Topic 7- Analysis Of Variance

Question 1

F statistic characteristics (4)

Answer 1

Can only be positive

Positively skewed

Degrees of freedom contained by numerator and denominator

Asymptotic (never reaches X-axis)

Question 2

Topic 1: Equal variances test

Example of where mean on two samples are equal, but variances are not equal

Answer 2

Shares, rate of return may be the same, but there may be more variation in one than the other (spread differently)

Question 3

2 assumptions for equal variance test

Answer 3

populations are normally distributed

level of measurement is interval or ratio, in order to find variances.

Question 4

Equal variances test steps (4)

Answer 4

1.Set up null/alternative hypothesis

H₀:σ²₁=σ²₂ (variance₁=variance₂)
H₁:σ²₁≉σ²₂ (variances arent equal!)

Both can be rearranged to

H₀:σ²₁/σ²₂ =1
H₁:σ²₁/σ²₂ ≉1

2.Significance level

3. Select test statistic (F statistic, shown on next card)
4. Decide if we reject (F>Fcrit= reject, as in critical region)

Question 5

F statistic formula

What is the smallest value of F

Answer 5

σ²₁/σ²₂ ~ Fn₁-1, Fn₂-1

Larger sample variance in numerator. Smallest value of F is 1. When 1, null cannot be rejected!

Question 6

If σ²₁=3.9 and σ²₂=3.5 and sample₁=10 sample₂=8

Calculate statistic, critical value and decide to reject/not.

Answer 6

3.9²/3.5²=1.24
(Always put larger of the 2 variances in numerator)
(Lowest possible value is 1)

To find critical value, remember
Fn₁-1, Fn₂-1

So 10-1, 8-1
9, 7

Use table to find the critical value. We find CV is 3.68

1.24<3.68 so we cannot reject the null (doesn’t lie in CV)

This means no difference in variation in the equal variances test!

Question 7

What is ANOVA and what does it do

Answer 7

Equal variance test tests whether 2 samples have equal variances

ANOVA (analysis of variances) test tests whether several means are equal SIMULTANEOUSLY

Question 8

We want to test the equality of the output of 3 different factories. The treatment here is the differences in the factories, e.g. location, size etc.

How do we do pairwise comparisons?

Answer 8

Factory 1 v Factory 2 (confidence 95%)
Factory 1 v Factory 3 (confidence 95%)
Factory 2 v Factory 3 (confidence 95%)

Question 9

What would the total confidence level be?

Answer 9

0.95³=0.86

(3 95% confidences, so less confidence)

Question 10

Example cont:
Sample mean of:
Factory 1=410.83
Factory 2=401.57
Factory 3=421.20

Grand mean=410.11(NOT SAMPLE MEANS ADDED/3!)

Answer 10

.

Question 11

3 assumptions for ANOVA test

Answer 11

Sampled populations must have normal distribution
Populations have equal SD (σ)
Samples random and independent.

Question 12

Steps for ANOVA test

Answer 12

1.State null/alt
Null: population means are equal
H₀:μ₁=μ₂=μ₃
H₁: means are not equal.

2. Significance level
3. Test statistic F
4. Decision rule (reject if F statistic result is in CR)

Question 13

F statistic for ANOVA

Answer 13

F= SST/(k-1)
/ ~Fk-1,n-k
SSE/(n-k)

SST: treatment variation
SSE: random variation
K: no of population sampled (3 factories in this case)
N: no of observations (we use 18 in this case)

So critical value will be found in
3-1,18-3
2,15

2,15=3.68 is the critical region

Question 14

SSE and SST, and what if they are of similar size

Answer 14

SSE- sum of squares of all deviations within each factory from factory average
SST- sum of squares of all deviations of factory means from overall average

If variations are of similar size, we assume the effect of specific factory characteristics on output, is no greater than the effect of any random event.

Question 15

SSTotal formual= SSE+SST

Answer 15

Σ(Xi-grand mean)²

Xi is each observation from all factories (18 of them)

Question 16

SST formula

Answer 16

Σfactory observations Σ(factory mean-grand mean)²

E.g factory 1 had 6 observations and mean of 410.83
So 6(410.83-410.11)² , do for all factories.

Question 17

2 ways to work out SSE (double check 2nd method on slide 26 video)

Answer 17

1.As we have SSTotal and SST and SSTotal=SST+SSE,

SSTotal-SST=SSE

2.Σ(Xi-factory mean)²

Question 18

So when do we use an F test

Answer 18

Equal variances test (if 2 samples have equal variances)

ANOVA test (if several means are equal)