PSYCH 240 1st Edition Lecture 1Section 9: Dependent Samples T-Test cont.Dependent T-Test-The dependent t-test is just a special case of the single sample t-test-To conduct the test, you first create difference scores by subtracting each score in one sample from its corresponding score in the other sample-After this, you do a single sample t-test to see if difference scores could come from a population with a mean of zero-The null hypothesis is that the two samples come from identical populations, which is equivalent to saying that the population mean of the difference scores is zero-If you randomly sampled sets of two from the same population distribution, half of the time the first score would be higher and half of the time the second score would be higher.If you subtracted the scores in each pair, half of the time you'd get a negative number and half of the time a positive number-If you get a huge number of samples, the positives and negatives will cancel out and the mean of the differences will be zero-The alternative hypothesis is either that the two samples come from populations with different means (non-directional) or that the population mean for one sample is higher than the other (directional)-If the alternative is directional, you have to be careful about how the difference scores were created to know whether you need a positive or negative critical value-Low/High-Population Mean: the hypothesis, NOT how the results happened to come outoIf the low-population-mean condition is subtracted from the high-population-mean condition, then the alternative hypothesis is that the difference scores have a mean above zerooIf the high-population-mean condition is subtracted from the low-population-mean condition, then the alternative hypothesis is that the difference scores have a mean below zero-ExampleoIn an experiment, subjects were given letter strings and asked to decide if they were words (“flick”) or non-words (“flig”). For each subject, some of the words were preceded by a related word (“axle” ; “tire”) while others were preceded by an unrelated word (“book” ; “sparrow”). The response time (RT) data appear below. Do these data provide evidence that RTs are higher for unrelated than related words (alpha = .05)? oDifference score M = 39.75oDifference score S = 37.38-Step 1:Null Hypothesis: Scores in the related and unrelated conditions come from identical population, so the difference scores come from a population with a mean of zeroAlternative Hypothesis: Scores in the unrelated condition come from a population with a higher mean that the related condition, so the difference scores come from a population with a mean greater than zero-Step 2:N = 4 difference scores-N is not the number of scores, but the number of DIFFERENCE scoresT distribution with 3 degrees of freedom-Step 3:Critical value(s): 2.353-Step 4:SM = 37.38/square root of 4 = 18.69t=39.75 - 0/18.69 = 2.13-Step 5:Non-significantThe test failed to find specific evidence that response times are higher for unrelated words than related words Assumptions of the Dependent-Samples T-Test-The test assumes that the population distribution of difference scores is a normal distribution Clicker Questions-A researcher wants to know if self-confidence is different for drummers and singers. He samples 24 bands and asks the drummer and singer from each to rate their level of experience. The samples for drummers and singers should be consideredoDependent Objectives-Know the difference between dependent and independent samples, and be able distinguish research scenarios with dependent versus independent samples. -Be able to define within-subject and between-subject experimental designs, and understand the relationship of this distinction to the distinction between dependent and independent samples. -Be able to perform a dependent samples t-test – including all 5 steps of hypothesistesting – given a research scenario, the mean and variability of the sample difference scores, the sample size (from which you must derive the degrees of freedom), and the desired alpha value. -Know that the dependent-samples t test assumes that difference scores are normally distributed at the population level.Sec. 10: Confidence IntervalsConfidence Intervals-For all 2014 graduated from US colleges, what was the average student loan debt at the time of graduation?-We couldn’t possibly get data from every single 2014 college graduate to exactly answer this question - that population is too big-We can, however, take a sample from this population and get a rough answer-If we approached this problem with a single sample t-test, we would pick a hypothetical population mean to test, say $30,000-The null hypothesis states that this value is the true mean of the population that our sample came from-The alternative hypothesis says that it isn't (lets stick to a non-directional null hypothesisfor this discussion)-If our sample mean was far enough away from $30,000, the test would be significant. We would take this as evidence for the alternative hypothesis. -If our sample mean was pretty close to $30,000, the test would be nonsignifcant. We could not interpret this as evidence for either hypothesis. -Neither test result is really all that informative if you want to know the average student loan debt. -In this section we will consider a more informative hypothesis testing procedure: the confidence interval. -To make a confidence interval, we conceptually run a hypothesis test for EVERY POSSIBLEhypothesized population mean-The confidence interval summarizes which hypothesized population means produce significant versus non-significant-Any potential population mean inside the interval has a nonsignificant result. -Any potential population mean outside the interval has a significant result. -So if a potential population mean is outside of the interval, that means that our sample provides specific evidence against that value as the true population mean. -95% confidence intervals use an alpha of .05 to classify results as significant versus non-significant. -99% confidence intervals use an alpha of .01. -You always use a 2-tailed critical value for confidence intervals. We want to rule out population means that seem too high OR too low for our sample. If the hypothesized population mean is really close to the sample mean, then you will get anon-significant test. oThat is,