select the correct answer.

raw scores on a certain standardized test one year were normally distributed, with a mean of 156 and a standard deviation of 23. if 48,592 students took the test, about how many of the students scored less than 96?

table shows values to the left of the z-score

19,312
60
220
18,627

Question

select the correct answer.

raw scores on a certain standardized test one year were normally distributed, with a mean of 156 and a standard deviation of 23. if 48,592 students took the test, about how many of the students scored less than 96?

table shows values to the left of the z-score

19,312
60
220
18,627

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Normal Distribution Calculations", "Z-Score Calculation" ], "new_concepts": [], "current_concepts": [ "Normal Distribution Calculations", "Z-Score Calculation" ] } ## Calculate the z-score for a raw score of 96 $$ \begin{aligned} \mu &= 156 \ \sigma &= 23 \ x &= 96 \ z &= \frac{x - \mu}{\sigma} = \frac{96 - 156}{23} = \frac{-60}{23} \approx -2.6087 \end{aligned} $$ ## Find the cumulative probability from the z-table $$ \begin{aligned} z &\approx -2.61 \ P(Z < -2.61) &= 0.00453 \end{aligned} $$ ## Calculate the expected number of students $$ \begin{aligned} N &= 48,592 \ \text{Number of students} &= N \times P(Z < -2.61) \ &= 48,592 \times 0.00453 \ &\approx 220.12 \approx 220 \end{aligned} $$

19,312

60

220

18,627

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Normal Distribution Calculations" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the z-score for a raw score of 96

Find the cumulative probability from the z-table

Calculate the expected number of students

Answer: