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The t-distribution William Gosset lived from 1876 to 1937 Gosset invented the t -test to handle small samples for quality control in brewing. He wrote under the name "Student".

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t test-origin Founder WS Gosset Wrote under the pseudonym Student Mostly worked in tea (t) time ? Hence known as Student's t test. Preferable when the n < 60 Certainly if n < 30

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Types One sample compare with population Unpaired compare with control Paired same subjects: pre-post

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T-test 1.Test for single mean Whether the sample mean is equal to the predefined population mean ?. Test for difference in means 2. Test for difference in means Whether the CD4 level of patients taking treatment A is equal to CD4 level of patients taking treatment B ? Test for paired observation 3. Test for paired observation Whether the treatment conferred any significant benefit ?

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Test direction One tailed t test Two tailed test

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Developing the Pooled-Variance t Test (Part 1) Setting Up the Hypothesis: H 0 : 1 - 2 = 0 H 1 : 1 - 2 0 H 0 : 1 = 2 H 1 : 1 2 OR Two Tail

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Developing the Pooled-Variance t Test (Part 1) Setting Up the Hypothesis: H 0 : 1 2 H 1 : 1 > 2 H 0 : 1 - 2 = 0 H 1 : 1 - 2 0 H 0 : 1 = 2 H 1 : 1 2 H 0 : 1 - 2 0 H 1 : 1 - 2 > 0 OR Right Tail Two Tail

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Developing the Pooled-Variance t Test (Part 1) Setting Up the Hypothesis: H 0 : 1 2 H 1 : 1 > 2 H 0 : 1 - 2 = 0 H 1 : 1 - 2 0 H 0 : 1 = 2 H 1 : 1 2 H 0 : 1 2 H 0 : 1 - 2 0 H 1 : 1 - 2 > 0 H 0 : 1 - 2 H 1 : 1 - 2 < 0 OR Left Tail Right Tail Two Tail H 1 : 1 < 2

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Mean systolic BP in nephritis is significantly higher than of normal person 100 110 120 130 140 0.05

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Mean systolic BP in nephritis is significantly different from that of normal person Mean systolic BP in nephritis is significantly different from that of normal person 0.025 100 110 120 130 140

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Statistical Analysis control group mean treatment group mean Is there a difference?

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What does difference mean? medium variability high variability low variability The mean difference is the same for all three cases

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What does difference mean? medium variability high variability low variability Which one shows the greatest difference?

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What does difference mean? a statistical difference is a function of the difference between means relative to the variability a small difference between means with large variability could be due to chance like a signal-to-noise ratio low variability Which one shows the greatest difference?

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So we estimate low variability signal noise difference between group means variability of groups = X T - X C SE(X T - X C ) = = t-value __ __

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Two sample t-test Difference between means Sample size Variability of data t-test t t t t + +

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Probability - p With t we check the probability Reject or do not reject Null hypothesis You reject if p < 0.05 or still less Difference between means (groups) is more & more significant if p is less & less

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-1.96 0 Area =.025 Area =.005 Z -2.575 Area =.025 Area =.005 1.96 2.575 Determining the p-Value

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.9 5 t 0 f(t) -1.961.96.025 red area = rejection region for 2-sided test

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Assumptions Normal distribution Equal variance Random sampling

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t-Statistic When the sampled population is normally distributed, the t statistic is Student t distributed with n-1 degrees of freedom.

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T- test for single mean The following are the weight (mg) of each of 20 rats drawn at random from a large stock. Is it likely that the mean weight of these 20 rats are similar to the mean weight ( 24 mg) of the whole stock ? 9 18 21 26 14 18 22 27 15 19 22 29 15 19 24 30 16 20 24 32

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Steps for test for single mean 1.Questioned to be answered Is the Mean weight of the sample of 20 rats is 24 mg? N=20, =21.0 mg, sd=5.91, =24.0 mg 2. Null Hypothesis The mean weight of rats is 24 mg. That is, The sample mean is equal to population mean. 3. Test statistics --- t (n-1) df 4. Comparison with theoretical value if tab t (n-1) < cal t (n-1) reject Ho, if tab t (n-1) > cal t (n-1) accept Ho, 5. Inference

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t –test for single mean Test statistics n=20, =21.0 mg, sd=5.91, =24.0 mg n=20, =21.0 mg, sd=5.91, =24.0 mg t = t.05, 19 = 2.093 Accept H 0 if t = 2.093 Inference : There is no evidence that the sample is taken from the population with mean weight of 24 gm There is no evidence that the sample is taken from the population with mean weight of 24 gm

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Given below are the 24 hrs total energy expenditure (MJ/day) in groups of lean and obese women. Examine whether the obese womens mean energy expenditure is significantly higher ?. Lean Lean 6.1 7.0 7.5 7.5 5.5 7.6 7.9 8.1 8.1 8.1 8.4 10.2 10.9 T-test for difference in means Obese Obese 8.8 9.2 9.2 8.8 9.2 9.2 9.7 9.7 10.0 9.7 9.7 10.0 11.5 11.8 12.8 11.5 11.8 12.8

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Null Hypothesis Obese womens mean energy expenditure is equal to the lean womens energy expenditure. Obese womens mean energy expenditure is equal to the lean womens energy expenditure. Data Summary lean Obese lean Obese N 13 9 8.10 10.30 8.10 10.30 S 1.38 1.25

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H 0 : 1 - 2 = 0 ( 1 = 2 ) H 1 : 1 - 2 0 ( 1 2 ) = 0.05 = 0.05 df = 13 + 9 - 2 = 20 Critical Value(s): t 02.086-2.086.025 Reject H 0 0.025 Solution

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t XX S nSnS nn dfnn P 1212 211 2 22 2 12 1 2 11 11 2 Hypothesized Difference (usually zero when testing for equal means) Compute the Test Statistic: ()) ( ()() ()() n1n1 n2n2 __ Calculating the Test Statistic:

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Developing the Pooled-Variance t Test Calculate the Pooled Sample Variances as an Estimate of the Common Populations Variance: = Pooled-Variance = Variance of Sample 1 = Variance of sample 2 = Size of Sample 1 = Size of Sample 2

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S nSnS nn P 2 11 2 22 2 12 22 11 11 131 1 3891125 13 1 91 1 765... (( (( ( ( ( ( ) ) ) ) ) )) ) First, estimate the common variance as a weighted average of the two sample variances using the degrees of freedom as weights

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t XX S nn P 1212 2 12 10.3 0 176 1 + 13 1919 3.82 8.1. Calculating the Test Statistic: ( (())) tab t 9+13-2 =20 dff = t 0.05,20 =2.086

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T-test for difference in means Inference : The cal t (3.82) is higher than tab t at 0.05, 20. ie 2.086. This implies that there is a evidence that the mean energy expenditure in obese group is significantly (p<0.05) higher than that of lean group

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Example Suppose we want to test the effectiveness of a program designed to increase scores on the quantitative section of the Graduate Record Exam (GRE). We test the program on a group of 8 students. Prior to entering the program, each student takes a practice quantitative GRE; after completing the program, each student takes another practice exam. Based on their performance, was the program effective?

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Each subject contributes 2 scores: repeated measures design StudentBefore ProgramAfter Program 1520555 2490510 3600585 4620645 5580630 6560550 7610645 8480520

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Can represent each student with a single score: the difference (D) between the scores Student Before ProgramAfter Program D 152055535 249051020 3600585-15 462064525 558063050 6560550-10 761064535 848052040

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Approach: test the effectiveness of program by testing significance of D Null hypothesis: There is no difference in the scores of before and after program Alternative hypothesis: program is effective scores after program will be higher than scores before program average D will be greater than zero H 0 : µ D = 0 H 1 : µ D > 0

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Student Before Program After ProgramDD2D2 1520555351225 249051020400 3600585-15225 462064525625 5580630502500 6560550-10100 7610645351225 8480520401600 D = 180D 2 = 7900 So, need to know D and D 2 :

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Recall that for single samples: For related samples: where: and

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Standard deviation of D: Mean of D: Standard error:

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Under H 0, µ D = 0, so: From Table B.2: for α = 0.05, one-tailed, with df = 7, t critical = 1.895 2.714 > 1.895 reject H 0 The program is effective.

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t-Value t is a measure of: How difficult is it to believe the null hypothesis? High t Difficult to believe the null hypothesis - accept that there is a real difference. Low t Easy to believe the null hypothesis - have not proved any difference.

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In Conclusion ! Student s t-test will be used: Student s t-test will be used: --- When Sample size is small --- When Sample size is small and for the following situations: and for the following situations: (1) to compare the single sample mean (1) to compare the single sample mean with the population mean with the population mean (2) to compare the sample means of (2) to compare the sample means of two indpendent samples two indpendent samples (3) to compare the sample means of paired samples (3) to compare the sample means of paired samples

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-1 Introduction to Statistics Chapter 10 Estimation and Hypothesis.

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-1 Introduction to Statistics Chapter 10 Estimation and Hypothesis.

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