1. Comparison of two samples
F-test a two-sample t-test
(both tests assume data normality)

2. I have two samples and I want to know, if they differ from each other
Ten plants have been watered with normal water and ten with water enriched with nutrients and after one year I investigate their weight (or number of stomata on leaves)
I have x individual of one species and y individuals of the other one and I want to know, if their pistil lengths differ from each other

3. I have two solutions
The samples (their parent population/distribution) can differ from each other either in variance or in mean
or in both...
Even two samples of one population will differ both in variance and in mean. Thus I am interested in, if the two samples differ so much, that it is improbable to be taken from the same population.

4. F-test - test of homogenity of variance
H0: 12 = 22, alternative HA: 12  22
We suppose
numerator
d.f.
denominator
Critical value for test on 5% is thus 97.5% quantile, or

5. I have to multiply value of this area by two to obtain P-value
“Probability”.

6. Critical value – depends on two degrees of freedom
Example:
Two-way test for variance ratio for hypothesis
&
Data represent numbers of moths captured during one night by eleven traps of type 1 or by eight traps of type 2.
Type 1 trap
Type 2 trap
moth2
moth2
moth2
moth2
Critical value – depends on two degrees of freedom
Thus we don’t refuse H0.

7. If I conclude that variances don’t differ from each other, I can compute pooled variance
For moths
sp2=( ) / (10 + 7) = moth2.

8. But we compare means more often than variances
We test null hypothesis H0: 1 = 2 against
alternative one HA: 1  2.
Classic t-test
Mean difference
Standard error of mean difference

9. Standard error of mean difference is computed with help of collective variance s2p
I suppose homogeneity of variances
Resulting formula is then

10. Assumptions of t-test are
Data normality
homogeneity of variances

11. Take notice value of standard error decreases (and power of test increases) with number of observations in groups; if we have constant total number of observations, then the error is smallest if the groups are of the same size.

12. Number of degrees of freedom is sum of degrees of freedom for both samples, thus (n1-1) + (n2-1) = n1 + n2 - 2.

13. Two-sample t-test for two-tailed hypothesis H0: 1 = 2 and HA: 1  2 (which can also be written as H0: 1 - 2 = 0 a HA: 1 - 2  0). Data are times of sedimentation (in minutes) for human blood after vaccination of two different medicines (B, G).
Medicine B vaccinated: 8.8, 8.4, 7.9, 8.7, 9.1, 9.6
Medicine G vaccinated: 9.9, 9.0, 11.1, 9.6, 8.7, 10.4, 9.5
n1 = 6 n2 = 7
df1 = 5 df2 = 6
X1= 8.75 min X2 = 9.74 min
SS1= min SS2 = min
sp2 = min2
t0.05(2),=t0.05(2),11 = Thus we reject H0.
0.02 < P(t  2.475) < 0.05

14. Nowadays we better find area of “tail” and (because it is two-tailed test), we multiply the result by 2.
This area size is – it means that
P=0.0308

15. If homogeneity of variances is disturbed, Welch approximate t can be used
Several other approximations exist
with approximate number of degrees of freedom

16. The same number of observations in both groups is not assumption for t-test
But test robustness against violation of homogeneity of variance is decreasing with very imbalanced numbers of observations (and test for homogeneity will be desperately weak)
Power of the test is decreasing with imbalanced group sizes too

17. Violation of data normality
Group means and not original values are in the t-test formula – so, the means are expected to have the normal distribution
Central limit theorem – they will, if they are based on great number of observations
With increasing number of observations increases not only power of test, but also robustness to violation of normality [comp. with normality tests!]

18. Similar for one-sample t-test, even here can we execute the one-tailed test
Two-tailed test – I test null hypothesis
H0: 1 = 2 against alternative HA: 1  2.
One-tailed test – I test null hypothesis
H0: 1 > 2 against alternative HA: 1 < 2
(or vice versa)

19. MAKE DIFFERENCE
test one-tailed – two-tailed – how null hypothesis is formulated
one-sample t-test and two-sample (pair) one – how is experimental or observation set-up