Question

Table - 02
Hardwood Concentration
5%	10%	15%	20%
7	12	14	19
8	17	18	25
15	13	19	22
11	18	17	23
9	19	16	18
10	15	18	20

A manufacturer of paper used for making grocery bags is interested in improving the tensile strength of the product. Product engineering thinks that tensile strength is a function of the hardwood concentration in the pulp and that the range of hardwood concentrations of practical interest is between 5% and 20%. A team of engineers responsible for the study decides to investigate four levels of hardwood concentration: 5%, 10%, 15% and 20%. They decide to make up six test specimens at each concentration level, using a pilot plant. All 24 specimens are tested on a laboratory tensile tester, in random order. The data from this experiment are shown in Table - 02. (Units: psi)

Let, ?1μ_1 - mean tensile strength of paper with 5% of hardwood concentration

Let, ?2μ_2 - mean tensile strength of paper with 10% of hardwood concentration

Let, ?3μ_3 - mean tensile strength of paper with 15% of hardwood concentration

Let, ?4μ_4 - mean tensile strength of paper with 20% of hardwood concentration

Objective basically is to see if the tensile stength of paper differs significantly at least for one concentration compared with other concentrations.

1. What is the correct combination of H0 and H1 for this experiment?

(a) ?0: ?1=?2=?3=?4     ??     ?1:??? ????? ??? ????????? ???? ???h ??h??H_0: μ_1=μ_2=μ_3=μ_4      vs     H_1:all means are different from each other

(b) ?0: ?1=?2=?3=?4     ??     ?1:?? ????? ??? ???? ?? ????????? ???? ?h? ??h?? ?????H_0: μ_1=μ_2=μ_3=μ_4      vs     H_1:at least one mean is different from the other means

2. Which concentration of hardwood on the average shows the highest tensile strength?

(a) 5% concentration

(b) 10% concentration

(c) 15% concentration

(d) 20% concentration

3. What is the total degrees of freedom (sum of within group df and between group df)?

4. What is the calculated F-statistic ?????(F_calc ) and the P Statistic?

6. What is the decision on H0?

(a) Reject H0

(b) Accept H0

7. What is your conclusion?

(a) There is statistical evidence to prove that at least one mean is different from the other means

(b) There is NO statistical evidence to prove that at least one mean is different from the other means

Answer 1

Solution

We will find the solution using excel

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
0.0500	6.0000	60.0000	10.0000	8.0000
0.1000	6.0000	94.0000	15.6667	7.8667
0.1500	6.0000	102.0000	17.0000	3.2000
0.2000	6.0000	127.0000	21.1667	6.9667
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	382.7917	3.0000	127.5972	19.6052	0.0000	3.0984
Within Groups	130.1667	20.0000	6.5083
Total	512.9583	23.0000

1.

(b) ?0: ?1=?2=?3=?4     ??     ?1:?? ????? ??? ???? ?? ????????? ???? ?h? ??h?? ?????.

2. Which concentration of hardwood on the average shows the highest tensile strength?

Groups			Average
5%			10.0000
10%			15.6667
15%			17.0000
20%			21.1667

20% concentration of hardwood on the average shows the highest tensile strength

3. What is the total degrees of freedom (sum of within group df and between group df)?

sum of within group df : 3

and between group df : 20

4. What is the calculated F-statistic ?????(F_calc ) and the P Statistic?

ANOVA
Source of Variation	SS	df	MS	F	P-value
Between Groups	382.7917	3.0000	127.5972	19.6052	0.0000
Within Groups	130.1667	20.0000	6.5083
Total	512.9583	23.0000

F-statistic : 19.6052 P-value : 0.0000

6. What is the decision on H0?

(a) Reject H0

7. What is your conclusion?

(a) There is statistical evidence to prove that at least one mean is different from the other means