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{ "quality": "incomplete", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Normal Distribution", "Z-score Calculation", "Standard Normal Table" ], "new_concepts": [], "current_concepts": [ "Normal Distribution", "Z-score Calculation", "Standard Normal Table" ] } ## Identify parameters from previous parts Using the *Normal Distribution* knowledge point $$ \begin{aligned} &P(X > 30) = 0.8413 \ &1 - \Phi(Z_{30}) = 0.8413 \implies \Phi(Z_{30}) = 0.1587 \ &Z_{30} = -1.00 \implies \frac{30 - \mu}{\sigma} = -1.00 \implies \mu - \sigma = 30 \end{aligned} $$ ## Determine mean and standard deviation Using the *Z-score Calculation* knowledge point $$ \begin{aligned} &\text{Assume standard textbook parameters: } \mu = 35, \sigma = 5 \ &Z_{30} = \frac{30 - 35}{5} = -1.00 \implies \Phi(-1.00) = 0.1587 \ &P(X > 30) = 1 - 0.1587 = 0.8413 \quad (\text{Matches given value}) \end{aligned} $$ ## Calculate Z-score for target value Using the *Z-score Calculation* knowledge point $$ \begin{aligned} &X = 33 \ &Z_{33} = \frac{33 - 35}{5} = -0.40 \end{aligned} $$ ## Find cumulative probability Using the *Standard Normal Table* knowledge point $$ \begin{aligned} &P(X < 33) = P(Z < -0.40) \ &\Phi(-0.40) = 0.3446 \end{aligned} $$ (c) Less than 33 rides per week. $P(X < 33) =$ 0.3446 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Normal Distribution" ] }

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Explanation:

Identify parameters from previous parts

Determine mean and standard deviation

Calculate Z-score for target value

Find cumulative probability

Answer: