Question

A statistical program is recommended.

Consider the following data for two variables, x and y.

x	22	24	26	30	35	40
y	12	20	32	36	39	36

(a). Develop an estimated regression equation for the data of the form

ŷ = b₀ + b₁x.

(b). Use the results from part (a) to test for a significant relationship between x and y. Use α = 0.05.

Find the value of the test statistic.

Find the p-value.

Is the relationship between x and y significant?

(c) Develop a scatter diagram for the data.

Does the scatter diagram suggest an estimated regression equation of the form ŷ = b₀ + b₁x + b₂x²? Explain.

(d). Develop an estimated regression equation for the data of the form ŷ = b₀ + b₁x + b₂x².

(e) Use the results from part (d) to test for a significant relationship between x, x², and y. Use α = 0.05. Is the relationship between x, x², and y significant?

Find the value of the test statistic.

Find the p-value.

(f). Use the model from part (d) to predict the value of y when x = 25.

Answer 1

Perform linear regression n R

Rcode is

x   <- c(22,   24,   26,   30,   35,   40)
y <- c(12,   20,   32,   36,   39,   36)  

reg<-lm(y ~ x)
coeff=coefficients(reg)
coeff
# equation of the line :
eq = paste0("y = ", round(coeff[2],4), "*x ", round(coeff[1],4))
eq
attach(TrackRecords)
# plot
plot( x,y, main=eq)
abline(reg, col="blue")

(a). Develop an estimated regression equation for the data of the form

ŷ = b₀ + b₁x.
y=-7.6006+1.2463*x

(b). Use the results from part (a) to test for a significant relationship between x and y. Use α = 0.05.

using Rcode:

summary(reg)

Ouptut:

Call:
lm(formula = y ~ x)

Residuals:
1 2 3 4 5 6
-7.819 -2.312 7.196 6.210 2.978 -6.253

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -7.6006 13.9489 -0.545 0.6148
x 1.2463 0.4624 2.696 0.0543 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.155 on 4 degrees of freedom
Multiple R-squared: 0.645,   Adjusted R-squared: 0.5562
F-statistic: 7.266 on 1 and 4 DF, p-value: 0.05434

the value of the test statistic.

t=2.696

the p-value=0.0543

Ho:slope=0

Ha:slope not =0

t=2.696

the p-value=0.0543

p>0.05

Fail to Reject Ho:

Accept Ho.

relationship between x and y significant is not significant

(c) Develop a scatter diagram for the data.

Rcode:

plot( x,y, main="Scatterplot")

Output:

From scatterplot we observe:

Form:linear

Direction:positive

strength:strong

Yes

The scatter diagram suggest an estimated regression equation of the form ŷ = b₀ + b_{1 x}