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# One Sample T Test

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### One Sample T Test

1. 1. Week 8: ● Hypothesis Testing with One-Sample t-test 1
2. 2. One Sample t-Test  The shortcoming of the z test is that it requires more information than is usually available.  To do a z-test, we need to know the value of population standard deviation to be able to compute standard error. But it is rarely known. 2
3. 3. When the population variance is unknown, we use one sample t-test What if σ is unknown? •Can’t compute z test statistics (z score) x−µ Population standard σ deviation must be known Z= n • Can compute t statistic t= x−µ Sample standard deviation s must be known n 3
4. 4. Z-test or t-test?  Do children with low self-esteem show significantly more depression than children in general? The average depression score for the general population is 90, with a deviation of 14.  Do children with low-self esteem take on a leadership role in a group project significantly less than two times a semester?  Is the average GPA of freshman admitted to OSU significantly higher or lower than 3.0 in 2007? 4
5. 5. Hypothesis testing with a one-sample t-test  State the hypotheses  Ho: μ = hypothesized value  H1: μ ≠ hypothesized value  Set the criteria for rejecting Ho  Alpha level  Critical t value 5
6. 6. Determining the criteria for rejecting the Ho  The t- value is used just like a z-statistic: if the value of t exceeds some threshold or critical valued, tα , then an effect is detected (i.e. the null hypothesis of no difference is rejected) 6
7. 7. Table C.3 (p. 638 in text) 7
8. 8. Degrees of freedom for One Sample t-test  Degrees freedom (d.f.) is computed as the one less than the sample size (the denominator of the standard deviation): df = n - 1 8
9. 9. Finding Critical Values The t-distribution for df = 3, 2-tailed α = 0.10 9
10. 10. Finding Critical Values The t-distribution for df =15, 2-tailed α = 0.05 10
11. 11. Finding Critical Values The t-distribution for df =15, one-tailed α = 0.05 11
12. 12. Hypothesis testing with a one-sample t-test  Compute the test statistic (t-statistic) x−µ t= s n  Make statistical decision and draw conclusion  t ≥ t critical value, reject null hypothesis  t < t critical value, fail to reject null hypothesis 12
13. 13. Decision about Ho 13
14. 14. One Sample t-test Example  You are conducting an experiment to see if a given therapy works to reduce test anxiety in a sample of college students. A standard measure of test anxiety is known to produce a µ = 20. In the sample you draw of 81 the mean = 18 with s = 9.  Use an alpha level of .05 14
15. 15. Write hypotheses  Ho: The average test anxiety in the sample of college students will not be statistically significantly different than 20.  Ho: μ = 20  H1 = The average test anxiety in the sample of college students will be statistically significantly lower than 20.  H1: μ < 20 15
16. 16. Set Criterion  αone-tailed = .05  df = n – 1  81 – 1 = 80  t critical value = - 1.671  Use closest and most conservative value if exact value not given 16
17. 17. Compute test statistic (t statistic) t = 18 – 20 = -2 = -2 9 / √81 1 17
18. 18. Compare to criteria and make decision  t-statistic of -2 exceeds your critical value of -1.671.  Reject the null hypothesis and conclude that average test anxiety in the sample of college students is statistically significantly lower than 20, t = -2.0, p < .05. 18