2. the ages of subscribers to a certain newspaper are normally distributed with mean 35.5 years and standard deviation 4.8. use the table provided on pages 9-10 to evaluate the following.
a. 4 pts $p(30

Question

2. the ages of subscribers to a certain newspaper are normally distributed with mean 35.5 years and standard deviation 4.8. use the table provided on pages 9-10 to evaluate the following.
a. 4 pts $p(30 < x < 40)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Normal Distribution", "Z-score Standardization" ], "current_concepts": [ "Normal Distribution", "Z-score Standardization" ] } ## Standardize the boundary values to Z-scores $$ \begin{aligned} &\mu = 35.5,\quad \sigma = 4.8\ &Z_1 = \frac{30 - 35.5}{4.8} = \frac{-5.5}{4.8} \approx -1.15\ &Z_2 = \frac{40 - 35.5}{4.8} = \frac{4.5}{4.8} \approx 0.94 \end{aligned} $$ ## Find the cumulative probabilities from the standard normal table $$ \begin{aligned} &P(Z < 0.94) \approx 0.8264\ &P(Z < -1.15) \approx 0.1251 \end{aligned} $$ ## Calculate the final interval probability $$ \begin{aligned} &P(-1.15 < Z < 0.94) = P(Z < 0.94) - P(Z < -1.15)\ &P(-1.15 < Z < 0.94) \approx 0.8264 - 0.1251 = 0.7013 \end{aligned} $$ $0.7013$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Statistics", "Normal Distribution" ] }

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Question

Explanation:

Standardize the boundary values to Z-scores

Find the cumulative probabilities from the standard normal table

Calculate the final interval probability

Answer: