Questions: A normal population has mean μ=39 and standard deviation σ=16. Find the values that separate the middle 85% of the population from the top and bottom 7.5% The values that separate the middle 85% of the population above from the top and bottom 7.5% are and . Enter the answers in ascending order and round to one decimal place.

Solution Steps

We are tasked with finding the values that separate the middle \( 85\% \) of a normally distributed population from the top and bottom \( 7.5\% \). Given the population parameters:

• Mean \( \mu = 39 \)
• Standard deviation \( \sigma = 16 \)

To find the values that separate the middle \( 85\% \), we need to identify the z-scores corresponding to the cumulative probabilities:

• Lower bound: \( 7.5\% \) (or \( 0.075 \))
• Upper bound: \( 92.5\% \) (or \( 0.925 \))

Using the standard normal distribution, we find the z-scores:

• \( z_{lower} = \Phi^{-1}(0.075) \)
• \( z_{upper} = \Phi^{-1}(0.925) \)

The original values corresponding to these z-scores can be calculated using the formula: \[ X = \mu + z \cdot \sigma \] Thus, we compute:

• For the lower bound: \[ X_{lower} = 39 + z_{lower} \cdot 16 \]
• For the upper bound: \[ X_{upper} = 39 + z_{upper} \cdot 16 \]

After performing the calculations, we find:

• \( X_{lower} = 16.0 \)
• \( X_{upper} = 62.0 \)

Final Answer