Questions: A population of values has a normal distribution with μ=233.5 and σ=22.7. You intend to draw a random sample of size n=210. Find P43, which is the mean separating the bottom 43% means from the top 57% means. P43 (for sample means) = Enter your answers as numbers accurate to 1 decimal place. Answers obtained using exact z scores or z-scores rounded to 3 decimal places are accepted.

Solution Steps

To find the Z-score corresponding to the 43rd percentile, we use the formula:

\[ z = \frac{X - \mu}{\sigma} \]

For the 43rd percentile, we have:

\[ z = \frac{0.43 - 0}{1} = 0.43 \]

Thus, the Z-score for the 43rd percentile is \( z = 0.43 \).

The standard error of the mean (SEM) is calculated using the formula:

\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} \]

Substituting the given values:

\[ \text{SEM} = \frac{22.7}{\sqrt{210}} \approx 1.5664 \]

To find \( P_{43} \), which is the mean separating the bottom 43% of sample means from the top 57%, we use the formula:

\[ P_{43} = \mu + z \cdot \text{SEM} \]

Substituting the values we have:

\[ P_{43} = 233.5 + 0.43 \cdot 1.5664 \approx 234.2 \]

Final Answer