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{ "quality": "blurry", "question_count": 3, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Confidence Interval Width", "Confidence Level Relationship" ], "current_concepts": [ "Confidence Interval Width", "Confidence Level Relationship", "Statistical Inference Assumptions" ] } ## Analyze the relationship between confidence level and interval width To determine which confidence level ($90\%$, $95\%$, or $99\%$) corresponds to each interval, we examine how the confidence level affects the width of a confidence interval. A higher confidence level requires a larger critical value (e.g., $z^*$ or $t^*$), which increases the margin of error: $$ \text{Margin of Error} = \text{Critical Value} \times \text{Standard Error} $$ Since the sample size ($n = 30$) and sample standard deviation remain constant, a higher confidence level directly results in a wider confidence interval. Therefore: - The narrowest interval corresponds to the lowest confidence level ($90\%$). - The widest interval corresponds to the highest confidence level ($99\%$). - The intermediate interval corresponds to the middle confidence level ($95\%$). ## Calculate the width of each given interval Let's calculate the width (upper limit minus lower limit) for each of the three intervals: 1. **First Interval:** Between $7.3$ and $8.7$ hours $$ \text{Width} = 8.7 - 7.3 = 1.4 \text{ hours} $$ 2. **Second Interval:** Between $7.0$ and $9.0$ hours $$ \text{Width} = 9.0 - 7.0 = 2.0 \text{ hours} $$ 3. **Third Interval:** Between $7.2$ and $8.8$ hours $$ \text{Width} = 8.8 - 7.2 = 1.6 \text{ hours} $$ ## Match the intervals to their confidence levels Now we rank the intervals by their widths and match them to the confidence levels ($90\%$, $95\%$, and $99\%$): - **Narrowest interval:** Width of $1.4$ hours (Between $7.3$ and $8.7$ hours) matches the lowest confidence level, which is **$90\%$**. - **Intermediate interval:** Width of $1.6$ hours (Between $7.2$ and $8.8$ hours) matches the middle confidence level, which is **$95\%$**. - **Widest interval:** Width of $2.0$ hours (Between $7.0$ and $9.0$ hours) matches the highest confidence level, which is **$99\%$**. ## Question 1 Between 7.3 and 8.7 hours of sleep: $90\%$ ## Question 2 Between 7.0 and 9.0 hours of sleep: $99\%$ ## Question 3 Between 7.2 and 8.8 hours of sleep: $95\%$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Confidence Interval Width" ] }

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Analyze the relationship between confidence level and interval width

Calculate the width of each given interval

Match the intervals to their confidence levels

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