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Outline. Type of t-test Z-test versus t-test Assumptions of the t-test One sample t-test Paired sample t-test F-test for equal variance Independent sample t-test: equal variance Independent sample t-test: unequal variance Comparing proportions. Type of the T-test.

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  1. Outline • Type of t-test • Z-test versus t-test • Assumptions of the t-test • One sample t-test • Paired sample t-test • F-test for equal variance • Independent sample t-test: equal variance • Independent sample t-test: unequal variance • Comparing proportions (c) 2007 IUPUI SPEA K300 (4392)

  2. Type of the T-test • One-sample t-test compares one sample mean with a hypothesized value • Paired sample t-test (dependent sample) compares the means of two dependent variables • Independent sample t-test compares the means of two independent variables • Equal variance • Unequal variance (c) 2007 IUPUI SPEA K300 (4392)

  3. Z-test and T-test • When σ is known, not likely in most cases, conduct the z-test • When σ is not known, conduct the t-test • What if N is large (large sample)? The z-test and t-test produce almost the same result. Therefore, t-test is more useful and practical. • Most software packages support the t-test with p-values reported. (c) 2007 IUPUI SPEA K300 (4392)

  4. Comparison of the z-test and t-test • Q 3, p.410, Revenue of large business • N=50, xbar=31.5, s=28.7, α=.05 • H0: µ=25, Ha: µ≠25 • Critical value: 1.96 (z), 2.01(t) • p-value: .1094 (z) and .1158 (t) • Test statistic: 1.601 (c) 2007 IUPUI SPEA K300 (4392)

  5. Assumptions of the T-test • Normality, otherwise comparison is not valid. Nonparametric methods are used. • Independence of (between) samples, otherwise the paired t-test is used. • Equal Variance, otherwise the pooled variance is not valid and approximation of degrees of freedom is needed. (c) 2007 IUPUI SPEA K300 (4392)

  6. One sample t-test • Compare a sample mean with a particular (hypothesized) value • H0: µ=c, Ha: µ≠c, where c is a particular value • Degrees of freedom: n-1 • This is exactly what we did for past two weeks (c) 2007 IUPUI SPEA K300 (4392)

  7. Paired sample t-test 1 • Compare two paired (matched) samples. • Ex. Compare means of pre- and post- scores given a treatment. We want to know the effect of treatment. • Ex. Compare means of midterm and final exam of K300. • Each subject has data points (pre- and post, or midterm and final) (c) 2007 IUPUI SPEA K300 (4392)

  8. Paired sample t-test 2 • Compute d=x1-x2 (order does not matter) • H0: µd=c, Ha: µd≠c, where c is a particular value (often 0) • Degrees of freedom: n-1 (c) 2007 IUPUI SPEA K300 (4392)

  9. Paired sample t-test 3: Example • Example 9-13, p. 495. Cholesterol levels • H0: µd=0, Ha: µd≠0 • N=5, dbar=16.7, std err=25.4, • Test size=.01, df=4, critical value=2.015 • Test statistic is 1.61, which is smaller than CV • Do not reject the null hypothesis. 1.61 is likely when the null hypothesis is true. (c) 2007 IUPUI SPEA K300 (4392)

  10. Independent sample t-test • Compare two independent samples • Ex. Compare means of personal income between Indiana and Ohio • Ex. Compare means of GPA between SPEA and Kelley School • Each variable include different subjects that are not related at all (c) 2007 IUPUI SPEA K300 (4392)

  11. How to get standard error? • If variances of two sample are equal, use the pooled variance. • Otherwise, you have to use individual variance to get the standard error of the mean difference (µ1-µ2) • How do we know two variances are equal? • (Folded form) F test is the answer. (c) 2007 IUPUI SPEA K300 (4392)

  12. F-test for equal variance • Compute variances of two samples • Conduct the F-test as follows. • Larger variance should be the numerator so that F is always greater than or equal to 1. • Look up the F distribution table with two degrees of freedom. • If H0 of equal variance is not rejected, two samples have the same variance. (c) 2007 IUPUI SPEA K300 (4392)

  13. Independent sample t-test: Equal variance • Compare means of two independent samples that have the same variance • The null hypothesis is µ1-µ2=c (often 0) • Degrees of freedom is n1+n2-2 (c) 2007 IUPUI SPEA K300 (4392)

  14. Independent sample t-test: Equal variance • Example 9-10, p.484 • X1bar=$26,800, s1=$600, n1=10 • X2bar=$25,400, s2=$450, n2=8 • F-test: F 1.78 is smaller than CV 4.82; do not reject the null hypothesis of equal variance at the .01 level. • Therefore, we can use the pooled variance. (c) 2007 IUPUI SPEA K300 (4392)

  15. Independent sample t-test: Equal variance • X1bar=$26,800, s1=$600, n1=10 • X2bar=$25,400, s2=$450, n2=8 • Since 5.47>2.58 and p-value <.01, reject the H0 at the .01 level. (c) 2007 IUPUI SPEA K300 (4392)

  16. Independent sample t-test: Unequal variance • Compare means of two independent samples that have different variances (if the null hypothesis of the F-test is rejected) • The null hypothesis is µ1-µ2=c (often 0) • Individual variances need to be used • Degrees of freedom is approximated; not necessarily an integer (c) 2007 IUPUI SPEA K300 (4392)

  17. Independent sample t-test: Unequal variance • Approximation of degrees of freedom • Not necessarily an integer • Satterthwait’s approximation (common) • Cochran-Cox’s approximation • Welch’s approximation (c) 2007 IUPUI SPEA K300 (4392)

  18. Independent sample t-test: Unequal variance • Example 9-9, p.483 • X1bar=191, s1=38, n1=8 • X2bar=199, s2=12, n2=10 • F-test: F 10.03 (7, 9) is larger than CV 4.20, indicating unequal variances. Reject H0 of equal variance at the .05 level. • Therefore, we have to use individual variances (c) 2007 IUPUI SPEA K300 (4392)

  19. Independent sample t-test: Unequal variance • Example 9-9, p.483 • X1bar=191, s1=38, n1=8 • X2bar=199, s2=12, n2=10 • Test statistics |-.57| is small. • Textbook uses CV 2.365 for 7 (8-1) degrees of freedom and does not reject the null hypothesis • However, we need the approximation of degrees of freedom to get more reliable df. (c) 2007 IUPUI SPEA K300 (4392)

  20. Independent sample t-test: Unequal variance • Example 9-9, p.483 • X1bar=191, s1=38, n1=8 • X2bar=199, s2=12, n2=10 • -.57~t(8.1213), CV is about 2.306. Df is not 16 but 8 • Therefore, do not reject the null hypothesis (c) 2007 IUPUI SPEA K300 (4392)

  21. Comparing proportions 1 • Compare proportions of two binary variables • The test statistic is normally distributed (not t distribution) • Think about normal approximation of a binomial distribution when N is large. (c) 2007 IUPUI SPEA K300 (4392)

  22. Comparing proportions 2 • Example 9-15, p. 505, Vaccination rates • N1=34, n2=24, alpha=.05 • P1hat=.35=12/34, p2hat=.71=17/24 • P1pooled=(12+17)/(34+24)=.5 • Z |-2.7| is larger than CV 1.96, reject H0. (c) 2007 IUPUI SPEA K300 (4392)

  23. Comparing proportions 3 • Proportions are represented by binary variables that have either 0 or 1. • The mean of a binary variable is a proportion • What if we conduct two independent sample t-test? • If N is large, z-test and t-test produce the same result. (c) 2007 IUPUI SPEA K300 (4392)

  24. Summary of Comparing Means (c) 2007 IUPUI SPEA K300 (4392)

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