Farmer Adam grows potatoes - Edexcel - A-Level Maths Statistics - Question 7 - 2018 - Paper 1

The upper quartile, $q_3$, corresponds to the 75th percentile. To find the z-score for the 75th percentile, we look it up in the z-table, which gives approximately $z = 0.674$.

Using the z-score formula:

$q_3 = \mu + z \cdot \sigma$

Substituting the values:

$q_3 = 140 + 0.674 \cdot 40 \approx 140 + 26.96 = 166.96 \text{ g}$

To find this probability, we need to calculate:

$P(X > q_3) = 1 - P(X < q_3)$

From the previous step, we found:

$P(X < q_3) = 0.75$

Thus, the probability that a randomly selected potato weighs more than $q_3$ is:

$P(X > q_3) = 1 - 0.75 = 0.25$

The lower quartile, $q_1$, corresponds to the 25th percentile. The z-score for the 25th percentile from the z-table is approximately $z = -0.674$.

Using the z-score formula:

$q_1 = \mu + z \cdot \sigma$

Substituting the values:

$q_1 = 140 + (-0.674) \cdot 40 \approx 140 - 26.96 = 113.04 \text{ g}$

Let’s denote:

• $P(X < q_1)$ as $p1$,
• $P(X > q_3)$ as $p3$,
• $P(q_1 < X < q_3)$ as $p2$.

From previous calculations, we have:

• $p1 \approx 0.25$ (from $q_1$)
• $p3 \approx 0.25$ (from $q_3$)

Thus,

$p2 = 1 - (p1 + p3) = 1 - (0.25 + 0.25) = 0.5$

The probability of selecting three potatoes where one is in each category is:

$P = p1 \times p2 \times p3 \times 3! = (0.25 \times 0.25 \times 0.5) \times 6 = 0.1875 \approx 18.75\%$