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## HIM 3200 Chapters 8 & 9 Hypothesis Testing and T-Tests

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**HIM 3200Chapters 8 & 9Hypothesis Testing and T-Tests**Dr. Burton**6.5**40% 40% 5% 5% 5% 5% BP Z 95 101 -1.65 -1.28 120 0 • 145 • 1.28 1.65**6.12**.45 .05 x Z 500 0 • 665 • 1.65 x = 500 + (1.65)100 = 665 5% of 1,000,000 = 50,000**6.12**.10 .30 .10 x Z 500 0 525 .25 • 628 • 1.28 x = 500 + (.25)100 = 525 x = 500 + (1.28)100 = 628 30% of 1,000,000 = 300,000**6.12**.45 .45 .05 .05 335 -1.65 • 665 • 1.65 x Z 500 0 x = 500 + (1.65)100 = 665 x = 500 + (-1.65)100 = 335 90% betweem 335 and 665**6.12**.4332 .50 .0668 350 -1.50 x Z 500 0 • 628 • 1.28 Z = 350 – 500/100 = -1.50 .0668 or about 7% would score less than 350**6.19** = 200 = 25 < 165 = 165 -200/25 = -1.4 = .5 - .4192 = .0808 1 - .0808 = .9192**7.2a**% Height Z 60 0 57 63 57 - 60 Z= -1.5 = .4332 10 / 25 .8664 = 86.8% x - t = s / n 63 - 60 Z= 1.5 = .4332 10 / 25**7.2b**% 58 -1.0 60 0 Height Z x - 58 - 60 t = Z = -1.0 = .5000 - .3413 s / n 10 / 25 .1587 = 16%**7.2c**% 61 60 0 0.5 Height Z x - 61 - 60 t = Z = 0.50 s / n 10 / 25 = 0.50 - .1915 = .3085 = 30.9%**7.16a**% 2400 0 2.0 2500 Height Z x - 2500 - 2400 t = Z = 2.0 400 / 64 s / n = 0.50 - .4772 = .0228 = 2.3%**7.16b**% Height Z 2400 0 2300 2500 2300 - 2400 Z= -2.0 = .4772 400 / 64 x - .9544 = 95.4% t = s / n 2500 - 2400 Z= 2.0 = .4772 400 / 64**7.16c**% 2350 2400 0 Height Z x - 2350 - 2400 t = Z = -1.0 400 / 64 s / n = 0.50 - .3413 = .1587 = 16%**Hypothesis Testing**Hypothesis: A statement of belief… Null Hypothesis, H0: …there is no difference between the population mean and the hypothesized value 0. Alternative Hypothesis, Ha: …reject the null hypothesis and accept that there is a difference between the population mean and the hypothesized value 0.**Probabilities of Type I and Type II errors**Truth H0 True H0 False a b Type II Error Correct results Accept H0 Test result 1 - c d Type I Error Correct results Reject H0 1 - Differences H0 True = statistically insignificant H0 False = statistically significant Accept H0 = statistically insignificant Reject H0 = statistically significant http://en.wikipedia.org/wiki/False_positive** = 0.05**0.025 0.025 Probability Distribution for a two-tailed test 1.96 SE -3 -2 -1 1 2 3 0 SE SE Magnitude of (XE – XC) XE < XC XE > XC**Probability Distribution for a one-tailed test** = 0.05 1.645 SE -3 -2 -1 0 1 2 3 SE SE Magnitude of (XE – XC) XE < XC XE > XC**Box 10 - 5t =**Distance between the means Variation around the means A**Box 10 - 5t =**Distance between the means Variation around the means A B**Box 10 - 5t =**Distance between the means Variation around the means A B C**t-Tests**• Students t-test is used if: • two samples come from two different groups. • e.g. A group of students and a group of professors • Paired t-test is used if: • two samples from the sample group. • e.g. a pre and post test on the same group of subjects.**One-Tailed vs. Two Tailed Tests**• The Key Question: “Am I interested in the deviation from the mean of the sample from the mean of the population in one or both directions.” • If you want to determine whether one mean is significantly from the other, perform a two-tailed test. • If you want to determine whether one mean is significantly larger, or significantly smaller, perform a one-tailed test.**t-Test(Two Tailed)Independent Sample means**xA - xB - 0 t = [ ( 1/NA ) + ( 1/NB) ] Sp d f = N A + N B - 2**Sample A (A – Mean)2**26 34.34 24 14.90 18 4.58 17 9.86 18 4.58 20 .02 18 4.58 Mean = 20.14 A2 = 2913 N = 7 (A – Mean)2 = 72.86 Var = 12.14 s = 3.48 Sample B (B – Mean)2 38 113.85 26 1.77 24 11.09 24 11.09 30 7.13 22 28.41 Mean = 27.33 B2 = 4656 N = 6 (B – Mean)2 = 173.34 Var = 34.67 s = 5.89 Independent Sample Means**Standard error of the difference between the means (SED)**Theoretical A 2 B 2 SED of E - C = + Population NB NA Estimate of the s A 2 s B 2 Sample SED of xE - xC = + NB NA**Pooled estimate of the SED (SEDp)**Estimate of the 1 1 SEDp of xA - xB = Sp + NB NA s2(nA-1) + s2 (nB – 1) Sp = nA + n B - 2 +34.67(5) 12.14 (6) Sp = = 22.38 = 4.73 7 + 6 - 2**t-Test(Two Tailed)**xA - xB - 0 t = [ ( 1/NA ) + ( 1/NB) ] Sp 20.14 - 27.33 - 0 = = -2.73 ( 1/7 ) + ( 1/6) 4.73 d f = N E + N C - 2 = 11 Critical Value 95% = 2.201**One-tailed and two-tailed t-tests**• A two-tailed test is generally recommended because differences in either direction need to be known.**d = D/N**= d 2 / N - 1 S d2 d 2 = D 2 – ( D) 2 / N Pairedt-test d - 0 d - 0 = ------------- t paired = t p = S d2 Standard error of d N df = N - 1**= 3.7 / 41.5667 / 10**= 3.7 / 4. 15667 df = N – 1 = 9 0.05 > 1.833 Pre/post attitude assessment N = 10 Student Before After Difference D squared Total 171 208 37 511 d - 0 d - 0 = ------------- t paired = t p = S d2 Standard error of d N d = D/N = 37/10 = 3.7 d 2 = D 2 – ( D) 2 / N = 511 - 1369/10 = 374.1 = 3.7 / 2.0387 = d 2 / N - 1 S d2 = 1.815 = 374.1 / 10 – 1 = 41.5667**Probabilities of Type I and Type II errors**Truth H0 True H0 False Type II Error Correct results Accept H0 Test result 1 - Type I Error Correct results Reject H0 1 - Differences H0 True = statistically insignificant H0 False = statistically significant Accept H0 = statistically insignificant Reject H0 = statistically significant**Disease status**Present Absent Total Present a b a + b Risk Factor Status Absent c d c + d a + c b + d a+b+c+d Total Standard 2 X 2 table a = subjects with both the risk factor and the disease b = subjects with the risk factor but not the disease c = subjects with the disease but not the risk factor d = subjects with neither the risk factor nor the disease a + b = all subjects with the risk factor c + d = all subjects without the risk factor a + c = all subjects with the disease b + d = all subjects without the disease a + b + c + d = all study subjects**Disease status**Present Absent Total Present a b a + b Risk Factor Status Absent c d c + d a + c b + d a+b+c+d Total Standard 2 X 2 table Sensitivity = a/a+c Specificity = d/b+d**Diabetic Screening Program**Disease status Diabetic Nondiabetic Total >125mg/100ml 5 13 18 Risk Factor Status Sensitivity = a/a+c = 100 X 5/6 = 83.3% (16.7% false neg.) Specificity = d/b+d = 100 X 81/94 = 86.2%(13.8% false pos.) <125mg/100ml 1 81 82 6 94 100 Total