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 the probability, we first convert the height to a z-score using the formula: $z = \frac{x - \mu}{\sigma}$ Where:

• $x$ = 1.82 m
• $\mu$ = 1.78 m
• $\sigma$ = 0.23 m

Calculating: $z = \frac{1.82 - 1.78}{0.23} = \frac{0.04}{0.23} \approx 0.174$

Using the z-table, the probability of a z-score of 0.174 is approximately 0.4320. Hence, the probability of selecting a male athlete with a height of 1.82 m is approximately 0.432.

First, we will calculate the z-scores for both heights:

• For 1.70 m: $z_{1.70} = \frac{1.70 - 1.78}{0.23} = \frac{-0.08}{0.23} \approx -0.348$
• For 1.90 m: $z_{1.90} = \frac{1.90 - 1.78}{0.23} = \frac{0.12}{0.23} \approx 0.522$

Using the z-table:

• The probability for $z_{1.70} = -0.348$ is approximately 0.3643.
• The probability for $z_{1.90} = 0.522$ is approximately 0.7019.

Thus, the probability that a randomly selected male athlete's height is between 1.70 m and 1.90 m is: $P(1.70 < x < 1.90) = P(z_{1.90}) - P(z_{1.70}) \approx 0.7019 - 0.3643 \approx 0.3376$

If the heights of the athletes are independent, the probability that both heights fall within this range is the product of the individual probabilities calculated in part (b) (ii): $P(both) = P(1.70 < x < 1.90)^2 \approx (0.3376)^2 \approx 0.1135$

Given the data:

• The total height sum is $\\sum h = 69.2$ for $n = 40$ athletes. Thus, the mean is calculated as: $\bar{h} = \frac{69.2}{40} = 1.73 \, m$

For the standard deviation, we use: $s = \sqrt{\frac{(\\sum (h - \bar{h})^2)}{n - 1}}$ Substituting the values: $s = \sqrt{\frac{2.81}{39}} \approx 0.265$

From our calculations:

• The mean height of male athletes at the Summer Olympics is 1.78 m, while for the Winter Olympics it is 1.73 m. This indicates that athletes at the Summer Olympics tend to be taller on average.
• Furthermore, considering the standard deviations, the Summer Olympics have a standard deviation of 0.23 m compared to 0.265 m for the Winter Olympics, indicating that the heights of Summer Olympic athletes are less varied compared to those of their Winter Olympic counterparts.