question 27 points 3

consider a normal distribution with a mean of 32 and a standard deviation of 1.5. find the probability that a value in the distribution is greater than 33.

zero
0.2514

Question

question 27 points 3

consider a normal distribution with a mean of 32 and a standard deviation of 1.5. find the probability that a value in the distribution is greater than 33.

zero
0.2514

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 $$ \begin{aligned} &\mu = 32\ &\sigma = 1.5\ &X \sim N(32, 1.5^2) \end{aligned} $$ ## Calculate the Z-score Using the *Z-score Calculation* knowledge point $$ \begin{aligned} &x = 33\ &z = \frac{x - \mu}{\sigma} = \frac{33 - 32}{1.5} = \frac{1}{1.5} \approx 0.67 \end{aligned} $$ ## Find the probability Using the *Standard Normal Table* knowledge point $$ \begin{aligned} &P(X > 33) = P(Z > 0.67) = 1 - P(Z \le 0.67)\ &P(Z \le 0.67) \approx 0.7486\ &P(X > 33) \approx 1 - 0.7486 = 0.2514 \end{aligned} $$

zero

0.2514

{ "elements": [ { "type": "functiongraph", "params": [ { "js": "Math.exp(-Math.pow(x-32, 2)/(2*Math.pow(1.5, 2))) / (1.5 * Math.sqrt(2*Math.PI))", "latex": "\frac{1}{1.5\sqrt{2\pi}} e^{-\frac{(x-32)^2}{2(1.5)^2}}" }, 26, 38 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "Normal Curve" } }, { "type": "inequality", "params": [ { "x_val": 33, "latex": "X > 33" }, [34, 0.05] ], "properties": { "isVertical": true, "inverse": true, "strict": true, "fillColor": "rgba(140, 85, 242, 0.3)" } }, { "type": "point", "params": [[32, 0]], "properties": { "name": "\mu = 32", "withLabel": true, "color": "#583C87" } }, { "type": "point", "params": [[33, 0]], "properties": { "name": "X = 33", "withLabel": true, "color": "#F2557F" } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Standard Normal Distribution" ] }

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QUESTION IMAGE

Question

Explanation:

Identify given parameters

Calculate the Z-score

Find the probability

Answer: