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 adult height, we need to find the z-score that corresponds to the 60th percentile. From standard normal distribution tables, the z-score for the 60th percentile is approximately:

$z \approx 0.253$

We then use the z-score formula again, rearranging it to find the height $X$:

$X = \mu + z \sigma$

For adult females, $\mu = 162$ cm and $\sigma = 7.5$ cm. Plugging the values in, we get:

$X = 162 + 0.253 \times 7.5 \approx 162 + 1.8975 \approx 163.9$

Therefore, Sarah's estimated height as an adult is approximately 163.9 cm.

Let $\mu$ represent the mean height of adult males. Since 90% of adult males are taller than the mean height of adult females, we set up the equation:

$P(X > 162) = 0.90$

This implies that:

$P(X \leq 162) = 0.10$

Using the z-score table: the z-score corresponding to 0.10 is approximately -1.2816. We can use the z-score formula to find the mean height of adult males:

$z = \frac{X - \mu}{\sigma}$

Substituting the known values, we have:

$-1.2816 = \frac{162 - \mu}{9}$

Rearranging gives:

$\mu = 162 + (-1.2816) \times 9 \approx 162 - 11.5344 \approx 150.4656$

Thus, the estimated mean height of an adult male is approximately 150.47 cm.