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CLASS NOTES: Introduction to the t Statistic / t Ratio / t Test 1st set: Basic t-test nd 2 set: t-test for independent measures 3rd set: t-test for repeated measures CONCEPT CALCULATION/EXAMPLES APPLICATION * As opposed to comparing sample scores w/ population scores, for tscores, we are now comparing 2 samples. * A t-statistic is often times preferred over a z-statistic. The zscore formula requires more information than is usually available (i.e. population data). * Remember that the reason for conducting hypothesis testing is to gain knowledge about an unknown population. * We have talked about sampling w/ replacement being a requirement for a true experiment. In these cases, when sampling w/ replacement is not feasible, you select a large population relative to smaller sample sizes. The t Statistic : An Alternative to z Requirements for Using the t Test 1. The samples have been randomly selected. 2. The traits being measured do not depart significantly from normality w/in the population. 3. The standard deviations of the two samples must be fairly similar. 4. The two samples are independent of each other. 5. Comparisons are made only b/t measures of the same trait. 6. The sample scores provide at least interval data or ratio data. Hypothesis Testing & the Single Sample t-Test State your hypothesis Step 1: Null Hypothesis: Ho = there is no difference The null hypothesis is always stated first following the alternative hypothesis. Alternative Hypothesis: H1 = there is a difference Locate your critical region Step 2: α = .05 two-tailed df = 9 Gather your sample & follow thru w/ the calculations Calculate the mean Step 3: M1 = ΣX1 n1 Remember that the alpha level is determined by the researcher. Whether or not you are using a one- or two-tailed test is determined by how you write your hypothesis statements (whether or not you use directionality terms in your statements) Calculate standard deviation Estimated Standard Error: Measures the difference b/t the sample mean & the population mean. Here, you see 2 formulas for the Estimated SE. Both give you the same outcome, but the second one requires less calculation. Calculate your t-ratio s1 = ΣX12 – (ΣX1)2 n___ n-1 Remember we are using values & formulas related to “sample means,” thus the ‘sample’ SD formula & the SE of M. sM or SEM = s2 n OR SEM = s n This second formula listed here is the formula we used when calculating your final SE of M value. SM or SEM is the notation for the estimated standard error for t tests. Remember that s stands for your sample standard deviation. t=M–µ SEM This particular SEM formula requires the least amount of calculations & is similar to the formula we used when calculating z-scores for our sample means. Although we are using formulas similar to those that we used in calculating sample means, for a single sample-t though, the population mean is generally available to the researcher. t-ratio: Used to test hypotheses t = sample mean - pop. Mean about an unknown population (data) (hypothesized mean µ when the value of σ is ______________from Ho)___ unknown. The formula for the tEstimated standard error statistic has the same structure as (computed from the z-score formula, except that the the sample data) t-statistic uses the estimated standard error in the denominator. t=M–µ * Using t tests, we do not know the SEM population standard deviation of that distribution since our population is unknown. Step 1: Compute the mean from * The t ratio tells us specifically So, the only difference b/t the zscore formula & the t-score formula is that the z-score uses the actual population variance & the tscore uses the corresponding sample variance when the population is unknown. how far the sample mean deviates from the population mean in units of standard errors of the mean. * We will always be comparing two sample means: a known sample mean w/ an assumed population mean. * In the calculation of the single-t ratio, the assumed population parameter is presented. your sample means set. Step 2: Subtract your estimated population mean from mean of the sample means.. Step 3: Calculate your estimated standard error. Step 4: Divide your numerator by your denominator. Critical Region for the t Test Look in Appendix B in the back of your text book to find the critical regions for a t-distribution. Your critical region is based upon your df of degrees of freedom & your chosen alpha level for either a one-tailed or two-tailed test. Make a decision Draw out a distribution & mark on that distribution where 1) your tvalue, or cut-off point falls (this is the t-value you obtained from your table in the back of the text) & then mark where your calculated tratio falls. If your t-ratio falls w/in the “critical region” (or the space beginning w/ the t-value cut-off point & beyond in the tail), then you will “reject the null.” If your calculated t-ratio falls outside of the “critical region” (prior to the cut-off point & beyond, or w/in the “null” area – the area outside of the critical region), then you will “fail to reject the null.” Hypothesis Tests & Effect Size The value you obtain from the tdistribution table will also be a tvalue. But this t-value represents the cut-off point between your critical region & your null region. Not to be confused with your calculated t-ratio. Step 4: Either reject the null (there is a difference b/t the sample & population values) Or fail to reject the null (there is no difference b/t the sample & population values) Whatever your outcome, you need to write first whether or not you reject or fail to reject the null, & then you need to repeat the hypothesis (from Step 1 in the hypothesis testing process) statement that associates w/ your outcome. A large value for the t-statistic (either positive or negative) indicates that the obtained difference is greater than would be expected by chance. Measuring Effect Size: Whenever a treatment effect is found to be statistically significant, it is recommended that you report a measure of the absolute magnitude of the effect (which the 2 measures are Cohens-d & r2. These will account for the percentage of variance. Use estimated-d formula Estimated Cohen’s d = d=t n Percentage of Variance r2 = ____t2__ t2 + df This formula expresses your effect size in terms of percentage. r2 represents the percentage of variance. t represents your tstatistic whereas df as you know, represents your degrees of freedom. OUTLINE OF MATERIAL I. II. t-Statistic: Basic principals Hypothesis testing for the t-statistic A. State your hypothesis 1. Null: Ho : M = µ 2. Alternative: H1 : M ≠ µ B. Locate the critical region 1. Select your alpha level 2. Determine your df value 3. Draw out your distribution & locate where your critical regions based upon your choice of a one-tailed or two-tailed test C. Collect the sample data & compute the t statistic 1. Sample mean(s) 2. Standard deviation(s) 3. Estimated Standard Error(s) 4. Estimated Standard Error of Difference (When 2 samples are being used) 5. t-Ratio D. Make a decision to either reject the Null hypothesis or fail to reject the null hypothesis E. Calculate your effect size CLASS NOTES: The t Test for Two Independent Samples CONCEPT CALCULATION/EXAMPLES APPLICATION Independent Measures Design / Between Measures Design: Uses a separate sample for each treatment condition (or for each population). * The goal of independentmeasures research study is to evaluate the mean difference b/t 2 populations (or b/t 2 treatment conditions). * It allows us to make a probability statement regarding whether two independently selected samples represent a single population. Example: *There is a risk involved in independent/ b/t measures designs where some individuals in one sample may be mismatched to individuals in the other sample (i.e., one may have a higher IQ, etc..) that can affect the outcome. Men vs. Women *The two separate samples are equal in size (n1 = n2) Stating your Hypothesis for the Independent Measures t- test The independent-measures tstatistic uses the data from 2 separate samples to help decide whether there is a significant mean difference b/t two populations or b/t 2 treatment conditions. µ1 = notation for population 1 µ2 = notation for population 2 Null hypothesis: Ho = µ1 - µ2 = 0 Remember that your null hypothesis (Ho) states that there is no change or no effect (there is no difference b/t the means of the 2 independent samples) Alternative Hypothesis: H1 = µ1 - µ2 ≠ 0 Your alternative hypothesis (H1) states that there is a change or there is an effect (there is a difference b/t the means of the 2 independent samples; the means are not equal) Remember your inferential statistics. You are using your samples to make inferences about unknown or theoretical populations. Estimated Standard Error Calculating for the standard error of both the first sample (represented by a subscript of “1”) σM or SEM1 = _s1_ n σM or SEM2 = _s2_ & the second sample (represented by a subscript of “2”) Standard Error of Difference Pooled Variance: Corrects for biases in sample variances by combining two sample variances into a single value. The variances are obtained by averaging or pooling the two sample variances using a “procedure” that allows the bigger sample to carry more weight in determining the final value. * This “procedure” or formula is structured so that each sample is weighted by the magnitude of its df value; thus, the larger sample carries more weight in determining the final value. Formulas for an IndependentMeasures Hypothesis Test: Uses n SED = SE2M1 + SE2M2 SED = (n1 – 1)s12 + (n2 – 1)s22 x (n1 + n2 – 2) 1 + 1 n1 n2 Step 1: Calculate the n value for each sample Step 2: Calculate the standard deviation for each sample Step 3: For the numerator in the first bracket, subtract 1 from the n from your first sample. Step 4: Multiply that number by its’ sample standard deviation Step 5: Repeat steps 3 & 4 for the second sample. Step 6: For the denominator in the first bracket, add your two n values from each sample & then subtract 2 from this value. Step 6: For the next bracket, divide the number 1 by the n value of your first sample. Step 7: Do the same for your second sample. Step 8: Add these 2 values together Step 9: Multiply the value from your first bracket to the value from your second bracket Step 10: Square root this remaining value t = M1 – M2 SED We will use the standard error of difference since we are working w/ 2 different sample groups, representing a “growth” or addition to our SE of M formulas. * If sample sizes are not equal, then the results are biased * If the variances are obtained from a large sample, it will be a more accurate estimate of σ2 than the variance obtained from a small sample. * The pooled variance is actually an average of the 2 sample variances & the value of the pooled variance will always be located b/t the 2 sample variances. t= sample __mean diff. - population mean diff._ the difference b/t two sample means to evaluate the difference b/t two population means. Estimated standard error of the difference Your subscript “1” represents data from your first sample. Your subscript “2” represents data from your second sample. The basic structure of the t-statistic is the same for both the independent measures & the singlesample hypothesis tests. Step 1: Find the mean for each group Step 2: Find the standard deviation for each group M1 = ΣX1 n1 s1 = s2 = Step 3: Find the estimated standard error of the mean for each group. Step 4: Find the estimated standard error of difference for both groups combined. The Estimated Standard Error of Difference (SED): Distribution of Differences with two populations. * Used as an estimate of the real standard error (σM) when the value of σ (the population standard deviation) is unknown. * SED is based on the estimated standard errors that have been obtained from each of the two samples. * The estimated standard error of difference is based on the information contained in just two samples. ΣX12 – (ΣX1)2 n___ n-1 Since we are working w/ 2 samples, then we use the sample computational standard deviation formula. ΣX22 – (ΣX2)2 n___ n-1 SEM1 = s n SED = M2= ΣX2 n2 SEM2 = s n SE2M1 + SE2M2 Step 1: Take the standard error of your first sample & square it. Step 2: Then you add to that the standard error of the second sample & square that number. Step 3: Last but not least, you square root that final number to come up w/ your estimated sample error of difference. Or the alternative formula for estimated standard error when using pooled variance: Remember that “s” represents your standard deviation for samples. Remember your standard error of the mean formula. * Since we measure only one pair of samples to generate this value, the estimated standard error of difference is a statistic & not a parameter. * It is computed from the sample variance or sample standard deviation & provides an estimate of the standard distance b/t a sample mean M & the population mean µ. Step 5: Calculate the t-ratio Degrees of Freedom: The degrees of freedom for the independentmeasures t-statistic are determined by the degrees of freedom values for the 2 separate samples. SEP = (n1 – 1)s12 + (n2 – 1)s22 x (n1 + n2 – 2) 1 + 1 n1 n2 t = M1 – M2 SED df for the first sample + df for the second sample = df1 + df2 = (n1 – 1) + (n2 – 1) OR Your total number of values in your whole study minus 2 nT - 2 Final step in the hypothesis Step 4: testing process: State your Either reject the null (there is a decision Remember that to reject the null difference b/t the first sample & the hypothesis, then your calculated tsecond sample) ratio must fall beyond your t-value Or fail to reject the null (there is no obtained w/in the table & w/in the difference b/t the first sample & the critical region, whereas to “fail to second sample) reject the null,” your calculated tratio must fall outside of the critical region & w/in the “null” region (the rest of the distribution that falls outside of the critical region). Effect Size for Independent Measures t-Statistic If your sample sizes are equal, then you can use the paired-t formula: * The subscript “p” in this formula of estimated standard error represents the “pooled variance” that is used in the independentmeasures research design t-statistic. * This is used when the sample sizes used are not equal. Your mean from sample 1 minus your mean of sample 2 divided by the Standard Error of the Difference. * Remember your Degrees of Freedom or df: It describes the number of scores in a sample that are independent & free to vary. * Remember also that your critical values region is based upon your df & your set alpha level for either a one-tailed or two-tailed test. The symbol nT represents the “total” number of ‘n’ values (subscript “T” representing “total”) Whatever your outcome, you need to write first whether or not you reject or fail to reject the null, & then you need to repeat the hypothesis (from Step 1 in the hypothesis testing process) statement that associates w/ your outcome. d=t n Estimated d = t S2p Percentage of Variance r2 = ____t2__ t2 + df n is representing the number of pairs of samples. This is the formula for the effect size. Your subscript “p” comes from your pooled variance. This formula expresses your effect size in terms of percentage. r2 (for now) represents the percentage of variance. t represents your t-statistic whereas df as you know, represents your degrees of freedom. OUTLINE OF MATERIAL I. II. t-Statistic for Independent Samples / Basic principals Hypothesis testing for the t-statistic A. State your hypothesis 1. Null: Ho = µ1 - µ2 = 0 2. Alternative: H1 = µ1 - µ2 ≠ 0 B. Locate the critical region 1. Select your alpha level 2. Determine your df value 3. Draw out your distribution & locate where your critical regions based upon your choice of a one-tailed or two-tailed test C. Collect the sample data & compute the t statistic 1. Sample mean(s) 2. Standard deviation(s) 3. Estimated Standard Error(s) 4. Estimated Standard Error of Difference (When 2 samples are being used) 5. Pooled variance (When 2 samples are not equal in size) 6. t-Ratio D. Make a decision to either reject the Null hypothesis or fail to reject the null hypothesis E. Calculate your effect size CLASS NOTES: The t Test for Two Related Samples / Repeated Measures Design CONCEPT CALCULATION/EXAMPLES Repeated Measures Design: A single sample of individuals is measured more than once on the same dependent variable. The same Example: A sample group is measured for symptoms reported before therapy APPLICATION * Remember your definition of dependent variable (The variable that is observed in order to assess the effect of the treatment) samples are used in all of the treatment conditions. 2 sets of data are obtained from the same sample of individuals. & again after therapy. * The goal is to use a sample of difference scores to answer questions about the general population. Basically, we would like to know what would happen if every individual in the population were measured in two treatment conditions (X1 & X2) & the difference score (D) were computed for everyone. So, we are therefore also interest in the mean for the population of difference scores. * The main advantage of a repeated-measures study is that it uses exactly the same individuals in all treatment conditions & there is no risk of a difference in participants from one sample to another. * A repeated measures design typically requires fewer subjects than an independent-measures design & uses the subjects more efficiently. * It works well for studying learning, development or other changes that take place over time. * Observations must be independent * Population distribution of difference scores must be normal. Formulas for Repeated Measures Hypothesis Test: Uses the difference b/t two sample means to evaluate the difference b/t two population means. Step 1: Find the mean for each group Step 2: Find the standard deviation for each group Step 3: Find the estimated standard error of the mean for each group. Step 4: Find the estimated standard error of difference for both groups combined. Step 5: Calculate your t-statistic M1 = ΣX1 n1 M2= ΣX2 n2 s= ΣX12 – (ΣX1)2 n___ n-1 s= ΣX22 – (ΣX2)2 n___ n-1 SEM1 = s n SED = SEM2 = s n Although we are using one sample group, we are obtaining 2 sets of separate values, before & after. Therefore, we use the sample computational standard deviation formula. These steps are the same as those presented in the above processes SE2M1 + SE2M2 t = M1 – M2 The mean of your first set of values minus the mean of your second set SED Hypothesis Testing Process State your Hypothesis Step 1: Null hypothesis: Ho = µD = 0 Alternative Hypothesis: H1 = µD ≠ 0 Set the criteria for a decision by selecting your alpha level & locating your critical regions by drawing out your distribution & using your t-distribution table of values divided by your Standard Error of the Difference. Often times in a related samples ttest, the researcher has a specific prediction regarding the direction of the treatment effect, therefore he or she will often use a directional or one-tailed test. You must write out your hypothesis statements. Step 2: two-tailed α = .05 df = (n - 1) = 4 Collect your data & compute your sample statistics Step 3: Calculate the means for the samples (ex. Before/after) M1 = ΣX1 n1 M2= ΣX2 n2 Calculate the standard deviation for each value set s= ΣX12 – (ΣX1)2 n__ n-1 s= ΣX22 – (ΣX2)2 n___ n-1 Calculate the standard error of the mean for each sample SEM1 = s SEM2 = s As opposed to the independent measures-t which combines the degrees of freedom for each sample set, in the repeated measures-t, since we only have one sample set, then the df formula remains n-1. n n Calculate the standard error of difference SED = SE2M1 + SE2M2 Calculate the t-ratio t = M1 – M2 SED Make a decision Step 4: Either reject the null (there is a difference) Or fail to reject the null (there is no difference) The mean from the first set of values minus the mean from the second set of values divided by the Standard Error of the Difference. Whatever your outcome, you need to write first whether or not you reject or fail to reject the null, & then you need to repeat the hypothesis (from Step 1 in the hypothesis testing process) statement that associates w/ your outcome. OUTLINE OF MATERIAL I. II. t-Statistic for repeated measures design: Basic principals Hypothesis testing for the t-statistic A. State your hypothesis 1. Null: Ho : M = µ 2. Alternative: H1 : M ≠ µ B. Locate the critical region 1. Select your alpha level 2. Determine your df value 3. Draw out your distribution & locate where your critical regions based upon your choice of a one-tailed or two-tailed test C. Collect the sample data & compute the t statistic 1. Sample mean(s) 2. Standard deviation(s) 3. Estimated Standard Error(s) 4. Estimated Standard Error of Difference (When 2 samples are being used) 5. t-Ratio D. Make a decision to either reject the Null hypothesis or fail to reject the null hypothesis E. Calculate your effect size