To find the probability of a sample mean being greater than 220, we use the Central Limit Theorem. Here's the process:

Step 1: Identify the Given Values

• Population mean ($\mu$): 219
• Population standard deviation ($\sigma$): 5.9
• Sample size ($n$): 70
• Sample mean ($\bar{x}$) threshold: 220

Step 2: Compute the Standard Error of the Mean (SEM)

The standard error of the mean is calculated as: $\text{SEM} = \frac{\sigma}{\sqrt{n}}$ $\text{SEM} = \frac{5.9}{\sqrt{70}} \approx \frac{5.9}{8.37} \approx 0.704$

Step 3: Compute the Z-Score

The Z-score formula for the sample mean is: $Z = \frac{\bar{x} - \mu}{\text{SEM}}$ $Z = \frac{220 - 219}{0.704} \approx \frac{1}{0.704} \approx 1.42$

Step 4: Find the Probability

Using the Z-score table or a statistical calculator, find the area to the left of $Z = 1.42$: $P(Z \leq 1.42) \approx 0.9222$

The probability of the sample mean being greater than 220 is the complement: $P(Z > 1.42) = 1 - 0.9222 = 0.0778$

Final Answer:

The probability of the sample mean being greater than 220 is approximately: \boxed{0.0778 \text{ (or 7.78%) }}

Would you like a step-by-step visual explanation or have questions about this calculation?

---

Related Questions:

1. How does increasing the sample size affect the standard error and the probability?
2. What is the interpretation of the Central Limit Theorem in this context?
3. Can you calculate the probability for a threshold of 221 instead of 220?
4. What assumptions are made about the population distribution in this scenario?
5. How would the calculation change if the population standard deviation were unknown?

Tip: Always check if the sample size $n$ is sufficiently large for the Central Limit Theorem to apply (typically $n \geq 30$).