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The Two Sample t

The Two Sample t. Review significance testing Review t distribution Introduce 2 Sample t test / SPSS . Significance Testing . State a Null Hypothesis Calculate the odds of obtaining your sample finding if the null hypothesis is correct

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The Two Sample t

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  1. The Two Sample t Review significance testing Review t distribution Introduce 2 Sample t test / SPSS

  2. Significance Testing • State a Null Hypothesis • Calculate the odds of obtaining your sample finding if the null hypothesis is correct • Compare this to the odds that you set ahead of time (e.g., alpha) • If odds are less than alpha, reject the null in favor of the research hypothesis • The sample finding would be so rare if the null is true that it makes more sense to reject the null hypothesis

  3. Significance the old fashioned way • Find the “critical value” of the test statistic for your sample outcome • Z tests always have the same critical values for given alpha values (e.g., .05 alpha  +/- 1.96) • Use if N >100 • t values change with sample size • Use if N < 100 • As N reaches 100, t and z values become almost identical • Compare the critical value with the obtained value  Are the odds of this sample outcome less than 5% (or 1% if alpha = .01)?

  4. Critical Values/Region for the z test( = .05)

  5. Directionality • Research hypothesis must be directional • Predict how the IV will relate to the DV • Males are more likely than females to… • Southern states should have lower scores…

  6. Non-Directional & Directional Hypotheses • Nondirectional • Ho: there is no effect: (X = µ) • H1: there IS an effect: (X ≠ µ) • APPLY 2-TAILED TEST • 2.5% chance of error in each tail • Directional • H1: sample mean is larger than population mean (X > µ) • Ho x ≤ µ • APPLY 1-TAILED TEST • 5% chance of error in one tail -1.96 1.96 1.65

  7. STUDENT’S t DISTRIBUTION • Find the t (critical) values in App. B of Healey • “degrees of freedom” • # of values in a distribution that are free to vary • Here, df = N-1 Practice: ALPHA TEST N t(Critical) .05 2-tailed 57 .01 1-tailed 25 .10 2-tailed 32 .05 1-tailed 15

  8. Example: Single sample means, smaller N • A random sample of 16 UMD students completed an IQ test. They scored an average of 104, with a standard deviation of 9. The IQ test has a national average of 100. IS the UMD students average different form the national average? • Professor Fred hypothesizes that Duluthians are more polite than the average American. A random sample of 50 Duluth residents yields an average score on the Fred politeness scale (FPS) of 31 (s = 9). The national average is known to be 29. Write the research an null hypotheses. What can you conclude? • USE ALPHA = .05 for both

  9. Answer #1 Conceptually • Under the null hypothesis (no difference between means), there is more than a 5% chance of obtaining a mean difference this large. Sampling distribution for one sample t-test (a hypothetical plot of an infinite number of mean differences, assuming null was correct) t (obtained) = 1.72 Critical Region Critical Region -2.131 (t-crit, df=15) 2.131(t-crit, df=15)

  10. “2-Sample” t test • Apply when… • You have a hypothesis that the means (or proportions) of a variable differ between 2 populations • Components • 2 representative samples – Don’t get confused here (usually both come from same “sample”) • One interval/ratio dependent variable • Examples • Do male and female differ in their aggression (# aggressive acts in past week)? • Is there a difference between MN & WI in the proportion who eat cheese every day? • Null Hypothesis (Ho) • The 2 pops. are not different in terms of the dependent variable

  11. 2-SAMPLE HYPOTHESIS TESTING • Assumptions: • Random (probability) sampling • Groups are independent • Homogeneity of variance • the amount of variability in the D.V. is about equal in each of the 2 groups • The sampling distribution of the difference between means is normal in shape

  12. 2-SAMPLE HYPOTHESIS TESTING • We rarely know population S.D.s • Therefore, for 2-sample t-testing, we must use 2 sample S.D.s, corrected for bias: • “Pooled Estimate” • Focus on the t statistic: t (obtained) = (X – X) σx-x • we’re finding the difference between the two means… …and standardizing this difference with the pooled estimate

  13. 2-SAMPLE HYPOTHESIS TESTING • 2-Sample Sampling Distribution • – difference between sample means (closer sample means will have differences closer to 0) • t-test for the difference between 2 sample means: • Does our observed difference between the sample means reflects a real difference in the population means or is due to sampling error? - t critical 0 t critical ASSUMING THE NULL!

  14. Applying the 2-Sample t Formula • Example: • Research Hypothesis (H1): • Soc. majors at UMD drink more beers per month than non-soc. majors • Random sample of 205 students: • Soc majors: N = 100, mean=16, s=2.0 • Non soc. majors: N = 105, mean=15, s=2.5 • Alpha = .01 • FORMULA: t(obtained) = X1 – X2 pooled estimate

  15. Answers • Null Hypothesis (Ho): Soc. majors at UMD DO NOT drink more beers per month than non-soc. Majors • Directional  1 tailed test • tcritical for 1 tailed test with 203 df and  = .05 (use infinity) = 1.645 • tobtained = 3.13 • Reject null hypothesis: there is less than a 1% chance of obtaining a mean difference of 1 beer if the null hypothesis is true. • 3.13 standard errors separates soc and non-soc majors with respect to their average beer consumption

  16. Example 2 • Dr. Phil believes that inmates with tattoos will get in more fights than inmates without tattoos. • Tattooed inmates  N = 25, s = 1.06, mean = 1.00 • Non-Tattooed inmates  N = 37, s =.5599, mean = 0.5278 • Null hypothesis? • Directional or non? • tcritical? • Difference between means? • Significant at the .01 level?

  17. Answers • Null hypothesis: • Inmates with tattoos will NOT get in more fights than inmates without tattoos. • Use a 1 or 2-tailed test? • One-tailed test because the theory predicts that inmates with tattoos will get into MORE fights. • tcritical (60 df) = 2.390 • tobtained = 2.23

  18. Answers • Reject the null? • No, because the t(obtained) (2.23) is less than the t(critical, one-tail, df=398) (1.658) • This t value indicates there are 2.23 standard error units that separate the two mean values • Greater than a 1% chance of getting this finding under null • SPSS DEMO

  19. 2-Sample Hypothesis Testing in SPSS • Independent Samples t Test Output: • Testing the Ho that there is no difference in number of adult arrests between a sample of individuals who were abused/neglected as children and a matched control group.

  20. Interpreting SPSS Output • Difference in mean # of adult arrests between those who were abused as children & control group

  21. Interpreting SPSS Output • t statistic, with degrees of freedom

  22. Interpreting SPSS Output • “Sig. (2 tailed)” • gives the actual probability of obtaining this finding if the null is correct • a.k.a. the “p value” – p = probability • The odds are NOT ZERO (if you get .ooo, interpret as <.001)

  23. “Sig.” & Probability • Number under “Sig.” column is the exact probability of obtaining that t-value (finding that mean difference) if the null is true • When probability > alpha, we do NOT reject H0 • When probability < alpha, we DO reject H0 • As the test statistics (here, “t”) increase, they indicate larger differences between our obtained finding and what is expected under null • Therefore, as the test statistic increases, the probability associated with it decreases

  24. Example 2: Education & Ageat which First Child is Born H0: There is no relationship between whether an individual has a college degree and his or her age when their first child is born.

  25. Education & Age at which First Child is Born • What is the mean difference in age? • What is the probability that this t statistic is due to sampling error? • Do we reject H0 at the alpha = .05 level? • Do we reject H0 at the alpha = .01 level?

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