- 00:00
DANIEL LITTLE: In this video, I willdiscuss using the T-test for the casewhen you have two independent samples of data.With two independent samples, we wantto know whether those samples came from the same populationwith the same population mean--this is the condition represented by the figureon the left-hand side of the screen--

- 00:22
DANIEL LITTLE [continued]: or whether the two samples come from different populationswith different means.And that's the situation representedby the figure on the right-hand side of the screen.The process that we use is exactly the sameas the process for the single sample T-test.That is we compute and observe t-value and degrees of freedom

- 00:43
DANIEL LITTLE [continued]: and then find the probability of that t-valueunder the t-distribution.The way in which we compute the t-value and the degreesof freedom, however, differs from the single sample t-test.Our null hypothesis is that the two sampleswere generated from the same population distribution.Another way to say the same thing

- 01:04
DANIEL LITTLE [continued]: is that we assume that the difference between the samplemeans actually equal 0.To compute the t-value, we subtract the two meansfrom each other and divide by the estimated standard error.So the first step is to subtract the two means from each other.So M1 represents the mean of our first sample,

- 01:29
DANIEL LITTLE [continued]: and M2 represents the mean of our second sample.The estimate of the standard errordepends on the variance and the sample size of both groups.So here S1 is the standard deviation of are first sample,

- 01:50
DANIEL LITTLE [continued]: and S2 is the standard deviation of our second sample.N1 is the sample size of our first sample,and N2 is the sample size of our second sample.

- 02:11
DANIEL LITTLE [continued]: If we use this formula, however, wherewe take each of our standard deviationsand square them to get the variance for eachof our samples and divide by the sample size,we do that for both of our samples,and then add them together and take the square root.We do get an estimate of the standard error.However, this formula only works whenever both of the samples

- 02:34
DANIEL LITTLE [continued]: have the same sample size, that is whenever N1 equals N2.If both groups do not have the same sample size,then this formula is biased towards the group, whichhas a larger sample size, towards the group whichhas a larger N.

- 02:57
DANIEL LITTLE [continued]: We can correct for this bias by using the pooled variance,instead of the sample variance.What the pooled variance representsis a weighted average of the sample variances,where the weights are actually determined by the sample size.So at the bottom of the screen, I'veindicated how we would compute the pooled variance

- 03:19
DANIEL LITTLE [continued]: for each of our two samples.Here we have an estimate of the degrees of freedomfor our sample on the left.We multiply that by the standard deviation of the sampleon the left squared, which is the varianceof our sample on the left.We do the same thing for the sample on the right.

- 03:41
DANIEL LITTLE [continued]: We multiply the degrees of freedomfor the sample of the right, the sample here,times the variance of that sample.And we add those two things together,and then we divide by the sum of the degrees of freedomfor both groups, the degrees of freedom for each sampleor the number of observations in this sample minus 1.

- 04:03
DANIEL LITTLE [continued]: What we get out of that is an estimateof the pooled variance, which we thensubstitute into our equation up hereto compute our estimated standard error of the mean.Using the pooled variance, we can thencompute the t-statistic for the independent samples T-test.

- 04:25
DANIEL LITTLE [continued]: Using the same formula as before,we subtract the mean of the first groupfrom the mean of the second group,and then we divide by our estimate of the standard errorof the mean.Our total degrees of freedom for this testis just the sum of the degrees of the freedom for each sample,that is the number of observations

- 04:48
DANIEL LITTLE [continued]: in our first sample minus 1 plus the number of observationsin our second sample minus 1.Having obtained the t-statistic and the degrees of freedom,we can then go to the t-distributionto obtain the p-value.The p-value represents the probabilityof our observed t-statistic under the null hypothesis

- 05:09
DANIEL LITTLE [continued]: assumption that the difference between the sample meansequals zero.In the next video, I will discusswhat to do if you have two samples which are notindependent from one another.

### Video Info

**Series Name:** Statistics for Psychology

**Episode:** 13

**Publisher:** University of Melbourne

**Publication Year:** 2014

**Video Type:**Tutorial

**Methods:** T-test, Degrees of freedom, Standard deviations, Standard error

**Keywords:** mathematical concepts

### Segment Info

**Segment Num.:** 1

**Persons Discussed:**

**Events Discussed:**

**Keywords:**

## Abstract

In chapter 13 of his series on statistics for psychology, Professor Daniel Little analyzes how to figure out if two data samples are from the same population or not.