PSYCH 240 1st Edition Lecture 14Section 8: Single-Sample T TestSingle-Sample T Test-For the single-sample t test, we will test the null hypothesis that a sample came from a population with a given mean-The alternative hypothesis will state that the sample did NOT come from a population with the hypothesized population mean-Sometimes the alternative hypothesis will be directional and specify that the sample came from a population with a higher or lower mean-Sometimes the alternative will be non-directional and just say a different mean without saying higher or lower-If the sample mean is far enough from the population mean specified in the null hypothesis, then we can take this as evidence against the null and for the alternative -We have used 7 scores to measure the typicality of a sample mean on a distribution of means:o-And we needed to know the population standard deviation to figure out the standard error of the mean:o-Use the sample scores to estimate the population standard deviationoWe can use 2 formulas from section two: - --Then we can use the sample standard deviation to estimate the standard deviation of the distribution of meanso-Now we can estimate how typical/atypical our sample mean is for a population with the mean specified in the null hypothesis, μMooThis is interpreted the same way as a z score:-T values above/below zero indicate that the sample mean is higher/lower than we would expect based on the null hypothesis-T values farther from zero indicate that the sample mean is more "surprising: under the null hypothesisoIf the t value is far enough from zero, we can take this as evidence against the null and for the alternative T-Distributions-T-distributions are theoretically constructed like this:oRandomly sample from a population of scores that follow a normal distribution with a mean equal to the value specified in the null hypothesisoCompute a t value using the mean and standard deviation of each sampleoAfter you do this a bunch of times, the t values will follow a t-distribution-We can use t -distributions as a comparison distribution showing what results we are likely to get if the null is true-We don’t need to know the population standard deviation-A t-distribution is symmetrical, unimodal, and bell-shaped like a normal distribution, but it has fatter tails-T-distributions get closer to a normal distribution as sample size increaseso-Knowing the t-distribution lets us figure out where to place the critical values to achieve the desired alpha value-You can get the critical values from a table of a computer program like R-If you are doing a 1-tailed test, then you need a critical value that puts proportions of t scores equal for alpha in the tails. This is the first three columns of your tableooChose the column that corresponds to the requested alpha value, and go to the row with the correct degrees of freedom (df)oIf the df's not on the table, go to the closest oneoThe critical values get closer to zero as the df increases because the t distributions are losing their fat tailsoThe table gives you the absolute value of the cutoff, and you have to figure out the sign. Use positive to test for an increase and negative to test for a decrease-If you are doing a 2-tailed test, then you have to divide alpha evenly between the high and low tail. This is the second three columns in your tableooThe tables give you the absolute value, and you will have two critical values, one positive and one negative Degrees of Freedom-The shape of the t-distribution depends on the degrees of freedom in your sample-Degrees of Freedom: number of scores that are free to vary in the calculation of a statistic-For the single sample t-test, the degrees of freedom a N - 1-To estimate the population standard deviation, we need to know all of the scores and the mean that they are varying fromo-If we knew the mean of the distribution that these scored came from, then the number of varying elements in these formulas would be the number of scores we have-But we DON'T actually know the mean of the distribution that our scores came from, wejust have one hypothesis that says it is a certain value and another hypothesis that says it is another value-To have a mean to use in the formula, we have to use the mean of out sample scores-This doesn’t actually add any information. If you already know all the scores in the sample, me telling you the mean of the sample doesn’t tell you anything new (you could have figured it out for yourself using the scores) -So if we considered the variability estimate to be based on N degrees of freedom, we would be pretending like we have more information than we actually do-We actually have N - 1 degrees of freedom because we "used up" one degree of freedom by using the sample scores to estimate the mean-Think of that 1 degree of freedom as the penalty that we have to pay for not knowing the true mean and having to estimate it Example-A developmental psychologist runs an experiment in which 9-month olds watch several small plays in which one puppet helps a bunny climb up a slope (the “helper”) while another puppet pushes the bunny back down the slope (the “blocker”). After watching each play, the babies are given the choice of playing with the helper or the blocker. If theplay had no influence on their choice, they would pick the helper .50 of the time. A sampleof 25 babies picked the helper .61 of the time on average with a standard deviation of .20. With an alpha of .01, did the play have any effect on the babies’ choices?oStep 1-Null hypothesis: The choice scores for babies who see the play come from a population with a mean of .5 (random guessing)-Alternative Hypothesis: the choice scores for babies who see the play come from a population with a mean different than .5oStep 2-N=25 babies-T distribution with 24 degrees of freedomoStep 3-Critical values(s) by using the table = -2.797, 2.797oStep 4-oStep 5---2.797 < 2.75 < 2.797-Non-significant-The test failed to produce specific evidence that the play influenced the babies' choices Assumptions of the Single-Sample T-Test-The test assumed that scores are normally distributed at the population level-If the scores do not follow a normal distribution, then the distribution of t values under the null hypothesis might not actually follow a t distribution-The test is usually very robust to non-normality, as long as the deviation from normal isn’t really extreme (such as a