A population of values has a normal distribution with $\mu=25.1$ and $\sigma=3.6$. You intend to draw a random sample of size $\mathrm{n}=213$. Please answer the following questions, and show your answers to 1 decimal place. Find $P_{25}$, which is the value $(X)$ separating the bottom $25 \%$ values from the top $75 \%$ values. $P_{25}$ (for population) $=$ Find $P_{25}$, which is the sample mean $(\bar{x})$ separating the bottom $25 \%$ sample means from the top $75 \%$ sample means. $P_{25}$ (for sample means) = Submit Question
Solution
Step 1 ：Given that the population of values has a normal distribution with a mean (\( \mu \)) of 25.1 and a standard deviation (\( \sigma \)) of 3.6. The sample size (\( n \)) is 213.
Step 2 ：We are asked to find the value (\( X \)) that separates the bottom 25% of the values from the top 75% of the values in the population. This is also known as the 25th percentile or \( P_{25} \).
Step 3 ：We can use the formula \( X = \mu + Z\sigma \), where \( Z \) is the z-score for the 25th percentile. The z-score for the 25th percentile is -0.674.
Step 4 ：Substituting the given values into the formula, we get \( X = 25.1 + (-0.674) \times 3.6 \).
Step 5 ：Solving the equation gives \( X \approx 22.7 \). So, \( P_{25} \) for the population is approximately 22.7.
Step 6 ：We are also asked to find the sample mean (\( \bar{x} \)) that separates the bottom 25% of the sample means from the top 75% of the sample means. This is also known as the 25th percentile or \( P_{25} \) for the sample means.
Step 7 ：We can use the formula \( \bar{x} = \mu + Z(SE) \), where \( SE \) is the standard error. The standard error is calculated as \( \sigma / \sqrt{n} \).
Step 8 ：Substituting the given values into the formula, we get \( SE = 3.6 / \sqrt{213} \).
Step 9 ：Solving the equation gives \( SE \approx 0.247 \).
Step 10 ：Substituting the values of \( \mu \), \( Z \), and \( SE \) into the formula for \( \bar{x} \), we get \( \bar{x} = 25.1 + (-0.674) \times 0.247 \).
Step 11 ：Solving the equation gives \( \bar{x} \approx 24.9 \). So, \( P_{25} \) for the sample means is approximately 24.9.
Step 12 ：Final Answer: \( P_{25} \) (for population) = \(\boxed{22.7}\)
Step 13 ：Final Answer: \( P_{25} \) (for sample means) = \(\boxed{24.9}\)
