Quizzes and exercises of single factor experiments
These quizzes and exercises help us review and practice what we obtained in this chapter. They are presented in "show-hide" form similar to the sections of a web page as we are already familiar. Quizzes are multiple choice questions. After choosing an option, an announcement of result appears. To return back to this page, we click "OK" on the announcement. Of course we can choose again.
In exercises, we are required to calculate a value then fill it in an empty rectangle. Note that in this rectangle, only value is acceptable, its unit is not required. After filling the result, we click on "Answer". If the answer is correct, the border of the rectangle is green and there is green "V" symbol in the adjacent square. If the answer is wrong, the border of the rectangle is red and there is red "X" symbol in the adjacent square. We can erase the answer and retry by clicking on "Retry".
For each exercise, there is a hidden solution. To show this solution, we click on "Solution" tab. But try to solve exercises by ourself, don't abuse these solutions.
Quiz 1
Analysis of variance is used to test
Quiz 2
In analysis of variance, alternative hypothesis is that:
Quiz 3
In analysis of variance, if `F_o< F`*, we conclude
Quiz 4
The variation of response between populations is characterized by:
Quiz 5
Two persons realize a same experiment which studies effect of factor A on response `Y`. Analysis of variance is used to analyze the data. The person smarter, more experience has
Quiz 6
In an experiment, we recognize that there is strong effect of random errors but the means of treatment are nearly the same. In analysis of variance, the value `F_o` is
Quiz 7
In analysis of variance, critical value `F`* depends on
Quiz 8
Two persons realize a same experiment which studies effect of factor A on response `Y`. Analysis of variance is used to analyze the data. The value `F_o` of the person smarter, more experience is
Quiz 9
Least Significant Difference (LSD) is determined
Quiz 10
Two persons realize a same experiment which studies effect of factor A on response `Y`. Assume that A does affect `Y`. The value `LSD` of the person smarter, more experience is
Exercise 1
In a biscuit factory, an experiment was realized to compare the productivity of 4 types of flour, each type of flour is tested in 5 batches. Some results of this experiment is shown in Table 1.
| Source of variation | Degree of freedom | `SS` | `MS` | `F_o` | `F`* |
|---|---|---|---|---|---|
| Type of flour | 114 | ||||
| Error | |||||
| Total | 178 |
a. Determine degrees of freedom `df_A`, `df_E`, `df_T`
| • `df_A=` |
| • `df_E` = |
| • `df_T` = |
b. Determine `SS_E`
| • `SS_E` = |
c. Determine `MS_A` and `MS_E`
| • `MS_A` = |
| • `MS_E` = |
d. Determine `F_o` and `F`* corresponding to confidence level of 95%.
| • `F_o` = |
| • `F`* = |
Quiz e
From the calculations above, we conclude
Solution
From the context : `a=4` ; `n=5` ; `SS_A=114` ; `SS_T=178` ; `alpha=0,05`
a. `df_A=a-1=3` ; `df_E=a(n-1)=16` ; ` df_T=an-1=19`
b. `SS_E=SS_T-SS_A=178-114=64`
c. `MS_A=SS_A//df_A=114/3=38`
`MS_E=SS_E//df_E=64//16=4`
d. So `F_o=MS_A//MS_E=38//4=9,50`
Besides `F`*`=F_(alpha,df_A,df_E)=F_(0,05,3,16)`
Using percentage point table of Fisher distribution `F`*`=3,239`
e. Because `F_o>F`* ; we reject Ho : at least two types of flour whose productivities are unequal.
These calculations are summarized in Table 2.
| Source of variation | Degree of freedom | `SS` | `MS` | `F_o` | `F`* |
|---|---|---|---|---|---|
| Type of flour | 3 | 114 | 38 | 9,5 | 3,239 |
| Error | 16 | 64 | 4 | ||
| Total | 19 | 178 |
Exercise 2
We would like to compare the effectiveness of 3 types of drug A, B and C in high blood pressure treatment. Each drug is tested on 6 patients. After 8 weeks, reductions of blood pressure of these patients are shown in Table 3.
| Drug A | Drug B | Drug C |
|---|---|---|
| 16 | 14 | 11 |
| 25 | 16 | 16 |
| 22 | 8 | 22 |
| 18 | 6 | 19 |
| 18 | 10 | 12 |
| 21 | 12 | 16 |
We compare the effectiveness of these drugs by analysis of variance with confidence level of 95%. The procedure is similar to that of Exercise 1.
a. Determine `df_A`, `df_E`, `df_T`.
| • `df_A` = |
| • `df_E` = |
| • `df_T` = |
b. Determine `SS_A`, `SS_E`, `SS_T`.
| • `SS_T` = |
| • `SS_A` = |
| • `SS_E` = |
c. Determine `MS_A`, `MS_E`.
| • `MS_A` = |
| • `MS_E` = |
d. Determine `F_o`, `F`*
| • `F_o` = |
| • `F`* = |
Quiz e
From these calculations, we conclude
f. What is the least significant difference (LSD) of blood pressure reduction between two types of drug?
| • `LSD` = |
g. Mean of blood pressure reduction (`dh`) of each drug:
| • `bar (dh)_A` = |
| • `bar (dh)_B` = |
| • `bar (dh)_C` = |
Quiz h
Which differences of two types of drug are insignificant statistically?
Solution
From the context : `a=3` ; `n=6` ; `alpha=0,05`
a. `df_A=a-1=2` ; `df_E=a(n-1)=15` ; `df_T=an-1=17`
b. Calculate similarly to exercises about analysis of variance:
`SS_T=454` ; `SS_A=244` ; `SS_E=210`
c. `MS_A=SS_A//df_A=244//2=122`
and `MS_E=SS_E//df_E=210//15=14`
d. So `F_o=MS_A//MS_E=122//14=8,714`
Besides : `F`*`=F_(alpha,df_A, df_E)=F_(0,05,2,15)`
Using percentage point table of Fisher distribution: `F`*`=3,682`
These calculations are summarized in Table 4.
| Source of variation | Degree of freedom | `SS` | `MS` | `F_o` | `F`* |
|---|---|---|---|---|---|
| Type of drug | 2 | 244 | 122 | 8,714 | 3,682 |
| Error | 15 | 210 | 14 | ||
| Total | 17 | 454 |
e. Because `F_o>F`* ; we reject Ho : at least two types of drug whose effectiveness are dissimilar in high blood pressure treatment.
f. Using percentage point table of Student's distribution: `t_(alpha//2,df_E)=t_(0,025,15)=2,1314`
Hence `LSD=t_(alpha//2,a(n-1))sqrt((2MS_E)/n)=2,1314xxsqrt((2xx14)/6)=4,604`
g. Mean blood pressure reduction of drug : `bar (dh)_A=20` ; `bar (dh)_B=11` ; `bar (dh)_C=16`
h. Difference of mean blood pressure reduction between two types of drug are:
- A and B : `bar (dh)_(AB)=9` (`>LSD`)
- A and C : `bar (dh)_(AC)=4` (`< LSD`)
- B and C : `bar (dh)_(BC)=5` (`>LSD`)
With confidence level of 95%, only the difference of effectiveness in high blood pressure treatment between A and C is insignificant statistically.
OK
This web page was last updated on 04 December 2018.