The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm - Edexcel - A-Level Maths Statistics - Question 6 - 2012 - Paper 2

To estimate Sarah's height as an adult, we find the z-score corresponding to the 60th percentile.
From z-tables, we find that the z-score for the 60th percentile is approximately $z \approx 0.2533$.
Using the formula for solving for height:

$x = \mu + z \cdot \sigma$

Where:

• $\mu = 162$ cm (mean of adult females)
• $\sigma = 7.5$ cm (standard deviation of adult females)
• $z \approx 0.2533$

So, substituting the values:

$x = 162 + 0.2533 \cdot 7.5 \approx 162 + 1.89975 \approx 163.9$
Thus, Sarah's estimated height is approximately 164 cm.

Given that 90% of adult males are taller than the mean height of adult females, we first find the z-score corresponding to the 10th percentile.
Looking it up, the z-score for the 10th percentile is approximately $z \approx -1.2816$.
Using the equation for the mean height of an adult male:

$\mu = \text{mean height of adult females} + z \cdot \sigma$

Where:

• Mean height of adult females is $162$ cm.
• Standard deviation for adult males is $9.0$ cm.
• Therefore:

$\mu = 162 + (-1.2816 \cdot 9.0) \approx 162 - 11.5344 \approx 150.4656$
Thus, the mean height of an adult male is approximately $174$ cm.