Unit 6 Test for Means If we don t have a specific idea about whether the true value is greater or less than 7 we d use our two sided alternative HA college 7 But if we think the true value is likely to be greater than 7 we d use the one sided alternative HA college 7 And if we think the true value is likely to be less than 7 we d use the one sided alternative HA college 7 when they do have a more specific hypothesis about the direction in which the mean may differ from the null value In fact psychologists usually use a two sided alternative even The sampling distribution of the sample mean will be N n if the sample size is However we don t know the population standard deviation We need the in N 7 n But you may recall from when we learned how to construct confidence intervals that we The z score this is a standardized score It expresses how far our sample mean is from the used the sample sd as an estimate of the population sd reasonably large null hypothesis in terms of standard errors of the mean Note that as your test statistic gets bigger the p value will get smaller A one sided alternative will always give you a smaller p value This may be regarded as cheating making it too easy to reject the null hypothesis unless you have a very good reason to adopt a one sided alternative absolute value to the values in the table Note that for a two sided test we ignore the sign of the z test statistic we compare its When we learned about confidence intervals we saw that when our sample is small n 50 we cannot assume that the sample standard deviation s is a perfectly good estimate of the population standard deviation Because of this uncertainty the sampling distribution of the sample mean is not perfectly Normal when the sample is small Instead it follows a t distribution which is a bit more taily than a Normal distribution How taily depends on the exact df n 1 To correct for this we used a critical value of t rather than z to construct our confidence interval t is a little bigger than z how much bigger depends on the degrees of freedom n 1 When our sample is small we do the same significance testing procedure as before except for two things First we compute a t test statistic instead of a z test statistic It is computed in just the same way Second we consult a t table rather than a z table to determine the probability of getting a t statistic as large as the one we got Note that when we report the results of a t test we report the df in parentheses right after the letter t t 19 1 07 p 20 Cases where we have only one sample of individuals but we have two observations from each individual In this case we have matched pairs data Cases where we have two genuinely distinct samples of individuals with one observation of each individual In this case we have two sample data When do we have matched pairs data In many many studies in psychology Any time we have before and after measurements on the same individual Any time we give the same individual two different tests Any time we test the same individual under two different experimental conditions When our two sets of measurements are from different individuals however we have a two sample or independent samples design In this case there is no one to one matching up of values from our two sets of measurements Thus we cannot compute difference scores for specific individuals We have two independent samples and we want to know if there is a difference between means The basic idea is the same However we will need to modify our formula for our test statistic First in our numerator instead of a single sample mean we have a difference between two sample means And our null hypothesis is also about a difference between means The quantity in the numerator is a difference of differences It is the difference between the observed difference between sample means the difference in population means according to the null hypothesis If your null hypothesis is that there is no difference in population means which is our usual null hypothesis the value of the second part is 0 so we can just forget about it we need to take the standard deviation and sample size of each sample into account Our SE term therefore will be a bit more complicated