Question

A study was made on the amount of converted sugar in a certain process at various temperatures. The data were coded and recorded as follows:

Temperature, x	converted sugar, y
1	8.1
1.1	7.8
1.2	8.5
1.3	9.8
1.4	9.5
1.5	8.9
1.6	8.6
1.7	10.2
1.8	9.3
1.9	9.2
2	10.5

a) Estimate the linear regression line.
b) Estimate the mean amount of converted sugar produced when the coded temperature is 1.75.
c) Plot the residuals versus temperature. Comment.
d) Compute SSE and estimate the variance.
e) Construct a 95% confidence interval for intercept;
f) Construct a 95% confidence interval for slope.
g) Use an ANOVA approach to test the hypothesis that slope = 0 against the alternative hypothesis slope ≠ 0 at the 0.05 level of significance.

Answer 1

Data

Temperature, x	converted sugar, y
1	8.1
1.1	7.8
1.2	8.5
1.3	9.8
1.4	9.5
1.5	8.9
1.6	8.6
1.7	10.2
1.8	9.3
1.9	9.2
2	10.5

Using Excel

a) Excel Steps for obtaining the estimates of regression coefficients

1) Go to "DATA". Select "Data Analysis".

2) Go to "Regression" and click on "OK".

3) Input Y range and X range by selecting data on converted sugar and temperature one by one. Tick on Label if labels are included in the input data.

4) Press "OK".

Excel Output

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.707026444
R Square	0.499886392
Adjusted R Square	0.444318214
Standard Error	0.632607239
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	1	3.600090909	3.600091	8.995911	0.014972903
Residual	9	3.601727273	0.400192
Total	10	7.201818182
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	6.413636364	0.924638018	6.936375	6.79E-05	4.321959853	8.505312874
Temperature, x	1.809090909	0.603167336	2.999318	0.014973	0.444631602	3.173550217

The estimated linear regression line is given by

b) the mean amount of converted sugar produced when the coded temperature is 1.75 is given by

c) Plot the residuals versus temperature

Using Excel

1) Go to "DATA". Select "Data Analysis".

2) Go to "Regression" and click on "OK".

3) Input Y range and X range by selecting data on converted sugar and temperature one by one. Tick on Label if labels are included in the input data.

4) For residual vs temperature plot, tick on "Residual Plots".

5) Click on "OK".

Output

Residuals
-0.122727273
-0.603636364
-0.084545455
1.034545455
0.553636364
-0.227272727
-0.708181818
0.710909091
-0.37
-0.650909091
0.468181818

Interpretation: The plot shows the random pattern which indicates there are no obvious model defects. The linear model is a good fit for the data.

d) Using the summary output in part 1, we have ANOVA table

ANOVA
	df	SS	MS	F	Significance F
Regression	1	3.600090909	3.600091	8.995911	0.014972903
Residual	9	3.601727273	0.400192
Total	10	7.201818182

Sum of square Error, SSE= 3.60107

Estimate of is given by

where n is the sample size.

Hence, the estimate of is given by,

e) Construct a 95% confidence interval for intercept

from the summary output of part (a) we have,

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	6.413636364	0.924638018	6.936375	6.79E-05	4.321959853	8.505313
Temperature, x	1.809090909	0.603167336	2.999318	0.014973	0.444631602	3.17355

95 % confidence interval for the intercept is (4.3219,8.5053).

f) Construct a 95% confidence interval for slope.

from the summary output of part (a) we have,

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	6.413636364	0.924638018	6.936375	6.79E-05	4.321959853	8.505313
Temperature, x	1.809090909	0.603167336	2.999318	0.014973	0.444631602	3.17355

95% confidence interval for slope is (0.4446,3.17355).

g) for testing slope =0, we have ANOVA table from part (a)

ANOVA
	df	SS	MS	F	Significance F
Regression	1	3.600090909	3.600091	8.995911	0.014972903
Residual	9	3.601727273	0.400192
Total	10	7.201818182

The test statistic for testing H0: Slope=0 against H1: Slope is not equal to 0 is

which is F= 8.9959

If H0: Slope= 0, the MSRegression and MSResidual are independently distributed and

follows F distribution with d.f 1 and 9.

The decision rule for H1: Slope 0 is to reject H0 if

F calculated > F critical.

F critical value is 5.12 at d.f (1,9) at 5% level of significance.

Conclusion: Since F = 8.9959 > 5.12= F critical, we reject H0. Hence, Slope 0, there is a significant contribution of the regressor in the model.