Final answer:
To separate the bottom 80% values from the top 20% values, we can use the z-score formula. The value separating the bottom 80% values from the top 20% values is 62.66. The sample mean separating the bottom 80% sample means from the top 20% sample means is 85.2829.
Step-by-step explanation:
To find the value separating the bottom 80% values from the top 20% values, we can use the z-score formula.
The z-score formula is given by: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
For the bottom 80% of values, the z-score will be -0.8416, because 80% of the values fall below a z-score of -0.8416.
We can then use the formula z = (x - μ) / σ to find the value that corresponds to this z-score. Rearranging the formula, we get x = z * σ + μ.
Plugging in the values, x = -0.8416 * 31.3 + 89.1 = 62.66.
To find the sample mean separating the bottom 80% sample means from the top 20% sample means, we can use the same approach as above. We'll use the z-score formula, but this time the mean and standard deviation will be for the sample means.
The sample mean has the same mean as the population mean (μ = 89.1), but the standard deviation is given by σ / √n, where n is the sample size (58). So, the standard deviation for the sample means is 31.3 / √58 = 4.1081.
Using the z-score formula with a z-score of -0.8416 and a standard deviation of 4.1081, we can find the sample mean that corresponds to the bottom 80% sample means. Plugging in the values, x = -0.8416 * 4.1081 + 89.1 = 85.2829.