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

To find $q_3$, we need to find the z-score corresponding to the 75th percentile of the normal distribution. The z-score for the upper quartile is approximately 0.674.

Now, we can convert this z-score back to the weight ($x$):

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

Substituting the values:

$x = 140 + 0.674 \cdot 40 = 140 + 26.96 = 166.96$

Thus, $q_3$ is approximately 167 g.

To find the probability that a randomly selected potato weighs more than $q_3$:

• Since we know that $q_3$ is the 75th percentile, the probability is: $P(X > q_3) = 1 - P(X < q_3) = 1 - 0.75 = 0.25$ Therefore, the probability that the weight of a randomly selected potato is more than $q_3$ is 0.25.

To find $q_1$, we find the z-score corresponding to the 25th percentile. This z-score is approximately -0.674.

We convert this z-score back to weight:

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

Now substituting the values:

$x = 140 + (-0.674) \cdot 40 = 140 - 26.96 = 113.04$

Thus, $q_1$ is approximately 113 g.

To find this probability, we consider three selected potatoes:

• Let $P_1$ = potato that weighs less than $q_1$,
• Let $P_2$ = potato that weighs more than $q_3$,
• Let $P_3$ = potato that has a weight between $q_1$ and $q_3$.

From previous calculations:

• $P(P_1) = P(X < q_1) = 0.25$
• $P(P_2) = P(X > q_3) = 0.25$
• The probability of selecting a potato with weight between $q_1$ and $q_3$ is: $P(q_1 < X < q_3) = 0.50$

Now, for three independent selections: The total probability is: $P(P_1 ext{ and } P_2 ext{ and } P_3) = P(P_1) \cdot P(P_2) \cdot P(P_3) = 0.25 \cdot 0.25 \cdot 0.50 = 0.03125$ Thus, the probability is approximately 0.03125.