Question
A farmer is growing a field of wheat plants. The height, \( H \) cm, of each plant can be modelled by a normal distribution with mean \( \mu \) and standard deviation \( \sigma \).
It is known that \( P(H < 94.6) = 0.288 \) and \( P(H > 98.1) = 0.434 \).
(a) Find the probability that the height of a randomly selected plant is between 94.6 cm and 98.1 cm. [2]
(b) Find the value of \( \mu \) and the value of \( \sigma \). [5]
(c) (i) Find the probability that exactly 34 plants are ready to harvest. [2]
(ii) Given that fewer than 49 plants are ready to harvest, find the probability that exactly 34 plants are ready to harvest. [4]
(d) Find the value of \( d \) where the interquartile range is 4.82 cm. [3]
▶️Answer/Explanation
(a) \[ P(94.6 < H < 98.1) = 1 – 0.288 – 0.434 = 0.278 \]
(b) Using the inverse normal function: \[ \mu = 97.3 \, \text{cm}, \quad \sigma = 4.82 \, \text{cm} \]
(c)(i) \[ P(X = 34) = \binom{100}{34} (0.434)^{34} (0.566)^{66} \approx 0.0133 \]
(c)(ii) \[ P(X = 34 | X < 49) = \frac{P(X = 34)}{P(X < 49)} \approx \frac{0.0133}{0.848} \approx 0.0157 \]
(d) \[ d = \frac{4.82}{1.349} \approx 3.57 \, \text{cm} \]