Question

A synthetic fiber manufacturer suspect that tensile strength is related to cotton percentage in the fiber. An experiment was conducted with five levels of cotton percentage and with five replicates in random order. The following tensile strength data was obtained. Does cotton percentage affect the tensile strength?
	% Cotton		Tensile Strength
	15	8	9	15	11	10
	20	12	17	12	18	18
	25	14	15	18	19	19
	30	19	25	20	19	23
	35	7	10	11	12	11

Project Questions

ANOVA

1. What type of experimental design is employed in this problem?

2. Identify the treatments and the dependent variable (Response).

3. Use a multiple boxplot in order to compare responses between different levels.

a. Are there differences between the mean of the responses in at least two of the levels?

b. Do you think this is the result of within-group variation or between-group variation?

4. State clearly the hypothesis that we are testing in this problem.

a. Set up the null and alternative.

5. Run ANOVA on the data and generate the output with plots.

a. Comment on the degree of freedom values for each source of variation. How do you calculate them?

b. Do we reject the hypothesis that we are testing? Why or why not?

i. If you reject, can you tell which level(s) is probably the one(s) that has the different mean? (Hint: use the boxplots from part 1a)

ii. If you are suspicious of only one level, repeat the ANOVA without that level and see if you fail to reject the null in the test.

b. Use the histogram of residuals to comment on the normality of error term (residuals) in this ANOVA model.

Answer 1

1. What type of experimental design is employed in this problem?

Ans: The type of experiment is Completly Randomized Design which is employed in this problem.

2. Identify the treatments and the dependent variable (Response).

Ans: The treatment is the cotton percentage and Tensile Strength is the dependent variable.

3. Use a multiple boxplot in order to compare responses between different levels.

Ans:

From the above box-plot, we can say that the mean Tensile Strength may different at the different Cotton percentage.

a. Are there differences between the mean of the responses in at least two of the levels?

Ans: Yes, there are differences between the mean of the responses in at least two of the levels.

b. Do you think this is the result of within-group variation or between-group variation?

Ans: between-group variation

4. State clearly the hypothesis that we are testing in this problem.

a. Set up the null and alternative.

Ans: Null Hypothesis: The mean Tensile Strength for all cotton percentage are equal.

Alternative Hypothesis: At least one level of cotton percentage has different mean Tensile Strength.

5. Run ANOVA on the data and generate the output with plots.

Ans:

One-way ANOVA: Strength versus Cotton

Source	DF	SS	MS	F	P
Cotton	4	424.64	106.16	15.84	0.000
Error	20	134	6.7
Total	24	558.64

S = 2.588 R-Sq = 76.01% R-Sq(adj) = 71.22%

b. Do we reject the hypothesis that we are testing? Why or why not?

Ans: We reject the null hypothesis because that estimated p-value is less than 0.05 level of significance.

i. If you reject, can you tell which level(s) is probably the one(s) that has the different mean? (Hint: use the boxplots from part 1a)

Ans: Level 30 is different from remaining.

b. Use the histogram of residuals to comment on the normality of error term (residuals) in this ANOVA model.

Ans:

The histogram is not symmetric. Hence, we are not satisfied the assumption of normality on error of the model.