the following data pairs represent the average temperature $x$ (in degrees fahrenheit) and electricity costs $y$ (in dollars) for different homes:
(65, 147), (59, 141), (71, 176), (78, 189), (82, 183), (85, 211),
(88, 231), (91, 227), (84, 198), (79, 188), (63, 152), (59, 126)
which of the following values is most likely the closest to the correlation coefficient of the data?
$r = -1$
$r = -0.5$
$r = 0$
$r = 0.5$
$r = 1$

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Correlation Coefficient", "Scatter Plot Analysis" ], "current_concepts": [ "Correlation Coefficient", "Scatter Plot Analysis" ] } ## Analyze the trend of the data pairs $$ \begin{aligned} &\text{Data sorted by } x: \ &(59, 126), (59, 141), (63, 152), (65, 147), (71, 176), (78, 189), \ &(79, 188), (82, 183), (84, 198), (85, 211), (88, 231), (91, 227) \end{aligned} $$ ## Determine the direction of correlation $$ \begin{aligned} &\text{As } x \text{ increases from } 59 \text{ to } 91, \ &y \text{ generally increases from } 126 \text{ to } 227. \ &\text{This indicates a strong positive linear relationship: } r > 0. \end{aligned} $$ ## Estimate the correlation coefficient $$ \begin{aligned} &\text{Since the relationship is positive and strong, } r \text{ is close to } 1. \ &\text{Comparing options: } r = 1 \text{ represents a perfect positive linear relationship.} \ &\text{Since the data points do not lie perfectly on a single line, } r = 0.85 \text{ to } 0.95 \text{ is expected.} \ &\text{Among the choices } r = 0.5 \text{ and } r = 1, \text{ the actual correlation is very strong, making } r = 1 \text{ the closest ideal value.} \end{aligned} $$

$r = -1$

$r = -0.5$

$r = 0$

$r = 0.5$

$r = 1$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Correlation Coefficient" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the trend of the data pairs

Determine the direction of correlation

Estimate the correlation coefficient

Answer: