question 2 points 3

given a normal distribution with $\mu = 60$ and $\sigma = 12$. find the probability that $x$ assumes a value between 36 and 84.

0.9426
0.9544

Question

question 2 points 3

given a normal distribution with $\mu = 60$ and $\sigma = 12$. find the probability that $x$ assumes a value between 36 and 84.

0.9426
0.9544

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Normal Distribution Properties", "Empirical Rule", "Z-score Calculation" ], "new_concepts": [], "current_concepts": [ "Normal Distribution Properties", "Empirical Rule", "Z-score Calculation" ] } ## Identify the given parameters Using the *Normal Distribution Properties* knowledge point $$ \begin{aligned} &\mu = 60\ &\sigma = 12\ &\text{Target interval: } 36 < X < 84 \end{aligned} $$ ## Calculate the Z-scores Using the *Z-score Calculation* knowledge point $$ \begin{aligned} &Z_1 = \frac{36 - 60}{12} = \frac{-24}{12} = -2\ &Z_2 = \frac{84 - 60}{12} = \frac{24}{12} = 2 \end{aligned} $$ ## Apply the Empirical Rule Using the *Empirical Rule* knowledge point $$ \begin{aligned} &P(36 < X < 84) = P(-2 < Z < 2)\ &P(-2 < Z < 2) \approx 0.9544 \end{aligned} $$

0.9426

0.9544

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QUESTION IMAGE

Question

Explanation:

Identify the given parameters

Calculate the Z-scores

Apply the Empirical Rule

Answer: