actual lengths of pregnancy terms for a particular species of mammal are nearly normally distributed about a mean pregnancy length with a standard deviation of 14 days. about what percentage of births would be expected to occur more than 28 days after the mean pregnancy length?

about % of births would be expected to occur more than 28 days after the mean pregnancy length.
(type an integer or a decimal.)

Question

actual lengths of pregnancy terms for a particular species of mammal are nearly normally distributed about a mean pregnancy length with a standard deviation of 14 days. about what percentage of births would be expected to occur more than 28 days after the mean pregnancy length?

about % of births would be expected to occur more than 28 days after the mean pregnancy length.
(type an integer or a decimal.)

about  % of births would be expected to occur more than 28 days after the mean pregnancy length.
(type an integer or a decimal.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Empirical Rule", "Normal Distribution" ], "new_concepts": [], "current_concepts": [ "Empirical Rule", "Normal Distribution" ] } ## Identify given parameters Using the *Normal Distribution* knowledge point $$ \begin{aligned} &\text{Distribution: } X \sim N(\mu, \sigma^2)\ &\text{Standard deviation: } \sigma = 14\text{ days}\ &\text{Target value: } X > \mu + 28 \end{aligned} $$ ## Calculate standard deviations from the mean Using the *Normal Distribution* knowledge point $$ \begin{aligned} &k = \frac{(\mu + 28) - \mu}{\sigma} = \frac{28}{14} = 2 \end{aligned} $$ ## Apply the Empirical Rule Using the *Empirical Rule* knowledge point $$ \begin{aligned} &P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 0.95 \quad (95\%)\ &P(X > \mu + 2\sigma) \approx \frac{1 - 0.95}{2} = 0.025 \quad (2.5\%) \end{aligned} $$ About 2.5% of births would be expected to occur more than 28 days after the mean pregnancy length. { "elements": [ { "type": "functiongraph", "params": [ { "js": "Math.exp(-x*x/2) / Math.sqrt(2*Math.PI)", "latex": "\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}" }, -4, 4 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "Normal Distribution", "withLabel": false } }, { "type": "inequality", "params": [ { "js": "0", "latex": "y \ge 0" }, [2.5, 0.05] ], "properties": { "inverse": true, "strict": false, "fillColor": "rgba(242, 85, 127, 0.4)", "strokeColor": "none" } }, { "type": "line", "params": [ [2, 0], [2, 1] ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2, "name": "\mu + 2\sigma", "withLabel": true, "labelPosition": [2, -0.05] } }, { "type": "point", "params": [ [0, 0] ], "properties": { "name": "\mu", "size": 3, "color": "#583C87", "withLabel": true } } ], "timestamps": [0.5] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Empirical Rule" ] }

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Question

Explanation:

Identify given parameters

Calculate standard deviations from the mean

Apply the Empirical Rule

Answer: