Answer :
To determine the best estimate for the correlation coefficient of the data, let's first understand what a correlation coefficient means. The correlation coefficient, represented as [tex]\( r \)[/tex], measures the strength and direction of a linear relationship between two variables.
The value of the correlation coefficient [tex]\( r \)[/tex] ranges from -1 to 1:
- An [tex]\( r \)[/tex] value of 1 indicates a perfect positive correlation, meaning as one variable increases, the other variable also increases.
- An [tex]\( r \)[/tex] value of -1 indicates a perfect negative correlation, meaning as one variable increases, the other variable decreases.
- An [tex]\( r \)[/tex] value of 0 indicates no correlation between the variables.
- Values closer to 1 or -1 indicate stronger correlations, whether positive or negative.
Given the options:
- Option A: [tex]\( r = 1.5 \)[/tex] is not a valid correlation coefficient because the value of [tex]\( r \)[/tex] must be between -1 and 1.
- Option B: [tex]\( r = 0.5 \)[/tex] indicates a moderate positive correlation.
- Option C: [tex]\( r = 0.96 \)[/tex] indicates a very strong positive correlation.
- Option D: [tex]\( r = -0.90 \)[/tex] indicates a strong negative correlation.
Since we are looking for the BEST estimate for the correlation between the number of math homework assignments Andy completes and his test score, a strong positive correlation would be ideal, implying that more homework is associated with higher test scores. The value [tex]\( r = 0.96 \)[/tex] is closest to 1 and indicates the strongest positive correlation among the valid options given.
Hence, the best estimate for the correlation coefficient of the data is:
[tex]\[ \boxed{0.96} \][/tex]
The value of the correlation coefficient [tex]\( r \)[/tex] ranges from -1 to 1:
- An [tex]\( r \)[/tex] value of 1 indicates a perfect positive correlation, meaning as one variable increases, the other variable also increases.
- An [tex]\( r \)[/tex] value of -1 indicates a perfect negative correlation, meaning as one variable increases, the other variable decreases.
- An [tex]\( r \)[/tex] value of 0 indicates no correlation between the variables.
- Values closer to 1 or -1 indicate stronger correlations, whether positive or negative.
Given the options:
- Option A: [tex]\( r = 1.5 \)[/tex] is not a valid correlation coefficient because the value of [tex]\( r \)[/tex] must be between -1 and 1.
- Option B: [tex]\( r = 0.5 \)[/tex] indicates a moderate positive correlation.
- Option C: [tex]\( r = 0.96 \)[/tex] indicates a very strong positive correlation.
- Option D: [tex]\( r = -0.90 \)[/tex] indicates a strong negative correlation.
Since we are looking for the BEST estimate for the correlation between the number of math homework assignments Andy completes and his test score, a strong positive correlation would be ideal, implying that more homework is associated with higher test scores. The value [tex]\( r = 0.96 \)[/tex] is closest to 1 and indicates the strongest positive correlation among the valid options given.
Hence, the best estimate for the correlation coefficient of the data is:
[tex]\[ \boxed{0.96} \][/tex]
