(a) find the five-number summary, and (b) draw a box-and-whisker plot of the data.

4 8 8 6 2 9 8 7 9 6 9 4 1 6 2 9 8 7 7 9

(a) \text{min} = 1 (simplify your answer.)
q_1 = 29 (simplify your answer.)

Question

(a) find the five-number summary, and (b) draw a box-and-whisker plot of the data.

4 8 8 6 2 9 8 7 9 6 9 4 1 6 2 9 8 7 7 9

(a) \text{min} = 1 (simplify your answer.)
q_1 = 29 (simplify your answer.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Five-Number Summary", "Box-and-Whisker Plot" ], "new_concepts": [], "current_concepts": [ "Five-Number Summary", "Box-and-Whisker Plot" ] } ## Sort the dataset Using the *Five-Number Summary* knowledge point $$ \begin{aligned} &\text{Original data: } 4, 8, 8, 6, 2, 9, 8, 7, 9, 6, 9, 4, 1, 6, 2, 9, 8, 7, 7, 9\ &\text{Sample size } n = 20\ &\text{Sorted data: } 1, 2, 2, 4, 4, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9 \end{aligned} $$ ## Find Min and Max Using the *Five-Number Summary* knowledge point $$ \begin{aligned} &\text{Minimum (Min)} = 1\ &\text{Maximum (Max)} = 9 \end{aligned} $$ ## Calculate the Median Using the *Five-Number Summary* knowledge point $$ \begin{aligned} &\text{Median position} = \frac{20 + 1}{2} = 10.5\ &\text{Median} = \frac{x_{10} + x_{11}}{2} = \frac{7 + 7}{2} = 7 \end{aligned} $$ ## Calculate Q1 and Q3 Using the *Five-Number Summary* knowledge point $$ \begin{aligned} &\text{Lower half (first 10 values): } 1, 2, 2, 4, 4, 6, 6, 6, 7, 7\ &Q_1 = \frac{x_5 + x_6}{2} = \frac{4 + 6}{2} = 5\ &\text{Upper half (last 10 values): } 7, 8, 8, 8, 8, 9, 9, 9, 9, 9\ &Q_3 = \frac{x_{15} + x_{16}}{2} = \frac{8 + 9}{2} = 8.5 \end{aligned} $$ ## Construct Box-and-Whisker Plot Using the *Box-and-Whisker Plot* knowledge point $$ \begin{aligned} &\text{Five-number summary: } [1, 5, 7, 8.5, 9] \end{aligned} $$ The five-number summary for the given data is: - **Minimum:** 1 - **$Q_1$:** 5 - **Median:** 7 - **$Q_3$:** 8.5 - **Maximum:** 9 { "elements": [ { "type": "segment", "params": [[1, 2], [5, 2]], "properties": { "strokeColor": "#583C87", "strokeWidth": 2 } }, { "type": "segment", "params": [[8.5, 2], [9, 2]], "properties": { "strokeColor": "#583C87", "strokeWidth": 2 } }, { "type": "segment", "params": [[1, 1.8], [1, 2.2]], "properties": { "strokeColor": "#583C87", "strokeWidth": 2 } }, { "type": "segment", "params": [[9, 1.8], [9, 2.2]], "properties": { "strokeColor": "#583C87", "strokeWidth": 2 } }, { "type": "polygon", "params": [[5, 1.5], [8.5, 1.5], [8.5, 2.5], [5, 2.5]], "properties": { "fillColor": "rgba(140, 85, 242, 0.3)", "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[7, 1.5], [7, 2.5]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 3 } }, { "type": "point", "params": [[1, 2]], "properties": { "name": "Min=1", "size": 3, "color": "#583C87", "withLabel": true, "labelPosition": [1, 1.3] } }, { "type": "point", "params": [[5, 2]], "properties": { "name": "Q1=5", "size": 3, "color": "#8C55F2", "withLabel": true, "labelPosition": [5, 1.3] } }, { "type": "point", "params": [[7, 2]], "properties": { "name": "Med=7", "size": 3, "color": "#F2557F", "withLabel": true, "labelPosition": [7, 1.3] } }, { "type": "point", "params": [[8.5, 2]], "properties": { "name": "Q3=8.5", "size": 3, "color": "#8C55F2", "withLabel": true, "labelPosition": [8.5, 1.3] } }, { "type": "point", "params": [[9, 2]], "properties": { "name": "Max=9", "size": 3, "color": "#583C87", "withLabel": true, "labelPosition": [9, 1.3] } } ], "timestamps": [0.1] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Statistics", "Five-Number Summary" ] }

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QUESTION IMAGE

Question

Explanation:

Sort the dataset

Find Min and Max

Calculate the Median

Calculate Q1 and Q3

Construct Box-and-Whisker Plot

Answer: