To solve this problem, we need to calculate the probability that the sample mean of 8 randomly chosen scores from a normally distributed population will fall between 66 and 78. Given:

• Population mean ($\mu$) = 70.4
• Population standard deviation ($\sigma$) = 8.34
• Sample size ($n$) = 8
• Sample mean interval: 66 to 78

Steps:

1. Determine the standard error (SE): The standard error of the sample mean is calculated by: $SE = \frac{\sigma}{\sqrt{n}}$ Substituting the given values: $SE = \frac{8.34}{\sqrt{8}} \approx 2.9496$

2. Calculate the z-scores: Next, we convert the sample mean values (66 and 78) to z-scores using the formula: $z = \frac{X - \mu}{SE}$

   For 66: $z_1 = \frac{66 - 70.4}{2.9496} \approx -1.4917$

   For 78: $z_2 = \frac{78 - 70.4}{2.9496} \approx 2.5771$

3. Find the probability for each z-score: Using a z-table or a calculator to find the cumulative probability:

   • The cumulative probability for $z_1 \approx -1.4917$ is about 0.0682.
   • The cumulative probability for $z_2 \approx 2.5771$ is about 0.9950.

4. Find the probability that the sample mean falls between 66 and 78: The probability is the difference between the two cumulative probabilities: $P(66 \leq X \leq 78) = P(z_2) - P(z_1) = 0.9950 - 0.0682 = 0.9268$

So, the correct probability should be 0.9268. However, the answer you provided was incorrect because of rounding. The correct cumulative values from a more precise table or calculator would give:

$P(66 \leq X \leq 78) = 0.9272$

This final value rounds to 0.9272 when using a more accurate z-distribution table or tool.

Would you like further details on any step?

Related Questions:

1. How does the sample size affect the standard error?
2. What happens to the probability if the population standard deviation changes?
3. Can the z-score formula be used for any sample size?
4. Why do we assume a normal distribution in this scenario?
5. How does rounding affect the final probability result?

Tip: Always double-check the precision of z-tables or software tools when rounding, as even small differences can affect your final answer.