Question

The length of a fully grown hibiscus plants is normally distributed and has a mean of 35 cm and a standard deviation of 4 cm. i. When plants are chosen for sale, those shorter than 30 cm are rejected. Find the probability that a plant will be rejected. [4] ii. 20% of hibiscus plants is taller than xcm. Find the value of x. [4]

Answer 1

The probability that a plant will be rejected is 10.56% and the value of x such that 20% of hibiscus plants are taller is approximately 31.54 cm.

Step-by-step explanation:

To find the probability that a plant will be rejected, we need to calculate the z-score for the threshold value of 30 cm using the formula z = (X - μ) / σ, where X is the threshold value, μ is the mean, and σ is the standard deviation.

z = (30 - 35) / 4 = -1.25

Next, we need to find the cumulative probability for a z-score of -1.25 using a standard normal distribution table or a calculator. The cumulative probability for a z-score of -1.25 is approximately 0.1056.

Therefore, the probability that a plant will be rejected is 0.1056 or 10.56%.

To find the value of x such that 20% of hibiscus plants are taller, we need to find the z-score corresponding to a cumulative probability of 20%. Using a standard normal distribution table or a calculator, we find that the z-score is approximately -0.8416.

Now, we can use the z-score formula to find the value of x. Rearranging the formula, we get x = z * σ + μ.

x = -0.8416 * 4 + 35 = 31.5354

Therefore, the value of x is approximately 31.54 cm.