question 26 points 3
if $x$ follows a normal distribution with $\mu = 100$ and $\sigma = 10$, what is the probability that $x 90$?

99.37%
84.13%

Question

question 26 points 3
if $x$ follows a normal distribution with $\mu = 100$ and $\sigma = 10$, what is the probability that $x > 90$?

99.37%
84.13%

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", "Z-score Calculation", "Standard Normal Table" ], "new_concepts": [], "current_concepts": [ "Normal Distribution Properties", "Z-score Calculation", "Standard Normal Table" ] } ## Identify given parameters Using the *Normal Distribution Properties* knowledge point $$ \begin{aligned} &\mu = 100\ &\sigma = 10\ &\text{Target: } P(X > 90) \end{aligned} $$ ## Calculate the Z-score Using the *Z-score Calculation* knowledge point $$ \begin{aligned} &Z = \frac{X - \mu}{\sigma}\ &Z = \frac{90 - 100}{10} = -1 \end{aligned} $$ ## Find the probability Using the *Standard Normal Table* knowledge point $$ \begin{aligned} &P(X > 90) = P(Z > -1)\ &P(Z > -1) = 1 - P(Z \le -1)\ &P(Z \le -1) \approx 0.1587\ &P(Z > -1) \approx 1 - 0.1587 = 0.8413 = 84.13\% \end{aligned} $$

(A) 99.37%

(B) 84.13%

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QUESTION IMAGE

Question

Explanation:

Identify given parameters

Calculate the Z-score

Find the probability

Answer: