Answer: See Explanation
Explanation: To find the value separating the bottom 35% values from the top 65% values, we can use the z-score formula. The z-score measures the number of standard deviations an individual value is from the mean.
First, we need to find the z-score corresponding to the 35th percentile. The 35th percentile is the same as saying that 35% of the values are below it. To find the z-score, we use the cumulative distribution function (CDF) of the standard normal distribution.
Using a z-score table or a calculator, we find that the z-score corresponding to the 35th percentile is approximately -0.3853.
To find the corresponding value in the original distribution, we use the formula:
X = μ + (z * σ)
where X is the value we're looking for, μ is the mean, z is the z-score, and σ is the standard deviation.
Plugging in the values, we get:
X = 244.3 + (-0.3853 * 91.3) ≈ 209.9
Therefore, the value separating the bottom 35% values from the top 65% values is approximately 209.9.
Now, to find the sample mean separating the bottom 35% sample means from the top 65% sample means, we can use the same approach.
The sample mean follows a normal distribution with the same mean as the population mean (μ) and a standard deviation (σ/√n), where n is the sample size.
To find the z-score corresponding to the 35th percentile, we use the same steps as before but with the adjusted standard deviation:
Z = (X - μ) / (σ/√n)
where Z is the z-score, X is the sample mean we're looking for, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Using the z-score table or a calculator, we find that the z-score corresponding to the 35th percentile is approximately -0.3853 (same as before).
To find the corresponding sample mean, we rearrange the formula:
X = μ + (Z * (σ/√n))
Plugging in the values, we get:
X = 244.3 + (-0.3853 * (91.3/√207)) ≈ 240.2
Therefore, the sample mean separating the bottom 35% sample means from the top 65% sample means is approximately 240.2.