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 tree with a height of 5.4 ft is 1 standard deviation below the mean.
• a tree with a height of 4.6 ft is 1 standard deviation above the mean.
• a tree with a height of 5.8 ft is 2.5 standard deviations above the mean.
• a tree with a height of 6.2 ft is 3 standard deviations above the mean.

Explanation:

Step1: Recall z - score formula

The z - score formula is \(z=\frac{x - \mu}{\sigma}\), where \(x\) is the value from the dataset, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. We will calculate the z - score for each option.

Step2: Analyze Option 1

For \(x = 5.4\), \(\mu=5\), \(\sigma = 0.4\).
\(z=\frac{5.4 - 5}{0.4}=\frac{0.4}{0.4}=1\). A z - score of 1 means the value is 1 standard deviation above the mean, not below. So this option is false.

Step3: Analyze Option 2

For \(x = 4.6\), \(\mu = 5\), \(\sigma=0.4\).
\(z=\frac{4.6 - 5}{0.4}=\frac{- 0.4}{0.4}=- 1\). A z - score of - 1 means the value is 1 standard deviation below the mean, not above. So this option is false.

Step4: Analyze Option 3

For \(x = 5.8\), \(\mu=5\), \(\sigma = 0.4\).
\(z=\frac{5.8 - 5}{0.4}=\frac{0.8}{0.4}=2\). The z - score is 2, not 2.5. So this option is false.

Step5: Analyze Option 4

For \(x = 6.2\), \(\mu = 5\), \(\sigma=0.4\).
\(z=\frac{6.2 - 5}{0.4}=\frac{1.2}{0.4}=3\). A z - score of 3 means the value is 3 standard deviations above the mean. So this option is true.

Answer:

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