Question

The data below are for 30 people. The independent variable is “age” and the dependent variable is “systolic blood pressure.” Also, note that the variables are presented in the form of vectors that can be used in R.

age=c(39,47,45,47,65,46,67,42,67,56,64,56,59,34,42,48,45,17,20,19,36,50,39,21,44,53,63,29,25,69)

systolic.BP=c(144,20,138,145,162,142,170,124,158,154,162,150,140,110,128,130,135,114,116,124,136,142,120,120,160,158,144,130,125,175)

1. Using R, develop and show a scatterplot of systolic blood pressure (dependent variable) by age (independent variable), and calculate the correlation between these two variables.
2. Assume that these data are “straight enough” to model using a linear regression line. Develop and show that model (write out the model in the terms of the problem), and also show in a plot the line that best fits these data.
3. Plot the residuals and comment on what you see as to how appropriate the model is.
4. Using a boxplot, determine if there are any outliers in systolic blood pressure. If so, point out which points are outliers, if any.
5. Assuming there is at least one outlier in systolic blood pressure, remove that outlier and re-do parts a) through c) again using the remaining data without the outlier(s). State and comment on this second model.
6. In your second model, explain in the context of age and systolic blood pressure what the slope of your fitted line means. Also, for your second model, calculate R² (the coefficient of determination), and explain what that means in the context of your second model.

Answer 1

> age=c(39,47,45,47,65,46,67,42,67,56,64,56,59,34,42,48,45,17,20,19,36,50,39,21,44,53,63,29,25,69) ;age
[1] 39 47 45 47 65 46 67 42 67 56 64 56 59 34 42 48 45 17 20 19 36 50 39 21 44
[26] 53 63 29 25 69
> systolic.BP=c(144,20,138,145,162,142,170,124,158,154,162,150,140,110,128,130,135,114,116,124,136,142,120,120,160,158,144,130,125,175);systolic.BP
[1] 144 20 138 145 162 142 170 124 158 154 162 150 140 110 128 130 135 114 116
[20] 124 136 142 120 120 160 158 144 130 125 175

#a) Using R, develop and show a scatterplot of systolic blood pressure (dependent variable) by age (independent variable), and calculate the correlation between these two variables

Ans
> plot(age,systolic.BP)

> cor(age,systolic.BP)
[1] 0.5032292

#. b) Assume that these data are “straight enough” to model using a linear regression line. Develop and show that model (write out the model in the terms of the problem), and also show in a plot the line that best fits these data

Ans:
> mod1=lm(systolic.BP~age);mod1

Call:
lm(formula = systolic.BP ~ age)

Coefficients:
(Intercept) age
94.5320 0.9158

> summary(mod1)

Call:
lm(formula = systolic.BP ~ age)

Residuals:
Min 1Q Median 3Q Max
-117.576 -3.934 4.760 8.765 25.171

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 94.5320 14.1390 6.686 2.95e-07 ***
age 0.9158 0.2972 3.081 0.00459 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 24.48 on 28 degrees of freedom
Multiple R-squared: 0.2532, Adjusted R-squared: 0.2266
F-statistic: 9.495 on 1 and 28 DF, p-value: 0.004588

> abline(lm(systolic.BP~age))

c) Plot the residuals and comment on what you see as to how appropriate the model is

Ans> residual=resid(mod1)
  
> plot(age,residual,ylab="residual",xlab="age"

>

#d) Using a boxplot, determine if there are any outliers in systolic blood pressure. If so, point out which points are outliers, if any.

outlier.values=boxplot(systolic.BP)$out;outlier.values
[1] 20
> boxplot(systolic.BP)
> # 20 is outlier in in systolic