Questions: In a survey of women in the United States (ages 20-29), the mean height was 64.1 inches with a population standard deviation of 2.71 inches. Women's heights are normally distributed. a) What height represents the 95th percentile? b) What height represents the first quartile?

Find the height that represents the 95th percentile

Identify the given information

• Mean height (μ) = 64.1 inches
• Population standard deviation (σ) = 2.71 inches
• Heights are normally distributed
• We need to find the 95th percentile

Determine the z-score for the 95th percentile

The 95th percentile corresponds to a z-score of 1.645 (from the standard normal table).

Calculate the height using the z-score formula

Height = μ + z·σ Height = 64.1 + 1.645(2.71) Height = 64.1 + 4.46 Height = 68.56 inches

\(\boxed{\text{The 95th percentile height is 68.56 inches}}\)

Find the height that represents the first quartile

Identify the given information

• Mean height (μ) = 64.1 inches
• Population standard deviation (σ) = 2.71 inches
• Heights are normally distributed
• We need to find the first quartile (25th percentile)

Determine the z-score for the 25th percentile

The 25th percentile corresponds to a z-score of -0.675 (from the standard normal table).

Calculate the height using the z-score formula

Height = μ + z·σ Height = 64.1 + (-0.675)(2.71) Height = 64.1 - 1.83 Height = 62.27 inches

\(\boxed{\text{The first quartile height is 62.27 inches}}\)

\(\boxed{\text{The 95th percentile height is 68.56 inches}}\) \(\boxed{\text{The first quartile height is 62.27 inches}}\)