Run a regression analysis on the following bivariate set of data with y as the response variable.

Question

x y

17.9 98.2

42.3 -14.4

19.4 94

18.8 70

59.2 7.9

27.3 34.4

8.6 135.4

2.3 137.8

24.7 80.1

33.9 54.8

-1.2 77.3

36.7 54

Find the correlation coefficient and report it accurate to three decimal places.

r = ______

What proportion of the variation in y can be explained by the variation in the values of x? Report answer as a percentage accurate to one decimal place. (If the answer is 0.84471, then it would be 84.5%...you would enter 84.5 without the percent symbol.)

r² = _____%

Based on the data, calculate the regression line (each value to three decimal places)

y = ___  x + ___

Predict what value (on average) for the response variable will be obtained from a value of 13.1 as the explanatory variable. Use a significance level of a = 0.05 to assess the strength of the linear correlation.

What is the predicted response value? (Report answer accurate to one decimal place.)

y = 

Answer 1

Hi Anchal,

Formula for correlation coefficient r is:

r=[sum(x_i -xbar)(y_i-ybar)]/[sqrt(sum(x_i-xbar)²*sum(y_i-ybar)²)]

xbar=x sample mean

ybar=y sample mean

x_i=Each individual x value

y_i=Each individual y value

As you can imagine, this calculation would be quite cumbersome even for the relatively small sample size of 12. It is best to use statistical software, an online calculator, or a graphics calculator in the TI-83 family. From online calculator:

r= -0.820

r² is exactly what it sounds like. Square the above value for r:

(-0.820)²=0.673=67.3%

Remember that a simple linear regression equates essentially to a slope, so we follow the traditional slope-intercept form y=mx + b. You may see this in statistics as Y=B₀ + B₁x. They are essentially the same. The b in the first form is the y-intercept. B₀ in the second form is the y-intercept. m in first form is slope. B₁ in second form is slope.

Anyway, recall formula for slope:

m=(y₂-y₁)/(x₂-x₁)

You can use any two pairs of x and y-coordinates you like. I'll go with the first pair in the set as (x₁, y₁) and the last pair as (x₂, y₂). Thus:

x₁=17.9

y₁=98.2

x₂=36.7

y₂=54

m=(54-98.2)/(36.7-17.9)

m= -2.351

Now, we need the intercept. Going back to our original formula: y=mx + b, again, we can substitute any ordered pair for x and y and the slope in for m. I'll go with the third pair (19.4, 94):

94= -2.351(19.4) + b

94= -45.609 + b

Add 45.609 to both sides:

b=139.609

Regression Line Equation: y= -2.351x +139.609

Explanatory variable means "x" variable, so we plug in 13.1 for x. Note that predicted values are usually denoted with a ^, read as "y-hat:"

y^= -2.351(13.1) + 139.609

y^=108.8

I hope this helps.