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t-Static 1. Single Sample or One Sample t-Test AKA student t-test. 2. Two Independent sample t-Test, AKA Between Subject Designs or Matched subjects Experiment. 3. Related Samples t-test or Repeated Measures Experiment AKA Within Subject Designs or Paired Sample TTest . 2 CHAP 9 t-Static Single Sample or One Sample t-Test t-test is used to test hypothesis about an unknown population mean, µ, when the . value of σ or σ² is unknown Ex. Is this class know more about STATS than the last year class? Mean for the last year class µ=80 Mean for this year class M=82 Note: We don’t know what the average STATS score should be for the population. We compare this year scores with the last year. 3 Degrees of Freedom df=n-1 4 Assumption of the t-test (Parametric Tests) 1.The Values in the sample must consist of independent observations 2. The population sample must be normal 3. Use a large sample n ≥ 30 5 6 Inferential Statistics t-Statistics: There are different types of t- Statistic 1. Single (one) Sample t-statistic/Test (chap 9) 2. Two independent sample t-test, Matched-Subject Experiment, or Between Subject Design t-test (chap10) 3.Repeated Measure Experiment, or Related/Paired Sample t-test (chap11) 7 FYI Steps in Hypothesis-Testing Step 1: State The Hypotheses H : µ ≤ 100 H : µ > 100 Statistics: 0 average 1 average Because the Population mean or µ is known the statistic of choice is z-Score 8 FYI Hypothesis Testing Step 2: Locate the Critical Region(s) or Set the Criteria for a Decision 9 FYI Directional Hypothesis Test 10 FYI None-directional Hypothesis Test 11 FYI Hypothesis Testing Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics Z Score for Research 12 FYI Hypothesis Testing Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics 13 Hypothesis Testing Step 4: Make a Decision 14 Calculations for t-test Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics M-μ t= s Sm Sm= Sm= M-μ √n t M=t.Sm+μ μ=M- Sm.t Sm= estimated standard error of the mean or 2 S /n 15 FYI Variability SS, Standard Deviations and Variances X 1 2 4 5 σ² = ss/N σ = √ss/N s = √ss/df s² = ss/n-1 or ss/df Pop Sample SS=Σx²-(Σx)²/N SS=Σ( x-μ)² Sum of Squared Deviation from Mean 16 FYI d=Effect Size for Z Use S instead of σ for t-test 17 Cohn’s d=Effect Size for t Use S instead of σ for t-test d = (M - µ)/s S= (M - µ)/d M= (d . s) + µ µ= (M – d) s 18 Percentage of Variance Accounted for by the Treatment (similar to Cohen’s d) Also known as ω² Omega Squared 2 t 2 r 2 t df 19 percentage of Variance accounted for by the Treatment Percentage of Variance Explained r 2 r2 r²=0.01-------- Small Effect r²=0.09-------- Medium Effect r²=0.25-------- Large Effect 20 Problems Infants, even newborns prefer to look at attractive faces (Slater, et al., 1998). In the study, infants from 1 to 6 days old were shown two photographs of women’s face. Previously, a group of adults had rated one of the faces as significantly more attractive than the other. The babies were positioned in front of a screen on which the photographs were presented. The pair of faces remained on the screen until the baby accumulated a total of 20 seconds of looking at one or the other. The number of seconds looking at the attractive face was recorded for each infant. 21 Problems Suppose that the study used a sample of n=9 infants and the data produced an average of M=13 seconds for attractive face with SS=72. Set the level of significance at α=.05 for two tails Note that all the available information comes from the sample. Specifically, we do not know the population mean μ or the population standard deviation σ. On the basis of this sample, can we conclude that infants prefer to look at attractive faces? 22 Null Hypothesis t-Statistic: If the Population mean or µ and the sigma are unknown the statistic of choice will be t-Static 1. Single (one) Sample t-statistic (test) Step 1 H : µ = 10 seconds H : µ ≠ 10 seconds 0 1 attractive attractive 23 None-directional Hypothesis Test 24 None-directional Hypothesis Test 25 Calculations for t-test Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics M-μ t= s Sm Sm= Sm= M-μ √n t M=t.Sm+μ μ=M- Sm.t Sm= estimated standard error of the mean or 2 S /n 26 Problems A psychologist has prepared an “Optimism Test” that is administered yearly to graduating college seniors. The test measures how each graduating class feels about its future. The higher the score, the more optimistic the class. Last year’s class had a mean score of μ=15. A sample of n=9 seniors from this year’s class was selected and tested.. 27 Problems The scores for these seniors are 7, 12, 11, 15, 7, 8, 15, 9, and 6, which produced a sample mean of M=10 with SS=94. On the basis of this sample, can the psychologist conclude that this year’s class has a different level of optimism? Note that this hypothesis test will use a t-statistic because the population variance σ² is not known. USE SPSS Set the level of significance at α=.01 for two tails 28 Null Hypothesis t-Statistic: If the Population mean or µ and the sigma are unknown the statistic of choice will be t-Static 1. Single (one) Sample t-statistic (test) Step 1 H : µ = 15 H : µ ≠ 15 0 1 optimism optimism 29 None-directional Hypothesis Test Step 2 30 Calculations for t-test Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics M-μ t= s Sm Sm= Sm= M-μ √n t M=t.Sm+μ μ=M- Sm.t Sm= estimated standard error of the mean or 2 S /n 31 t-Static 1. Single Sample or One Sample t-Test AKA student t-test. 2. Two Independent sample t-Test, AKA Between Subject Designs or Matched subjects Experiment. 3. Related Samples t-test or Repeated Measures Experiment AKA Within Subject Designs or Paired Sample TTest . 32 Chapter 10 Two Independent Sample t-test Matched-Subject Experiment, or Between Subject Design An independent-measures study uses a separate sample to represent each of the populations or treatment conditions being compared. 33 Independent Sample t-test An independent measures study uses a separate group of participants to represent each of the populations or treatment conditions being compared. 34 Two Independent Sample t-test Null Hypothesis: If the Population mean or µ is unknown the statistic of choice will be t-Static Two independent sample t-test, Matched-Subject Experiment, or Between Subject Design Step 1 H : µ -µ = 0 H : µ -µ ≠ 0 0 1 1 1 2 2 35 None-directional Hypothesis Test Step 2 36 STEP 3 37 Estimated Standard Error S(M -M ) 1 2 The estimated standard error measures how much difference is expected, on average, between a sample mean difference and the population mean difference. In a hypothesis test, µ1 -µ2 is set to zero and the standard error measures how much difference is expected between the two sample means. 38 Estimated Standard Error S (M1-M2)= 39 Pooled Variance s² P 40 Pooled Variance s² P 41 Step 4 42 Measuring d=Effect Size for the independent measures d M1 M 2 2 S p 43 Estimated d 44 Estimated d 45 Percentage of Variance Accounted for by the Treatment (similar to Cohen’s d) Also known as ω² Omega Squared 2 t 2 r 2 t df 46 FYI in Chap 15 We use the Point-Biserial Correlation (r) when one of our variable is dichotomous, in this case (1) watched Sesame St. (2) and didn’t watch Sesame St. 2 t 2 r 2 t df 47 Problems Research results suggest a relationship Between the TV viewing habits of 5-year-old children and their future performance in high school. For example, Anderson, Huston, Wright & Collins (1998) report that high school students who regularly watched Sesame Street as children had better grades in high school than their peers who did not watch Sesame Street. 48 Problems The researcher intends to examine this phenomenon using a sample of 20 high school students. She first surveys the students’ s parents to obtain information on the family’s TV viewing habits during the time that the students were 5 years old. Based on the survey results, the researcher selects a sample of n1=10 49 Problems students with a history of watching “Sesame Street“ and a sample of n2=10 students who did not watch the program. The average high school grade is recorded for each student and the data are as follows: Set the level of significance at α=.05 for two tails 50 Problems Average High School Grade Watched Sesame St (1). Did not Watch Sesame St.(2) 86 87 91 97 98 99 97 94 89 92 n1=10 M1=93 SS1=200 90 89 82 83 85 79 83 86 81 92 n2=10 M2= 85 SS2=160 51 Two Independent Sample t-test Null Hypothesis: Two independent sample t-test, Matched-Subject Experiment, or Between Subject Design- nondirectional or two-tailed test Step 1. H : µ -µ = 0 H : µ -µ ≠ 0 0 1 1 1 2 2 52 Two Independent Sample t-test Null Hypothesis: Two independent sample t-test, Matched-Subject Experiment, or Between Subject Design directional or one-tailed test Step 1. H : µ ≤µ Sesame St . 0 H : 1 µ Sesame St. No Sesame St. >µ No Sesame St. 53 Problems In recent years, psychologists have demonstrated repeatedly that using mental images can greatly improve memory. Here we present a hypothetical experiment designed to examine this phenomenon. The psychologist first prepares a list of 40 pairs of nouns (for example, dog/bicycle, grass/door, lamp/piano). Next, two groups of participants are obtained (two separate samples). Participants in one group are given the list for 5 minutes and instructed to 54 memorize the 40 noun pairs. Problems Participants in another group receive the same list of words, but in addition to the regular instruction, they are told to form a mental image for each pair of nouns (imagine a dog riding a bicycle, for example). Later each group is given a memory test in which they are given the first word from each pair and asked to recall the second word. The psychologist records the number of words correctly recalled for each individual. The data from this experiment are as follows: Set the level of significance at α=.01 for two tails 55 Problems Data (Number of words recalled) Group 1 (Images) Group 2 (No Images) 19 20 24 30 31 32 30 27 22 25 n1=10 M1=26 SS1=200 23 22 15 16 18 12 16 19 14 25 n2=10 M2= 18 SS2=160 56 57