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(c) 2007 IUPUI SPEA K300 (4392) 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

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(c) 2007 IUPUI SPEA K300 (4392) 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

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(c) 2007 IUPUI SPEA K300 (4392) 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.

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(c) 2007 IUPUI SPEA K300 (4392) 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 H 0 : µ=25, H a : µ25 Critical value: 1.96 (z), 2.01(t) p-value:.1094 (z) and.1158 (t) Test statistic: 1.601

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(c) 2007 IUPUI SPEA K300 (4392) 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.

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(c) 2007 IUPUI SPEA K300 (4392) One sample t-test Compare a sample mean with a particular (hypothesized) value H 0 : µ=c, H a : µc, where c is a particular value Degrees of freedom: n-1 This is exactly what we did for past two weeks

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(c) 2007 IUPUI SPEA K300 (4392) 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)

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(c) 2007 IUPUI SPEA K300 (4392) Paired sample t-test 2 Compute d=x 1 -x 2 (order does not matter) H 0 : µ d =c, H a : µ d c, where c is a particular value (often 0) Degrees of freedom: n-1

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(c) 2007 IUPUI SPEA K300 (4392) Paired sample t-test 3: Example Example 9-13, p. 495. Cholesterol levels H 0 : µ d =0, H a : µ d 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.

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(c) 2007 IUPUI SPEA K300 (4392) 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

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(c) 2007 IUPUI SPEA K300 (4392) 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.

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(c) 2007 IUPUI SPEA K300 (4392) 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 H 0 of equal variance is not rejected, two samples have the same variance.

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(c) 2007 IUPUI SPEA K300 (4392) 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

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(c) 2007 IUPUI SPEA K300 (4392) 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.

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(c) 2007 IUPUI SPEA K300 (4392) 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 H 0 at the.01 level.

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(c) 2007 IUPUI SPEA K300 (4392) 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

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(c) 2007 IUPUI SPEA K300 (4392) Independent sample t-test: Unequal variance Approximation of degrees of freedom Not necessarily an integer Satterthwaits approximation (common) Cochran-Coxs approximation Welchs approximation

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(c) 2007 IUPUI SPEA K300 (4392) 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 H 0 of equal variance at the.05 level. Therefore, we have to use individual variances

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(c) 2007 IUPUI SPEA K300 (4392) 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.

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(c) 2007 IUPUI SPEA K300 (4392) 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

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(c) 2007 IUPUI SPEA K300 (4392) 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.

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(c) 2007 IUPUI SPEA K300 (4392) 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 H 0.

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(c) 2007 IUPUI SPEA K300 (4392) 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.

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

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