Question

Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable)

X Y 2 12 3 9 6 8 7 7 8 6 7 5 9 2

a. develop the least squares estimated regression equation

b. At 95% confidence, perform a t test and determine whether or not the slope is significantly different from 0

c.Compute the coefficient of determination

d. Develop the 95% confidence interval for the mean value of y* and 95% prediction interval for individual value of y* when x*=5

Answer 1

To get the regression eq use lm function in R

and to get the predcition and confdience intervals use predict fucntion

Solution-A:

Rcode:
df=data.frame(X=c(2,3,6,7,8,7,9), Y=c(12,9,8,7,6,5,2))

regeq <- lm(Y~X,data=df)
summary(regeq)

Output:

we get

the least squares estimated regression equation from output

Y= 13.750 -1.125 *X

b. At 95% confidence, perform a t test and determine whether or not the slope is significantly different from 0

Ho:slope=0

Ha=slope not = 0

alpha=0,05

t statistic=coefficient of slope/std error

= -1.1250/0.2165

= -5.196

p=0.003478

p<0.05

Reject Ho

Accept Ha

Conclusion:

There is sufficient statistical evidence at 5% level of significance to conclude that the slope is significantly different from 0

c.Compute the coefficient of determination

From output:

Rsq=0.8438

=84.38% variation in Y is explained by X

Solution-d:

From output

95% confidence interval for x=5 is

6.682878 and 9.567122

Solution-d:

95% prediction interval for x=5 is

4.321119 and 11.92888