The heights of a group of athletes are modelled by a normal distribution with mean 180 cm and a standard deviation 5.2 cm - Edexcel - A-Level Maths Statistics - Question 7 - 2006 - Paper 1

Let $W$ be the weight of athletes, so $W \sim N(85, 7.1^2)$. To find the probability that an athlete weighs less than 97 kg:

$P(W < 97) = P\left(Z < \frac{97 - 85}{7.1}\right) = P(Z < 1.69)$

Using standard normal distribution tables or calculators, we find:

$P(Z < 1.69) \approx 0.9545$

Thus, the probability that a randomly chosen athlete weighs less than 97 kg is approximately 0.9545.

Given the assumption of independence between height and weight, we can calculate:

$P(H > 188 \text{ and } W > 97) = P(H > 188) \times P(W > 97)$

From part (a), we have: $P(H > 188) \approx 0.0618$

And: $P(W > 97) = 1 - P(W < 97) \approx 1 - 0.9545 = 0.0455$

Now, substituting these values into the equation:

$P(H > 188 \text{ and } W > 97) \approx 0.0618 \times 0.0455 \approx 0.00281$

Thus, the combined probability is approximately 0.00281.

Evidence suggests that height and weight are positively correlated, indicating that taller athletes generally weigh more. Thus, the assumption of independence is not sensible.