To estimate the degree of suspension of a suspension polyethylene, one extracts

Question

To estimate the degree of suspension of a suspension polyethylene, one extracts polyethylene by using Ethanol as a solving agent, and compare the gel contents (gel proportion). This method is called the gel proportion estimation method. In the book Design of Experiments for the Quality Improvement published by the Japanese Standards Association (1989), a study was conducted on the relationship between the amount of gel generated and two treatment factors, namely, extraction temperature and extraction time. The experiment was carried out at three specific temperatures 80, 100 and 120, whereas, the extraction times 4, 9 and 24 were selected at random from all possible extraction times. The data are as follows:

b) If appropriate, find point estimate of true effect of temperature level 80 on gel proportion. What is the standard error of your estimate? Find a 95% confidence interval for the true effect of temperature level 80 on gel proportion and draw your conclusions.

To estimate the degree of suspension of a suspension polyethylene, one extracts polyethylene by using Ethanol as a solving agent, and compare the gel contents (gel proportion). This method is called the gel proportion estimation method. In the book Design of Experiments for the Quality Improvement published by the Japanese Standards Association (1989), a study was conducted on the relationship between the amount of gel generated and two treatment factors, namely, extraction temperature and extraction time. The experiment was carried out at three specific temperatures 80, 100 and 120, whereas, the extraction times 4, 9 and 24 were selected at random from all possible extraction times. The data are as follows: a) Analyze the data completely and draw comprehensive conclusions [This involves all relevant statistical analyses including aposteriori analyses]. b) If appropriate, find point estimate of true effect of temperature level 80 on gel proportion. What is the standard error of your estimate? Find a 95% confidence interval for the true effect of temperature level 80 on gel proportion and draw your conclusions.

Explanation / Answer

Tests of Between-Subjects Effects

Dependent Variable:Gel

Source

Type III Sum of Squares

df

Mean Square

F

Sig.

Corrected Model

87.099a

8

10.887

73.306

.000

Intercept

314347.111

1

314347.111

2116551.621

.000

Time

71.841

2

35.920

241.857

.000

Temperature

14.541

2

7.270

48.952

.000

Time * Temperature

.718

4

.179

1.208

.330

Error

4.010

27

.149

Total

314438.220

36

Corrected Total

91.109

35

a. R Squared = .956 (Adjusted R Squared = .943)

The p-values for time and temperature are 0.000, which are extremely small. One can conclude that both main effects are significant. The Tukey’s posthoc test results are shown below.

Time

Multiple Comparisons

(I) Time

(J) Time

Mean Difference (I-J)

Std. Error

Sig.

95% Confidence Interval

Lower Bound

Upper Bound

4

9

.200

.1573

.423

-.190

.590

24

3.092*

.1573

.000

2.702

3.482

9

4

-.200

.1573

.423

-.590

.190

24

2.892*

.1573

.000

2.502

3.282

24

4

-3.092*

.1573

.000

-3.482

-2.702

9

-2.892*

.1573

.000

-3.282

-2.502

Based on observed means.

The error term is Mean Square(Error) = .149.

*. The mean difference is significant at the .05 level.

Temperature

Multiple Comparisons

(I) Temperature

(J) Temperature

Mean Difference (I-J)

Std. Error

Sig.

95% Confidence Interval

Lower Bound

Upper Bound

80

100

1.300*

.1573

.000

.910

1.690

120

1.392*

.1573

.000

1.002

1.782

100

80

-1.300*

.1573

.000

-1.690

-.910

120

.092

.1573

.830

-.298

.482

120

80

-1.392*

.1573

.000

-1.782

-1.002

100

-.092

.1573

.830

-.482

.298

Based on observed means.

The error term is Mean Square(Error) = .149.

*. The mean difference is significant at the .05 level.

From the above results we can conclude that time 24 is significantly different from 4 and 9 and temperature 80 is significantly different from 100 and 120.

2. The SPSS output of marginal means is shown below.

Temperature

Dependent Variable:Gel

Temperature

Mean

Std. Error

95% Confidence Interval

Lower Bound

Upper Bound

80

94.342

.111

94.113

94.570

100

93.042

.111

92.813

93.270

120

92.950

.111

92.722

93.178

The point estimate of true effect of temperature level 80 on gel proportion is 94.342 and a 95% confidence interval for the true effect of temperature level 80 on gel proportion is (94.113, 94.570). One can be 95% confident that the true effect of temperature level 80 on gel proportion lies between 94.113 and 94.570.

Tests of Between-Subjects Effects

Dependent Variable:Gel

Source

Type III Sum of Squares

df

Mean Square

F

Sig.

Corrected Model

87.099a

8

10.887

73.306

.000

Intercept

314347.111

1

314347.111

2116551.621

.000

Time

71.841

2

35.920

241.857

.000

Temperature

14.541

2

7.270

48.952

.000

Time * Temperature

.718

4

.179

1.208

.330

Error

4.010

27

.149

Total

314438.220

36

Corrected Total

91.109

35

a. R Squared = .956 (Adjusted R Squared = .943)

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