In a particular year, the height of a male athlete at the Summer Olympics has a mean 1.78 metres and standard deviation 0.23 metres - AQA - A-Level Maths Pure - Question 18 - 2022 - Paper 3

To find this probability, we first convert the height to a z-score using the formula:

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

where $x = 1.82$, $\mu = 1.78$, and $\sigma = 0.23$.

Calculating the z-score:

$z = \frac{1.82 - 1.78}{0.23} \approx 0.1739$

Using standard normal distribution tables, we find that the probability corresponding to this z-score is approximately 0.4320. Thus, the probability that the height is exactly 1.82 metres is the density function value, which can be calculated with:

$P(X = 1.82) \approx 0.4320$.

To calculate this probability, we find the z-scores for both heights.

For $x_1 = 1.70$:

$z_1 = \frac{1.70 - 1.78}{0.23} \approx -0.3478$

For $x_2 = 1.90$:

$z_2 = \frac{1.90 - 1.78}{0.23} \approx 0.5217$

Using the standard normal distribution table, we find:

• $P(Z < z_1) \approx 0.3656$
• $P(Z < z_2) \approx 0.6968$

Thus, the probability that the height is between 1.70 and 1.90 metres is:

$P(1.70 < X < 1.90) = P(Z < 0.5217) - P(Z < -0.3478) \approx 0.6968 - 0.3656 = 0.3312$.

Using the results from part (b)(ii), we found the probability that a randomly selected athlete's height is between 1.70 and 1.90 metres is approximately 0.3312. For two independent events (the heights of two athletes), we multiply the probabilities:

$P(A \cap B) = P(A) \times P(B) = (0.3312) \times (0.3312) \approx 0.1093$.

The mean height ($\bar{h}$) can be calculated as follows:

$\bar{h} = \frac{\sum h}{n} = \frac{69.2}{40} = 1.73 \text{ metres}$

For the standard deviation (s), use the formula:

$s = \sqrt{\frac{\sum (h - \bar{h})^2}{n - 1}}$

Substituting the values:

$s = \sqrt{\frac{2.81}{39}} \approx 0.265\text{ metres}$.

From part (c), we determined that the mean height for male athletes at the Winter Olympics is 1.73 metres, compared to the mean height of 1.78 metres at the Summer Olympics. Hence, on average, male athletes at the Summer Olympics are taller than those at the Winter Olympics. Additionally, the standard deviation at the Summer Olympics is 0.23, suggesting that the heights there are more consistent compared to the Winter Olympics, which have a standard deviation of approximately 0.265.