Question

1. what does a correlation of - 0.89 suggest about two variables? (a) that an increase in the explanatory variable causes the response variable to decrease (b) that an increase in the explanatory variable causes the response variable to increase (c) as the explanatory variable increases, the response variable to increase (d) as the explanatory variable increases, the response variable increases (e) as the explanatory variable decreases, the response variable decreases

c 13) for a study using 5 data points, the following values were obtained: $\bar{x}=5$ $\bar{y}=4$ $s_y = 40$ $s_x = 100$ $r = 0.5$ what is the equation of the least - square regression line? (a) $hat{y}=0.4x + 3.4$ (b) $hat{y}=1.25x + 3$ (c) $hat{y}=0.2x + 3$ (d) $hat{y}=3x + 0.2$ (e) $hat{y}=1.25x - 2.25$ slope=$r\frac{s_y}{s_x}=0.5\times\frac{40}{100}$

1. which of the following definitions is false? (a) a lurking variable is a variable other than x and y that simultaneously affects both variables, accounting for the apparent association between the two. (b) the correlation coefficient is a numerical measure of the direction, form, and strength of an association between two variables. (c) in a scatterplot, the y - axis is the response variable that you hope to predict or explain. (d) the residual is the observed y - value minus the predicted y - value. (e) the slope predicts changes in y - units per one unit change in x.

Explanation:

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The correlation coefficient \(r\) measures the strength and direction of a linear relationship between two variables. A negative \(r\) value means that as one variable (explanatory variable) increases, the other (response variable) decreases. Here \(r=- 0.89\) which is negative, indicating an inverse relationship.

Step1: Calculate the slope \(b_1\)

The formula for the slope of the least - square regression line is \(b_1=r\frac{S_y}{S_x}\). Given \(r = 0.5\), \(S_y=40\), and \(S_x = 100\), we have \(b_1=0.5\times\frac{40}{100}=0.2\).

Step2: Calculate the y - intercept \(b_0\)

The formula for the y - intercept is \(b_0=\bar{y}-b_1\bar{x}\). Given \(\bar{x}=5\), \(\bar{y}=4\), and \(b_1 = 0.2\), we get \(b_0=4-0.2\times5=4 - 1=3\).
The equation of the least - square regression line is \(\hat{y}=b_1x + b_0=0.2x+3\).

The correlation coefficient \(r\) measures the strength and direction of a linear relationship, not the form of the relationship (such as non - linear). So the statement that the correlation coefficient measures the direction, form, and strength of an association between two variables is false.

Answer:

E. As the explanatory variable increases, the response variable decreases

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