Exercise 4 - Correlation and Regression

Problem 1:

Consider round 1 and and 2 of the Sony open golf tournament (data set golfscores). Is there a statistically significant relationship between the scores?

Problem 2:

Consider round 1 and and 2 of the Sony open golf tournament (data set golfscores). What is the least squares regression equation with Sony 1 as the predictor variable? Draw the fitted line plot. Is there an indication of “regression to the mean”? Why?

Problem 3:

Consider the men’s long jump in the Olympics (longjump). How strong is the relationship between Year and LongJump?

Problem 4:

Consider the following data set:

x y
10 58
11 54
12 51
13 52
14 62
15 57
16 63
17 64
18 69
19 71
20 70

Find the least squares regression equation and use it to predict the y value for an observation with x=15

Solutions

Problem 1:

Consider round 1 and and 2 of the Sony open golf tournament (data set golfscores). Is there a statistically significant relationship between the scores?

Parameter: correlation coefficient

Problem Test for independence

Method: pearson.test

attach(golfscores)
  1. Parameter: Pearson’s correlation coefficient \(\rho\)
  2. Method: Test for Pearson’s correlation coefficient \(\rho\)
  3. Assumptions: relationship is linear and that there are no outliers.
  4. \(\alpha = 0.05\)
  5. H0: \(\rho =0\) (no relationship between Day of Year and Draft Number)
  6. Ha: \(\rho \ne 0\) (some relationship between Day of Year and Draft Number)
  7. p = 0.000
pearson.cor(Sony1, Sony2, rho.null=0)

## p value of test H0: rho=0 vs. Ha: rho <> 0:  0.000
  1. \(p<\alpha = 0.05\), so we reject the null hypothesis,
  2. There is a statistically significant relationship between Day of Year and Draft Number.

Assumptions: boxplots and scatterplot show no outliers. No non-linear relationship.

Problem 2:

Consider round 1 and and 2 of the Sony open golf tournament (data set golfscores). What is the least squares regression equation with Sony 1 as the predictor variable? Draw the fitted line plot. Is there an indication of “regression to the mean”? Why?

Parameter: regression coefficients

Problem: find model

Method: slr

slr(Sony2, Sony1)

## The least squares regression equation is: 
##  Sony2  = 45.836 + 0.348 Sony1 
## R^2 = 9.27%
splot(y=Sony2, x=Sony1, add.line=1)

the slope of the line (0.348) is between 0 and 1, so yes, there is an indication of regression to the mean.

Problem 3:

Consider the men’s long jump in the Olympics (longjump). How strong is the relationship between Year and LongJump?

Parameter: correlation coefficient

Problem: find correlation

Method: ????

the scatterplot of LongJump by Year shows a non-linear relationship, so we can’t answer this question (want to know? come to ESMA3102!)

attach(longjump)
splot(LongJump, Year)

Problem 4:

Consider the following data set:

kable(p4data)
x y
10 58
11 54
12 51
13 52
14 62
15 57
16 63
17 64
18 69
19 71
20 70

Find the least squares regression equation and use it to predict the y value for an observation with x=15

Parameter: regression coefficients

Problem: find model

Method: slr

slr(y=y, x=x)

## The least squares regression equation is: 
##  y  = 32.773 + 1.882 x 
## R^2 = 75.79%
32.773 + 1.882*15
## [1] 61.003

so y=61 is the prediction.