is approximately **0.0668** or **6.68%**.

Would you like further details or have any questions?

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Here are five related questions:
1. How does increasing the sample size affect the standard error of the mean?
2. What is the probability of obtaining a sample mean less than 70 in this scenario?
3. How would the probability change if the sample mean were 78 instead of 75?
4. What assumptions are made about the population when using the Z-score approach?
5. How does the Central Limit Theorem help in calculating probabilities for sample means?

**Tip**: When sample sizes are large (e.g., \( N > 30 \)), the sampling distribution of the sample mean approaches a normal distribution, even if the population distribution is not normal. This is due to the Central Limit Theorem.

For a population with a μ = 72 and σ = 20, what is the probability of obtaining a sample mean greater than 75 for a random sample of N = 100 scores?

Learn how to calculate the probability of obtaining a sample mean greater than 75 for a population with μ = 72 and σ = 20, using the Z-score formula and standard error. This problem demonstrates key concepts of normal distribution and sampling distribution. Suitable for advanced high school or college-level statistics students.

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