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

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

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.

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

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

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.

If I conclude that variances don’t differ from each other, I can compute pooled variance For moths sp2=(218.73 + 107.50) / (10 + 7) = 19.19 moth2.

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

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

Assumptions of t-test are Data normality homogeneity of variances

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.

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

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= 1.6950 min SS2 = 4.0171 min sp2 = 0.5193 min2 t0.05(2),=t0.05(2),11 = 2.201 Thus we reject H0. 0.02 < P(t  2.475) < 0.05

Nowadays we better find area of “tail” and (because it is two-tailed test), we multiply the result by 2. This area size is 0.0154 – it means that P=0.0308

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

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

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!]

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)

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