Question

A plant biologist studies the height of sunflowers. He measures a large sample of sunflowers and creates a probability distribution. The distribution is normal in shape with mean 112 cm and standard deviation 16 cm. What is the probability (approximately) that a sunflower will be less than 136 cm tall?

Answer 1

Probability that a sunflower will be less than 136 cm tall is 0.9332.

Explanation:

We are given that a plant biologist studies the height of sunflowers. He measures a large sample of sunflowers and creates a probability distribution.

The distribution is normal in shape with mean 112 cm and standard deviation 16 cm.

Let X = height of sunflowers

So, X ~ N(
`\mu=112,\sigma^(2) = 16^(2)`)

The z-score probability distribution for single selected value is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = population mean height = 112 cm

`\sigma` = standard deviation = 16 cm

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

So, probability that a sunflower will be less than 136 cm tall is given by = P(X < 136 cm)

P(X < 136 cm) = P(
`(X-\mu)/(\sigma)` <
`(136-112)/(16)` ) = P(Z < 1.50) = 0.9332

The above probabilities is calculated by looking at the value of x = 1.50 in the z table which has an area of 0.9332.

Hence, probability that a sunflower will be less than 136 cm tall is 0.9332.