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

Next, we need to find the probability that a randomly selected athlete weighs less than 97 kg.

Let $W$ be the weight of athletes, so $W \sim N(85, 7.1^2)$.

We need to find: P(W < 97) = P\left(Z < \frac{97 - 85}{7.1}\right)\text{, where } Z \text{ is a standard normal variable.}\ Calculating this:

1. Standardize the variable: $Z = \frac{97 - 85}{7.1} = \frac{12}{7.1} \approx 1.6901$

2. Use standard normal distribution tables or a calculator to find: $P(Z < 1.6901) \approx 0.9545$.

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

To find this combined probability, we will use the results from parts (a) and (b).

We know:

• $P(H > 188) \approx 0.0618$ (from part (a))
• $P(W > 97) = 1 - P(W < 97) \approx 1 - 0.9545 = 0.0455$ (from part (b)).

Since height and weight are independent:

The combined probability is: $P(H > 188 \text{ and } W > 97) = P(H > 188) \times P(W > 97)$

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

Thus, the probability that a randomly chosen athlete is taller than 188 cm and weighs more than 97 kg is approximately 0.00281.

The assumption of independence between height and weight is not sensible. Evidence suggests that height and weight are positively correlated. Typically, taller individuals tend to weigh more, primarily due to a larger body mass. Therefore, assuming these two variables to be independent can lead to inaccurate calculations and interpretations regarding the characteristics of athletes.