Final answer:
To find the 90th percentile for the sample means, divide the standard deviation of the population by the square root of the sample size. Multiply the standard deviation of the sample means by the z-score corresponding to the 90th percentile and add it to the population mean to find the sample mean at the 90th percentile.
Step-by-step explanation:
To find the 90th percentile for the sample means, we need to first find the standard deviation of the sample means. We can do this by dividing the standard deviation of the population by the square root of the sample size.
Standard deviation of the sample means = standard deviation of the population / square root of the sample size = 9 / sqrt(10) ≈ 2.84
Then, we can find the z-score corresponding to the 90th percentile, which is the z-score that leaves 90% of the area under the normal distribution curve to its left. We can use a z-score table or a calculator to find this value. In this case, the z-score is 1.28 (approximately).
Finally, we can find the sample mean corresponding to the 90th percentile by multiplying the standard deviation of the sample means by the z-score and adding it to the population mean. Sample mean = population mean + (z-score * standard deviation of the sample means) = 61 + (1.28 * 2.84) ≈ 65.87
Therefore, the 90th percentile for the sample means is approximately 65.87.