Hypothesis Testing and Comparison of Two Populations Dr. Burton

If the heights of male teenagers are normally distributed with a mean of 60 inches and standard deviation of 10, And the sample size was 25, what percentage of boy’s heights in inches would be: Between 57 and 63 Lass than or larger

7.2a 60 0 Height Z 5763 % Z = x -  s / n / / 25 Z= -1.5 =.4332 Z= 1.5 = = 86.8%

7.2b 60 0 Height Z 58 % Z = x -  s / n / 25 Z = -1.0 = = 16%

7.2c Height Z 61 % Z = x -  s / n / 25 Z = =.3085 = 30.9%= 0.50

Hypothesis Testing Hypothesis: A statement of belief… Null Hypothesis, H 0 : …there is no difference between the population mean  and the hypothesized value  0. Alternative Hypothesis, H a : …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 H 0 TrueH 0 False Accept H 0 Reject H 0 Type I Error Type II Error Correct results Correct results Truth Test result  1 -   1 -  H 0 True = statistically insignificant H 0 False = statistically significant Accept H 0 = statistically insignificant Reject H 0 = statistically significant Differences ab cd

SE Probability Distribution for a two-tailed test SE Magnitude of (X E – X C ) 1.96 SE X E < X C X E > X C  =

SE Probability Distribution for a one-tailed test SE Magnitude of (X E – X C ) SE X E < X C X E > X C  = 0.05

Box t = A Distance between the means Variation around the means

Box t = A B Distance between the means Variation around the means

Box t = A B C Distance between the means Variation around the means

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 x A - x B - 0 t = Sp [ ( 1/N A ) + ( 1/N B ) ] d f = N A + N B - 2

Independent Sample Means Sample A(A – Mean) Mean =  A 2 = 2913 N = 7 (A – Mean) 2 = Var = s = 3.48 Sample B(B – Mean) Mean =  B 2 = 4656 N = 6 (B – Mean) 2 = Var = s = 5.89

Standard error of the difference between the means (SED) SED of  E -  C = s A 2 Estimate of the s B 2 N AN A N BN B + SED of x E - x C =  A 2  B 2 N AN A N BN B + Theoretical Population Sample

Pooled estimate of the SED (SEDp) 1 Estimate of the 1 N AN A N BN B + SEDp of x A - x B = Sp s 2 (n A -1) + s 2 (n B – 1) Sp= n A + n B ( 6 ) ( 5 ) Sp= = = 4.73

t-Test (Two Tailed) d f = N E + N C - 2 = 11 x A - x B - 0 t = Sp [ ( 1/N A ) + ( 1/N B ) ] = 4.73 ( 1/7 ) + ( 1/6) = 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.

Paired t-test t paired = t p = d - 0 Standard error of d = d - 0 S d 2 N df = N - 1 d =  D/N  d 2 =  D 2 – (  D) 2 / N S d 2 =  d 2 / N - 1

Pre/post attitude assessment StudentBeforeAfterDifferenceD squared Total  D = 37  D 2 = 511

Pre/post attitude assessment StudentBeforeAfterDifference D squared Total t paired = t p = d - 0 Standard error of d = d - 0 S d 2 N d =  D/N N = 10  d 2 =  D 2 – (  D) 2 / N S d 2 =  d 2 / N - 1 = 37/10 = 3.7 = /10 = = / 10 – 1 = = 3.7 / = = 3.7 / / 10 = 3.7 / df = N – 1 = > 1.833

Probabilities of Type I and Type II errors H 0 TrueH 0 False Accept H 0 Reject H 0 Type I Error Type II Error Correct results Correct results Truth Test result  1 -   1 -  H 0 True = statistically insignificant H 0 False = statistically significant Accept H 0 = statistically insignificant Reject H 0 = statistically significant Differences

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 PresentAbsent Present Absent Disease status Risk Factor Status ab cd a + b c + d a + c b + d a+b+c+d Total

Standard 2 X 2 table Sensitivity = a/a+c Specificity = d/b+d PresentAbsent Present Absent Disease status Risk Factor Status ab cd a + b c + d a + c b + d a+b+c+d Total

Diabetic Screening Program 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.) DiabeticNondiabetic >125mg/100ml <125mg/100ml Disease status Risk Factor Status Total