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{ "quality": "blurry", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "One-Sample Proportion Confidence Interval", "Margin of Error for Proportions" ], "new_concepts": [], "current_concepts": [ "One-Sample Proportion Confidence Interval", "Margin of Error for Proportions", "Statistical Interpretation" ] } ## Calculate sample proportion and sample size $$ \begin{aligned} x &= 160 \quad (\text{yellow peas})\ n &= 448 + 160 = 608 \quad (\text{total offspring})\ \hat{p} &= \frac{160}{608} \approx 0.263158 \end{aligned} $$ ## Calculate standard error and margin of error for 98% confidence $$ \begin{aligned} z_{\alpha/2} &= 2.3263 \quad (\text{for } 98\% \text{ confidence})\ SE &= \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.263158 \times 0.736842}{608}} \approx 0.017866\ E &= z_{\alpha/2} \times SE = 2.3263 \times 0.017866 \approx 0.04156 \end{aligned} $$ ## Determine confidence interval limits and interpretation $$ \begin{aligned} \text{Lower Limit} &= \hat{p} - E = 0.263158 - 0.04156 \approx 0.222\ \text{Upper Limit} &= \hat{p} + E = 0.263158 + 0.04156 \approx 0.305\ \text{Interval} &: 0.222 < p < 0.305 \end{aligned} $$ Since the interval contains $0.25$ (or $25\%$), the results do not contradict the expectation. ## Question a Construct a 98% confidence interval. Express the percentages in decimal form. $0.222$ $< p <$ $0.305$ (Round to three decimal places as needed.) ## Question b Based on the confidence interval, do the results of the experiment appear to contradict the expectation that 25% of the offspring peas would be yellow? No, the confidence interval includes 0.25, so the true percentage could easily equal 25%

Yes, the confidence interval does not include 0.25, so the true percentage could not equal 25%

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Statistics", "One-Sample Proportion Confidence Interval" ] }

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Explanation:

Calculate sample proportion and sample size

Calculate standard error and margin of error for 98% confidence

Determine confidence interval limits and interpretation

Answer:

Question a

Question b