Question

the heights of a certain type of tree are approximately normally distributed with a mean height $mu = 5$ ft and a standard deviation $sigma = 0.4$ ft. which statement must be true?
a. a tree with a height of 5.4 ft is 1 standard deviation below the mean.
b. a tree with a height of 4.6 ft is 1 standard deviation above the mean.
c. a tree with a height of 5.8 ft is 2.5 standard deviations above the mean.
d. a tree with a height of 6.2 ft is 3 standard deviations above the mean.

Explanation:

Step1: Define z-score formula

The z-score formula is $z = \frac{x - \mu}{\sigma}$, where $x$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Test Option A

Substitute $x=5.4$, $\mu=5$, $\sigma=0.4$
$z = \frac{5.4 - 5}{0.4} = \frac{0.4}{0.4} = 1$
This is 1 standard deviation above the mean, not below.

Step3: Test Option B

Substitute $x=4.6$, $\mu=5$, $\sigma=0.4$
$z = \frac{4.6 - 5}{0.4} = \frac{-0.4}{0.4} = -1$
This is 1 standard deviation below the mean, not above.

Step4: Test Option C

Substitute $x=5.8$, $\mu=5$, $\sigma=0.4$
$z = \frac{5.8 - 5}{0.4} = \frac{0.8}{0.4} = 2$
This is 2 standard deviations above the mean, not 2.5.

Step5: Test Option D

Substitute $x=6.2$, $\mu=5$, $\sigma=0.4$
$z = \frac{6.2 - 5}{0.4} = \frac{1.2}{0.4} = 3$
This is 3 standard deviations above the mean.

Answer:

D. A tree with a height of 6.2 ft is 3 standard deviations above the mean.