Question

24. (Correlation) A researcher wants to determine if there is a linear relationship between height and weight. The following table represents the data collected. Display the data in a scatter plot on your calculator, draw a quick sketch below. Then find the linear regression and put the line of best fit on the sketch. Then state the value for the correlation coefficient and determine if this is significant correlation or no correlation using the table in the back of the book. Please state the critical value you found from the table. Age in 10 15 45 32 66 75 81 55 45 years Height 4 5.7 5.5 5.2 6 5.2 5.1 5.9 in feet 5.8

Answer 1

-----------------------------Please find R code

age=c(10,15,45,32,66,75,81,55,45)
height=c(4,5.7,5.5,5.2,6,5.2,5.1,5.9,5.8)
plot(age,height,main="Regression",col="red")
fit=lm(height~age)
fit$coefficients
abline(fit)
summary(fit)
cor(age,height)
cor.test(age,height)
qt(0.975,9-2) #Critical value

---------------------------------------------------------OUTPUT

summary(fit)

Call:
lm(formula = height ~ age)

Residuals:
Min 1Q Median 3Q Max
-1.0519 -0.4227 0.1408 0.4530 0.6042

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.964121 0.455303 10.903 1.21e-05 ***
age 0.008780 0.008645 1.016 0.344
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6107 on 7 degrees of freedom
Multiple R-squared: 0.1284,   Adjusted R-squared: 0.003943
F-statistic: 1.032 on 1 and 7 DF, p-value: 0.3436

cor.test(age,height)

   Pearson's product-moment correlation

data: age and height
t = 1.0157, df = 7, p-value = 0.3436
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.4012218 0.8259319
sample estimates:
cor
0.358399

> qt(0.975,9-2)
[1] 2.364624

-----------------------------------------------------------------------------

Scatter plot with line of best fit

height 4.0 4.5 50 5 60 이 age 10203040 506070 80 Regression

The line of best fit is :- Height = 4.964121 +  0.008780*(age)

Correlation is found to be 0.358399

P value from the output above is 0.3436,, hence we reject H0 and conclude correlation is not significant

Critical value at 7 degree of freedom with alpha 0.05 is  2.364624