Question

a1.s.id.c.5, a1.s.id.c.6, a1.s.id.c.7

1. the table shows the number of items sally bought each time she visited the mall and the money she spent.

the calculator screen shows the linear regression for the data in the table.
what type of correlation does the correlation coefficient indicate, and why?
a a strong positive correlation
since ( r^{2} approx 1 )
b a weak positive correlation
since a and ( b > 10 ), but ( r^{2} < 1 )
c a strong positive correlation
since ( r approx 1 )
d a strong positive correlation
since ( a > 10 )

Explanation:

Step1: Recall the concept of correlation coefficient \( r \)

The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. The value of \( r \) ranges from \(- 1\) to \(1\). If \( r = 1\), there is a perfect positive linear correlation; if \( r=-1\), there is a perfect negative linear correlation. When \(|r|\) is close to \(1\) (e.g., \(|r|\geq0.7\)), the correlation is strong. When \(|r|\) is close to \(0\) (e.g., \(|r|\leq0.3\)), the correlation is weak. A positive \(r\) value indicates a positive correlation (as \(x\) increases, \(y\) increases).

Step2: Analyze the given value of \( r \)

We are given \(r = 0.9954806834\approx1\). Since \(r>0\) and \(r\approx1\), this indicates a strong positive correlation.

Step3: Analyze other options

• Option A: \(r^{2}\) (the coefficient of determination) measures the proportion of the variance in the dependent variable that is predictable from the independent variable. While \(r^{2}\) is related to \(r\) (\(r^{2}=r\times r\)), the correlation coefficient for strength of linear - correlation is \(r\), not \(r^{2}\).
• Option B: \(a\) and \(b\) in the regression equation \(y = ax + b\) are the slope and y - intercept. They do not determine the strength of the correlation. Also, \(r^{2}\approx0.99\) is close to \(1\), not less than a value that would indicate a weak correlation.
• Option D: The slope \(a\) of the regression line \(y=ax + b\) indicates the rate of change of \(y\) with respect to \(x\), but it does not measure the strength of the correlation.

Answer:

C. a strong positive correlation since \(r\approx1\)