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 at the 60th percentile, we again standardize:

Using the z-value for the 60th percentile, which is approximately 0.25337, we rearrange the formula:

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

Where:

• $\mu = 162$ cm (mean height)
• $\sigma = 7.5$ cm (standard deviation)

Calculating her estimated height:

$x = 162 + (0.25337 \cdot 7.5) \approx 162 + 1.900275 \approx 163.9 \text{ cm}$

Therefore, Sarah's estimated height as an adult is approximately 164 cm.

Let the mean height of adult females be represented as $\mu_f$, which is equal to 162 cm.

We know that 90% of adult males are taller than $\mu_f$, meaning that:

$P(X > \mu_m) = 0.90$

This implies that:

$P(X \leq \mu_m) = 0.10$

Using the z-score table, the z-value for the lower 10% is approximately -1.2816.

Now, we can express $\mu_m$ in terms of the population mean and standard deviation:

$\mu_m = \mu_f + z \cdot \sigma_m$

Where:

• $\sigma_m = 9.0$ cm

Substituting the values:

$\mu_m = 162 + (-1.2816 \cdot 9.0) \approx 162 - 11.5344 \approx 173.533$

Thus, the mean height of an adult male is approximately 174 cm.