find the area of the shaded region. the graph to the right depicts iq scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.

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the area of the shaded region is (round to four decimal places as needed.)

Question

find the area of the shaded region. the graph to the right depicts iq scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.

click to view page 1 of the table. click to view page 2 of the table.

the area of the shaded region is   (round to four decimal places as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Normal Distribution", "Z-score Calculation", "Normal Curve Area" ], "new_concepts": [], "current_concepts": [ "Normal Distribution", "Z-score Calculation", "Normal Curve Area" ] } ## Calculate the z-scores for the boundaries $$ \begin{aligned} &\mu = 100,\quad \sigma = 15\ &z_1 = \frac{70 - 100}{15} = -2.00\ &z_2 = \frac{125 - 100}{15} = 1.67 \end{aligned} $$ ## Find the cumulative probabilities $$ \begin{aligned} &P(Z < -2.00) = 0.0228\ &P(Z < 1.67) = 0.9525 \end{aligned} $$ ## Compute the shaded area $$ \begin{aligned} &\text{Area} = P(Z < 1.67) - P(Z < -2.00)\ &\text{Area} = 0.9525 - 0.0228 = 0.9297 \end{aligned} $$ The area of the shaded region is 0.9297 (Round to four decimal places as needed.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Normal Curve Area" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the z-scores for the boundaries

Find the cumulative probabilities

Compute the shaded area

Answer: