Respuesta :
Answer:
1. Null hypothesis: [tex]\mu_{A}=\mu_{B}=\mu_{C}[/tex]
Alternative hypothesis: Not all the means are equal [tex]\mu_{i}\neq \mu_{j}, i,j=A,B,C[/tex]
2. D. 36
3. C. 34
4. B. 1.059
5. B. 8.02
Step-by-step explanation:
Analysis of variance (ANOVA) "is used to analyze the differences among group means in a sample".
The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"
Part 1
The hypothesis for this case are:
Null hypothesis: [tex]\mu_{A}=\mu_{B}=\mu_{C}[/tex]
Alternative hypothesis: Not all the means are equal [tex]\mu_{i}\neq \mu_{j}, i,j=A,B,C[/tex]
Part 2
In order to find the mean square between treatments (MSTR), we need to find first the sum of squares and the degrees of freedom.
If we assume that we have [tex]p[/tex] groups and on each group from [tex]j=1,\dots,p[/tex] we have [tex]n_j[/tex] individuals on each group we can define the following formulas of variation:
[tex]SS_{total}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x)^2 [/tex]
[tex]SS_{between}=SS_{model}=\sum_{j=1}^p n_j (\bar x_{j}-\bar x)^2 [/tex]
[tex]SS_{within}=SS_{error}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x_j)^2 [/tex]
And we have this property
[tex]SST=SS_{between}+SS_{within}[/tex]
We need to find the mean for each group first and the grand mean.
[tex]\bar X =\frac{\sum_{i=1}^n x_i}{n}[/tex]
If we apply the before formula we can find the mean for each group
[tex]\bar X_A = 27[/tex], [tex]\bar X_B = 24[/tex], [tex]\bar X_C = 30[/tex]. And the grand mean [tex]\bar X = 27[/tex]
Now we can find the sum of squares between:
[tex]SS_{between}=SS_{model}=\sum_{j=1}^p n_j (\bar x_{j}-\bar x)^2 [/tex]
Each group have a sample size of 4 so then [tex]n_j =4[/tex]
[tex]SS_{between}=SS_{model}=4(27-27)^2 +4(24-27)^2 +4(30-27)^2=72 [/tex]
The degrees of freedom for the variation Between is given by [tex]df_{between}=k-1=3-1=2[/tex], Where k the number of groups k=3.
Now we can find the mean square between treatments (MSTR) we just need to use this formula:
[tex]MSTR=\frac{SS_{between}}{k-1}=\frac{72}{2}=36[/tex]
D. 36
Part 3
For the mean square within treatments value first we need to find the sum of squares within and the degrees of freedom.
[tex]SS_{within}=SS_{error}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x_j)^2 [/tex]
[tex]SS_{error}=(20-27)^2 +(30-27)^2 +(25-27)^2 +(33-27)^2 +(22-24)^2 +(26-24)^2 +(20-24)^2 +(28-24)^2 +(40-30)^2 +(30-30)^2 +(28-30)^2 +(22-30)^2 =306[/tex]
And the degrees of freedom are given by:
[tex]df_{within}=N-k =3*4 -3 = 12-3=9[/tex]. N represent the total number of individuals we have 3 groups each one with a size of 4 individuals. And k the number of groups k=3.
And now we can find the mean square within treatments:
[tex]MSE=\frac{SS_{within}}{N-k}=\frac{306}{9}=34[/tex]
C. 34
Part 4
The test statistic F is given by this formula:
[tex]F=\frac{MSTR}{MSE}=\frac{36}{34}=1.059[/tex]
B. 1.059
Part 5
The critical value is from a F distribution with degrees of freedom in the numerator of 2 and on the denominator of 9 such that we have 0.01 of the area in the distribution on the right.
And we can use excel to find this critical value with this function:
"=F.INV(1-0.01,2,9)"
And we will see that the critical value is [tex]F_{crit}=8.02[/tex]
B. 8.02
