the heights of a certain type of tree are approximately normally distributed with a mean height $\mu = 5\text{ ft}$ and a standard deviation $\sigma = 0.4\text{ ft}$. which statement must be true?

a tree with a height of $5.4\text{ ft}$ is 1 standard deviation below the mean.
a tree with a height of $4.6\text{ ft}$ is 1 standard deviation above the mean.
a tree with a height of $5.8\text{ ft}$ is 2.5 standard deviations above the mean.
a tree with a height of $6.2\text{ ft}$ is 3 standard deviations above the mean.

Question

the heights of a certain type of tree are approximately normally distributed with a mean height $\mu = 5\text{ ft}$ and a standard deviation $\sigma = 0.4\text{ ft}$. which statement must be true?

a tree with a height of $5.4\text{ ft}$ is 1 standard deviation below the mean.
a tree with a height of $4.6\text{ ft}$ is 1 standard deviation above the mean.
a tree with a height of $5.8\text{ ft}$ is 2.5 standard deviations above the mean.
a tree with a height of $6.2\text{ ft}$ is 3 standard deviations above the mean.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Z-Score Formula" ], "new_concepts": [], "current_concepts": [ "Normal Distribution", "Z-Score Formula", "Standard Deviation Interval" ] } ## Identify given parameters Using the *Z-Score Formula* knowledge point $$ \begin{aligned} &\mu = 5\text{ ft}\ &\sigma = 0.4\text{ ft} \end{aligned} $$ ## Calculate z-scores for each option Using the *Z-Score Formula* knowledge point $$ \begin{aligned} &z = \frac{x - \mu}{\sigma}\ &\text{Option 1: } x = 5.4 \implies z = \frac{5.4 - 5}{0.4} = 1\ &\text{Option 2: } x = 4.6 \implies z = \frac{4.6 - 5}{0.4} = -1\ &\text{Option 3: } x = 5.8 \implies z = \frac{5.8 - 5}{0.4} = 2\ &\text{Option 4: } x = 6.2 \implies z = \frac{6.2 - 5}{0.4} = 3 \end{aligned} $$ ## Evaluate statements Using the *Z-Score Formula* knowledge point $$ \begin{aligned} &z = 1 \implies 1\text{ standard deviation above the mean}\ &z = -1 \implies 1\text{ standard deviation below the mean}\ &z = 2 \implies 2\text{ standard deviations above the mean}\ &z = 3 \implies 3\text{ standard deviations above the mean} \end{aligned} $$

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.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Z-Score Formula" ] }

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QUESTION IMAGE

Question

Explanation:

Identify given parameters

Calculate z-scores for each option

Evaluate statements

Answer: