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Using t-tests (independent samples) Statistics for the Social Sciences Psychology 340 Spring 2010

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PSY 340 Statistics for the Social Sciences Statistical analysis follows design The one-sample t-test can be used when: –1 sample –One score per subject –Population mean (μ) is known –but Population standard deviation (σ) is NOT known

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PSY 340 Statistics for the Social Sciences Independent samples What are we doing when we test the hypotheses? –Consider a new variation of our memory experiment example Memory treatment Memory patients Memory Test the memory treatment sample are the same as those in the population of memory patients. they aren’t the same as those in the population of memory patients H0:H0: HA:HA: Memory placebo Memory Test Compare these two means

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PSY 340 Statistics for the Social Sciences Statistical analysis follows design The independent samples t-test can be used when: –2 samples –Samples are independent

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PSY 340 Statistics for the Social Sciences Performing your statistical test Test statistic One-sample t Independent-samples t Observed (sample) means

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PSY 340 Statistics for the Social Sciences Performing your statistical test Test statistic One-sample t Independent-samples t Hypothesized population means from the Null hypothesis

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PSY 340 Statistics for the Social Sciences Performing your statistical test Test statistic One-sample t Independent-samples t Hypothesized population means from the Null hypothesis H0:H0: Memory performance by the treatment group is equal to memory performance by the no treatment group. So:

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PSY 340 Statistics for the Social Sciences Performing your statistical test Test statistic One-sample t Estimated standard error (difference expected by chance) estimate is based on one sample We have two samples, so the estimate is based on two samples The Estimate of the Standard Error is based on the variability of both samples

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PSY 340 Statistics for the Social Sciences Performing your statistical test “pooled variance” We combine the variance from the two samples Number of subjects in group A Number of subjects in group B

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PSY 340 Statistics for the Social Sciences variance Performing your statistical test “pooled variance” We combine the variance from the two samples Recall “weighted means,” need to use “weighted variances” here Variance (s 2 ) * degrees of freedom (df)

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PSY 340 Statistics for the Social Sciences Performing your statistical test Independent-samples t Compute your estimated standard error Compute your t-statistic Compute your degrees of freedom This is the one you use to look up your t crit

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PSY 340 Statistics for the Social Sciences Performing your statistical test Exp. group Control group 45 55 40 60 43 49 35 51 Need to compute the mean and variability for each sample Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he randomly assigns 8 patients to one of two samples. He then gives one sample the new treatment but not the other. Following the treatment period he gives both groups a memory test. The data are presented below. Use α = 0.05.

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PSY 340 Statistics for the Social Sciences Performing your statistical test Exp. group Control group 45 55 40 60 43 49 35 51 Need to compute the mean and variability for each sample Control group = 50 (45-50) 2 + (55-50) 2 + (40-50) 2 + (60-50) 2 = 250 SS = A Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he randomly assigns 8 patients to one of two samples. He then gives one sample the new treatment but not the other. Following the treatment period he gives both groups a memory test. The data are presented below. Use α = 0.05.

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PSY 340 Statistics for the Social Sciences Performing your statistical test Exp. group (43-44.5) 2 + (49- 44.5) 2 + (35- 44.5) 2 + (51- 44.5) 2 = 155 SS = B Exp. group Control group 45 55 40 60 43 49 35 51 Need to compute the mean and variability for each sample Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he randomly assigns 8 patients to one of two samples. He then gives one sample the new treatment but not the other. Following the treatment period he gives both groups a memory test. The data are presented below. Use α = 0.05. = 44.5

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PSY 340 Statistics for the Social Sciences Performing your statistical test Exp. group Control group 45 55 40 60 43 49 35 51 Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he randomly assigns 8 patients to one of two samples. He then gives one sample the new treatment but not the other. Following the treatment period he gives both groups a memory test. The data are presented below. Use α = 0.05. = 0.95

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PSY 340 Statistics for the Social Sciences Performing your statistical test T obs = 0.95 T crit = ±2.447 α = 0.05 Two-tailed Exp. group Control group 45 55 40 60 43 49 35 51 Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he randomly assigns 8 patients to one of two samples. He then gives one sample the new treatment but not the other. Following the treatment period he gives both groups a memory test. The data are presented below. Use α = 0.05. = 0.95

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PSY 340 Statistics for the Social Sciences Performing your statistical test T obs = 0.95 α = 0.05 Two-tailed T crit = ±2.447 Exp. group Control group 45 55 40 60 43 49 35 51 Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he randomly assigns 8 patients to one of two samples. He then gives one sample the new treatment but not the other. Following the treatment period he gives both groups a memory test. The data are presented below. Use α = 0.05. +2.45 = t crit - Fail to Reject H 0 t obs =0.95 = 0.95

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PSY 340 Statistics for the Social Sciences Assumptions: Independent samples t Each of the population distributions follows a normal curve (this is an assumption of all t-tests) –T-tests are fairly ‘robust’ against this assumption This means that the results generally still hold even if this assumption is violated Homogeneity of variance: The two populations have the same variance –SPSS tests this using Levene’s Test Two rows in the SPSS output –Us the top row if the p-value for the Levene’s test is greater than 0.05 –Use the bottom row if the p-value for the Levene’s test is less than 0.05 Tests the Null hypothesis that the two groups have equal variances

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PSY 340 Statistics for the Social Sciences Effect Size for the t Test for Independent Means Estimated effect size after a completed study “pooled standard deviation” not “pooled variance,” so take the square root of s P 2

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PSY 340 Statistics for the Social Sciences Power for the t Test for Independent Means (.05 significance level) 8-4 8-5

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PSY 340 Statistics for the Social Sciences Approximate Sample Size Needed for 80% Power (.05 significance level) 8-5

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PSY 340 Statistics for the Social Sciences Statistical Tests Summary DesignStatistical test (Estimated) Standard error One sample, σ known One sample, σ unknown Two related samples, σ unknown Two independent samples, σ unknown Next time

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PSY 340 Statistics for the Social Sciences Using SPSS: Independent samples t Entering the data –Different groups of observations go into SAME column e.g., Exp grp and control grp in a single column Person Cntrl-grp Exp-grp 1 2 3 4 45 55 40 60 43 49 35 51 Performing the analysis –Analyze -> Compare means -> independent samples t-test –Identify which columns have the observations (test variable) and which column has the group membership defined (grouping variable) –Define groups: what numbers correspond to the two groups? Reading the output –Means of the different groups, the mean difference, the computed-t, degrees of freedom, p-value (Sig.), Levene’s test –Separate column defines the group membership for each observation –e.g., exp grp = 0, control grp = 1

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