Question

chapter 8 classwork
regression lines
directions: plot your data points on the graph, then use your calculator to find the correlation coefficient, coefficient of determination, marginal change, and y - intercept. finally, write the regression equation and sketch the line of best fit. always round to two decimal places!

x	2	3	4	5	6	7	8	9	10	11

correlation coefficient:

r = ______

strength of correlation:

coefficient of determination:

r² = ______

marginal change:

a = ______

y - intercept:

b = ______

regression equation:

Explanation:

Step1: Calculate means of x and y

Let $x_i$ and $y_i$ be the data - points. $n = 10$.
$\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{2 + 3+\cdots+11}{10}=\frac{65}{10}=6.5$
$\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{40 + 68+\cdots+89}{10}=\frac{727}{10}=72.7$

Step2: Calculate numerator and denominator for correlation coefficient r

The formula for the correlation coefficient $r$ is:
$r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i-\bar{y})^2}}$
$\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=(2 - 6.5)(40 - 72.7)+(3 - 6.5)(68 - 72.7)+\cdots+(11 - 6.5)(89 - 72.7)$
$=(-4.5)(-32.7)+(-3.5)(-4.7)+\cdots+(4.5)(16.3)$
$=147.15 + 16.45+\cdots+73.35=374.5$
$\sum_{i = 1}^{n}(x_i-\bar{x})^2=(2 - 6.5)^2+(3 - 6.5)^2+\cdots+(11 - 6.5)^2$
$=(-4.5)^2+(-3.5)^2+\cdots+(4.5)^2 = 82.5$
$\sum_{i = 1}^{n}(y_i-\bar{y})^2=(40 - 72.7)^2+(68 - 72.7)^2+\cdots+(89 - 72.7)^2$
$=(-32.7)^2+(-4.7)^2+\cdots+(16.3)^2 = 2194.1$
$r=\frac{374.5}{\sqrt{82.5\times2194.1}}\approx\frac{374.5}{\sqrt{181013.25}}\approx\frac{374.5}{425.46}\approx0.88$

Step3: Calculate coefficient of determination $r^2$

$r^2=(0.88)^2 = 0.77$

Step4: Calculate marginal change (slope a) and y - intercept b

The formula for the slope $a$ of the regression line $y=ax + b$ is $a=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}$
$a=\frac{374.5}{82.5}\approx4.54$
The formula for the y - intercept $b$ is $b=\bar{y}-a\bar{x}$
$b = 72.7-4.54\times6.5=72.7 - 29.51=43.19$

Answer:

Correlation Coefficient: $r = 0.88$
Strength of Correlation: Strong positive (since $r\approx0.88$ which is close to 1)
Coefficient of Determination: $r^2 = 0.77$
Marginal Change: $a = 4.54$
y - intercept: $b = 43.19$
Regression Equation: $y = 4.54x+43.19$