PSYCH 240 1st Edition Lecture 182 Groups 2 Means-The null hypothesis is that the two samples come from populations with the same mean. -The alternative hypothesis is that the two samples do NOT come from populations with the same mean. -As the previous tests, sometimes the null will just say that the population means are different, and sometimes it will specify which population mean is higher. -Even if the two samples come from the exact same population, the sample means won’t be exactly the same. -The two means are from randomly selected samples of people, and this random sampling will introduce some degree of variation. -We need to know how big the difference in the sample means has to be to count as good evidence that the population means are different. -In other words, we need to make sure we have a difference between sample means that we would be unlikely to get if the population means were equal. -To figure out if we have evidence that the population means are different, we need to know what the distribution of differences between means looks like if the population means are actually the same. -In theory, this distribution is formed like this:1) Take two random samples from the same population. 2) Get the mean of each sample.3) Subtract one mean from the other. 4) Repeat 1-3 a huge number of times and make a distribution with all the differences that you get. Distribution of Mean Differences-Central Tendency: if we take both sample from the same population, then mean 1 will be higher and mean 2 on a random half of the sample and vice versa on the other halfoSo the mean of the distribution of differences between means, μDIFF, is zerooμDIFF is a population mean. It's what the mean would be if you had the difference between means for every possible pair of samples from the population-Shape: if the original population of scores is normal, then the distribution of differences between means will be normaloIf the sample sizes for both samples are above 30, then the distribution of differences between means will be very closs to normal regardless of the shape of the distribution of scoresThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.-Variability: here is how you find the population standard deviation of the distribution of differences between means (σDIFF)o Estimating the Population Variance-For all the problems that we will do, we won't actually know the population variability for the scores-We will have to estimate this using the scores in our two samples-You'll be given a variability estimate from both samples that was calculated with the formulas that you learned in section 2-The estimate from each sample had N-1 degrees of freedomodf1 = N1-1odf2 = N2-1 -If the null hypothesis is true, then both estimates are estimating the same population variance. Thus, we combine (or "pool") them to get one better estimate-The combined estimate reflects the degrees of freedom in both estimatesodfTOT = df1 - df2odfTOT: total degrees of freedom across both samples-To combine the two variance estimates, we take their weighted average:ooIf there are an equal number of scores in both examples (N1=N2)m then the pooled estimate is just a simple average of the two individual estimatesoWith unequal M, the larger sample has more influence on the overall estimateoWe want the pooled estimate to come out closed to the larger sample, because bigger sample sizes give better estimates-After you find s2POOLED, you can use this to estimate the population standard deviation of the distribution if differences between means (SDIFF)oComputing T-oThe t value tells us how much the difference between means that we observes (M1-M2)…oDeviates from the difference that we would expect if the null was true (μDIFF)…oRelative to how much sample difference if the null is true (SDIFF) Comparison Distribution-The comparison distribution tells us this:oIFoThe null hypothesis was true, and oWe took many pairs of samples, andoWe computes a t value for each pair of samples with the t formulaoTHENoT values would be common where the distribution is high and rare where it is low-The comparison distribution is a t distribution with dfTOT degrees of freedom-T values shape is normal if the null is