The time, in minutes, taken by men to run a marathon is modelled by a normal distribution with mean 240 minutes and standard deviation 40 minutes - Edexcel - A-Level Maths Statistics - Question 6 - 2016 - Paper 1

To determine the longest time Nathaniel can take to finish in the first 20% of male runners, we need to find the 20th percentile of the normal distribution:

$P(Z < z) = 0.20$

Using a Z-table, we find: $z \approx -0.8416$

Now, we can convert this Z-score back to the actual time using:

$X = \mu + z \cdot \sigma$ where:

• $\mu = 240$
• $\sigma = 40$
• $z \approx -0.8416$

Calculating: $X = 240 + (-0.8416) \cdot 40 \approx 240 - 33.664 \approx 206.336$

Thus, Nathaniel can take approximately 206 minutes to run the marathon to be in the top 20%.

To solve the conditional probability, we have:

$P(W < μ - 30 | W < μ) = \frac{P(W < μ - 30)}{P(W < μ)}$

Given: $P(W < μ + 30) = 0.82$. Thus,

$P(W < μ) = 1 - P(W > μ) = 1 - 0.82 = 0.18$

Also from the problem, we have: $P(W < μ - 30) = P(W < μ) * P(W < μ - 30 | W < μ)$ This leads to:

$P(W < μ - 30) = 0.18 * k$

Now, using the values derived, we know: $P(W < μ - 30) = P(W < μ - 30) = 0.36 \\ herefore \\ k \approx 0.36 / 0.18 = 2$ So, $P(W < μ - 30 | W < μ) = \frac{0.36}{0.18} = 0.36$.