Question

Question 5. In R there is a dataset called BOD. For more information type help(BOD) in R. We wish to build a regression line between Time (as the explanatory variate) and demand (as the response). Part A. Formally test whether the slope differs from zero. Use the 4 steps of an hypothesis test.
Part B. Build a 95% confidence interval for the slope parameter.

Part C. What can be said about the correlation between Time and demand?

Answer 1

Solution:

Rcode:

attach(BOD)
colnames(BOD)
regmod <- lm(demand~Time,data=BOD)
summary(regmod)
coefficients(regmod)
confint(regmod)
cor.test(demand,Time,data=BOD)

Output:

Call:
lm(formula = demand ~ Time, data = BOD)

Residuals:
1 2 3 4 5 6
-1.9429 -1.6643 5.3143 0.5929 -1.5286 -0.7714

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.5214 2.6589 3.205 0.0328 *
Time 1.7214 0.6387 2.695 0.0544 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.085 on 4 degrees of freedom
Multiple R-squared: 0.6449,   Adjusted R-squared: 0.5562
F-statistic: 7.265 on 1 and 4 DF, p-value: 0.05435

> coefficients(regmod)
(Intercept) Time
8.521429 1.721429
> confint(regmod)
2.5 % 97.5 %
(Intercept) 1.13900242 15.90385
Time -0.05177283 3.49463
> cor.test(demand,Time,data=BOD)

   Pearson's product-moment correlation

data: x and y
t = 2.6954, df = 4, p-value = 0.05435
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.02438414 0.97753316
sample estimates:
cor
0.8030693

Part A. Formally test whether the slope differs from zero. Use the 4 steps of an hypothesis test.

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.5214 2.6589 3.205 0.0328 *
Time 1.7214 0.6387 2.695 0.0544

Ho:

Ha:

alpha=0.05

test statistic from above output:

t= 2.695

p= 0.0544

p>0.05

Fail to reject null hypothesis.

Accept null hypothesis.

There is no statistical evidence at 5% level of significance to conclude that the slope differs from zero.

Part B. Build a 95% confidence interval for the slope parameter.

95 % confidence interval for slope is

Rcodde;

regmod <- lm(demand~Time,data=BOD)
confint(regmod)

Output:

> confint(regmod)
2.5 % 97.5 %
(Intercept) 1.13900242 15.90385
Time -0.05177283 3.49463

95% confidence interval for slope is

-0.05177283 and 3.49463

Part C. What can be said about the correlation between Time and demand?

cor.test(demand,Time,data=BOD)

Output:

Pearson's product-moment correlation

data: x and y
t = 2.6954, df = 4, p-value = 0.05435
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.02438414 0.97753316
sample estimates:
cor
0.8030693

correlation coefficient,r=0.8030693

There exists a strong positive relationship between time and demand.

as time increases ,demand increases and vice versa.

Regression line is

Demand =8.521 + 1.721 *Time

slope=1.721

y intercept=8.521