Question

You will complete a question about Correlation Examples and complete a Simple Linear Regression. For the Simple Linear Regression, make sure to complete the following steps:

Construct a scatter plot.

Find the equation of the regression line.

Predict the value of y for each of the x-values.
Use this resource: Regression

Give an example of two variables that have a positive linear correlation. Give an example of two variables that have a negative linear correlation. Give an example of two variables that have no correlation. Height and Weight: The height (in inches) and weights (in pounds) of eleven football players are shown in this table.

Height, x 62 63 66 68 70 72 73 74 74 75 75

Weight, y 195 190 250 220 250 255 260 275 280 295 300

x = 65 inches x = 69 inches x = 71 inches

Answer 1

• Simple Linear Regression

​ For this task i have used R software. Output and syntax are given below

At first i have defined the data using the syntax:

  > Height.x.<-c(62,63,66,68,70,72,73,74,74,75,75)
> weight.y.<-c(195,190,250,220,250,255,260,275,280,295,300)

Then i have drawn Scatter plot

Syntax: > plot(Height.x.,weight.y.)

The output is

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Next for the regression line i have used the syntax

  > model<-lm(weight.y.~Height.x.)
> summary(model)

Where 'lm' is the linear model function in R.

The output is

Call:

lm(formula = weight.y. ~ Height.x.)

Residuals:

Min 1Q Median 3Q Max

-15.9904 -9.8632 -0.4992 5.3824 28.5184

Coefficients:

Estimate Std. Error t value Pr(>|t|)   

(Intercept) -257.3083 63.7622 -4.035 0.00295 **

Height.x. 7.2544 0.9066 8.002 2.21e-05 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 13.68 on 9 degrees of freedom

Multiple R-squared: 0.8768, Adjusted R-squared: 0.8631

F-statistic: 64.02 on 1 and 9 DF, p-value: 2.21e-05

Hemce from the abouve output equation of regression line is

Weight(y)=-257.3083+ 7.2544 *Height(x)

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Prediction of the value of y for each of the x-values.

x	y
65	214.2272
69	243.2448
71	257.7536
62	192.4641
63	199.7185
66	221.4816
68	235.9904
70	250.4992
72	265.0080
73	272.2624
74	279.5168
74	279.5168
75	286.7712
75	286.7712

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• Correlation Examples

​

1. an example of two variables that have a positive linear correlation

   Height(x)	weight(y)
   64	57
   60	60
   67	73
   59	62
   69	68

     . cov(x,y)= 18.25, hence cor(x,y)=0.655 ----------------------------------------------------------------------------------------------------------------------

2. ​ An example of two variables that have a negative linear correlation   

   GPA(x)	Hours watching tv(y)
   2.5	15
   2.9	13
   3.5	5
   3.15	4
   3.9	12

   Here cov(x,y)= -1.0525 . hence cor(x,y)= -0.393 --------------------------------------------------------------------------------------------------------------------
3. an example of two variables that have no correlation.

   (x)	y=x2
   -3	9
   -1	1
   +1	1
   +3	9

Here sum of x is zero(=0)

Hence

Hence correlation is zero.

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THANK YOU.