select the correct answer.

the income distribution in a county is a normal distribution with a mean income of $10,000. the top 2.5% of the wage earners earn $18,000 or more. which sentence most closely summarizes this information?

95% of the wage earners earn less than $14,000 each.
16% of the wage earners earn less than $14,000 each.
68% of the wage earners earn less than $14,000 each.
84% of the wage earners earn less than $14,000 each.

Question

select the correct answer.

the income distribution in a county is a normal distribution with a mean income of $10,000. the top 2.5% of the wage earners earn $18,000 or more. which sentence most closely summarizes this information?

95% of the wage earners earn less than $14,000 each.
16% of the wage earners earn less than $14,000 each.
68% of the wage earners earn less than $14,000 each.
84% of the wage earners earn less than $14,000 each.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Empirical Rule", "Normal Distribution Calculations" ], "new_concepts": [], "current_concepts": [ "Empirical Rule", "Normal Distribution Calculations" ] } ## Determine the standard deviation using the empirical rule $$ \begin{aligned} &\mu = 10000 \ &P(X \ge 18000) = 0.025 \implies P(X < 18000) = 0.975 \ &z = 2 \implies \mu + 2\sigma = 18000 \ &10000 + 2\sigma = 18000 \implies \sigma = 4000 \end{aligned} $$ ## Calculate the z-score for the target income $$ \begin{aligned} &X = 14000 \ &z = \frac{14000 - 10000}{4000} = 1 \end{aligned} $$ ## Find the percentage of earners below this threshold $$ \begin{aligned} &P(X < 14000) = P(Z < 1) \ &P(Z < 1) = 0.5 + \frac{0.68}{2} = 0.84 \implies 84\% \end{aligned} $$

95% of the wage earners earn less than $14,000 each.

16% of the wage earners earn less than $14,000 each.

68% of the wage earners earn less than $14,000 each.

84% of the wage earners earn less than $14,000 each.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Empirical Rule" ] }

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Question

Explanation:

Determine the standard deviation using the empirical rule

Calculate the z-score for the target income

Find the percentage of earners below this threshold

Answer: