Consider the following data values of variables x and y: x 2 4 6 8 10 13 y 7 11 17 21 27 36 a. Determine the least squares regression line. b. Find the predicted value of y for x... Consider the following data values of variables x and y: x 2 4 6 8 10 13 y 7 11 17 21 27 36 a. Determine the least squares regression line. b. Find the predicted value of y for x = 9. c. What does the value of the slope of the regression line tell you? d. Calculate the coefficient of determination, and describe what this statistic tells you about the relationship between the two variables. e. Calculate the Pearson coefficient of correlation. What sign does it have? Why? f. What does the coefficient of correlation calculated in part (e) tell you about the direction and strength of the relationship between the two variables? Read more

Steps to Solve

1. Calculate the means of x and y First, we calculate the mean of the x values ($\bar{x}$) and the mean of the y values ($\bar{y}$).

   $\bar{x} = \frac{2+4+6+8+10+13}{6} = \frac{43}{6} \approx 7.167$

   $\bar{y} = \frac{7+11+17+21+27+36}{6} = \frac{119}{6} \approx 19.833$

2. Calculate the slope (b) of the regression line The formula for the slope $b$ is:

   $ b = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2}$

   We need to calculate $\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})$ and $\sum_{i=1}^{n}(x_i - \bar{x})^2$.

   $x_i$	$y_i$	$x_i - \bar{x}$	$y_i - \bar{y}$	$(x_i - \bar{x})(y_i - \bar{y})$	$(x_i - \bar{x})^2$
   2	7	-5.167	-12.833	66.306	26.698
   4	11	-3.167	-8.833	27.972	10.030
   6	17	-1.167	-2.833	3.306	1.362
   8	21	0.833	1.167	0.972	0.694
   10	27	2.833	7.167	20.306	8.026
   13	36	5.833	16.167	94.291	34.024
   				$\sum = 213.153$	$\sum = 80.834$

   Therefore, $b = \frac{213.153}{80.834} \approx 2.637$

3. Calculate the y-intercept (a) of the regression line The formula for the y-intercept $a$ is:

   $a = \bar{y} - b\bar{x}$

   $a = 19.833 - 2.637 * 7.167 \approx 19.833 - 18.891 \approx 0.942$

4. Write the least squares regression line The equation of the least squares regression line is given by:

   $\hat{y} = a + bx$

   $\hat{y} = 0.942 + 2.637x$

5. Find the predicted value of y for x = 9 Substitute $x = 9$ into the regression equation: $\hat{y} = 0.942 + 2.637 * 9 = 0.942 + 23.733 \approx 24.675$

6. Interpret the slope of the regression line The slope represents the average change in y for each unit change in x. In this case, for every increase of 1 in x, y increases by approximately 2.637.

7. Calculate the coefficient of determination ($R^2$) The formula for $R^2$ is:

   $R^2 = \frac{\sum_{i=1}^{n}(\hat{y}i - \bar{y})^2}{\sum{i=1}^{n}(y_i - \bar{y})^2}$ where $\hat{y}_i = a + bx_i$

   First, calculate $\hat{y}_i$ for each $x_i$:

   $x_i$	$y_i$	$\hat{y}_i = 0.942 + 2.637x_i$	$y_i - \bar{y}$	$\hat{y}_i - \bar{y}$	$(y_i - \bar{y})^2$	$(\hat{y}_i - \bar{y})^2$
   2	7	6.216	-12.833	-13.617	164.686	185.413
   4	11	11.490	-8.833	-8.343	78.022	69.606
   6	17	16.764	-2.833	-3.069	8.026	9.419
   8	21	22.038	1.167	2.205	1.362	4.862
   10	27	27.312	7.167	7.479	51.362	55.935
   13	36	35.223	16.167	15.390	261.378	236.852
   					$\sum = 564.736$	$\sum = 561.977$

   $R^2 = \frac{561.977}{564.736} \approx 0.995$

   $R^2$ describes the proportion of variance in dependent variable explained by independent variable. In this case, about 99.5% of the variance in $y$ is explained by $x$.

8. Calculate the Pearson correlation coefficient (r) The Pearson correlation coefficient $r$ is the square root of the coefficient of determination $R^2$, with the sign being the same as the slope $b$.

   $r = \sqrt{R^2} = \sqrt{0.995} \approx 0.997$

   Since the slope $b$ is positive, $r$ is also positive.

   The sign is positive because as $x$ increases, $y$ also tends to increase.

9. Interpret the Pearson correlation coefficient The correlation coefficient of 0.997 indicates a strong, positive, linear relationship between x and y. As x increases, y strongly tends to increase.