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Student’s t test This test was invented by a statistician working for the brewer Guinness. He was called WS Gosset (1867-1937), but preferred to keep anonymous so wrote under the name “Student”. This test was invented by a statistician working for the brewer Guinness. He was called WS Gosset (1867-1937), but preferred to keep anonymous so wrote under the name “Student”.

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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-Statistic When the sampled population is normally distributed, the t statistic is Student t distributed with n-1 degrees of freedom. 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 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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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 for the whole stock could be 24 mg, a value observed in some previous work?. 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 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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-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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Given below are the 24 hrs total energy expenditure (MJ/day) in groups of lean and obese women. Examine whether the obese women’s 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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Two sample t-test Difference between means Sample size Variability of data t-test t t t t + +

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T-test for difference in means Null Hypothesis Null Hypothesis Obese women’s mean energy expenditure is equal to the lean women’s energy expenditure. Obese women’s mean energy expenditure is equal to the lean women’s energy expenditure. Test statistics : 1, 2 - means of sample 1 and sample 2 1, 2 – sd of sample 1 and sample 2 n1, n2 – number of study subjects in sample 1 and sample 2 t(n1+n2-2)

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T-test for difference in means Data Summary lean Obese lean Obese N 13 9 8.10 10.30 8.10 10.30 S 1.38 1.25 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 tab t 9+13-2 =20 df = t 0.05,20 =2.086

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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 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 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 Approach: test the effectiveness of program by testing significance of D Alternative hypothesis: program is effective → scores after program will be higher than scores before program → average D will be greater than zero 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 = 180∑D 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 crit = 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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