6.4 Confidence Interval for the Difference between Two Population Means
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1 6.4 Confidence Interval for the Difference between Two Poulation Means If we draw two samles from two indeendent oulations with means and and variances σ and σ, resectively, and we want to construct the confidence interval for the difference between two oulation means -, then we have the following cases : A. When the oulations are normally distributed. When the variances are known and the samle sizes are large or small, then (-α)00% C.I. for the difference - is given by: (X X) Z /. When the variances are unknown but equal and the samle sizes are small, then (-α)00% C.I. for the difference - is given by: ( XX) t ( /, ) where, (n ) (n ) 3/6/07 TA 3: Biostatistics Dr. Ahmed Jaradat, AGU
2 B. When the oulations are nonnormal ) When the variances are known and the samle sizes are large, then (-α)00% C.I. for the difference - is given by: (X X) Z / ) When the variances are unknown and the samle sizes are large, then (-α)00% C.I. for the difference - is given by: X) Z / ( X 3/6/07 TA 3: Biostatistics Dr. Ahmed Jaradat, AGU
3 Examle 6.4. P77: The researcher team interested in the difference between serum uric and acid level in a atient with and without Down s syndrome. In a large hosital for the treatment of the mentally retarded, a samle of individual with Down s yndrome yielded a mean of 4.5 mg/00 ml. In a general hosital a samle of 5 normal individual of the same age and sex were found to have a mean value of 3.4. If it is reasonable to assume that the two oulation of values are normally distributed with variances equal to and.5, find the 95% C.I. for μ - μ. olution We need to construct a 95% confidence interval The two oulations are normally distributed The variances are known σ = and σ =.5 n =, X 4.5, n =4, X 3.4 =0.05 /=0.05, -/=0.975, Z =.96 95%confidence interval for is given by (X 5. X) Z / ( ) ,.939 TA 3: Biostatistics Dr. Ahmed Jaradat, AGU 3
4 Examle Page 80 The urose of a study by Granholm et al. was to determine the effectiveness of an integrated outatient dual-diagnosis treatment rogram for mentally ill subjects. The authors were addressing the roblem of substance abuse issues among eole with severe mental disorders. A retrosective chart review was carried out on 50 consecutive atient referrals to the ubstance Abuse / Mental Illness rogram at the VA an Diego Healthcare ystem. One of the outcome variables examined was the number of inatient treatment days for sychiatric disorder during the year following the end of the rogram. Among 8 subjects with schizohrenia, the meaumber of treatment days was 4.7 with a standard deviation of 9.3. For 0 subjects with biolar disorder, the meaumber of sychiatric disorder treatment days was 8.8 with a standard deviation of.5. We wish to construct a 95 ercent confidence interval for the difference between the means of the oulations reresented by these two samles. 3/6/07 TA 3: Biostatistics Dr. Ahmed Jaradat, AGU 4
5 Examle P80: olution: The two oulations are normally distributed The variances are unknown. X 4.7 n =8,, =9.3 n =0,, =.5 X 8.8 =0.05 /=0.05, -/=0.975, t (0.975, 6) = %confidence interval for is given by: ( XX) t n ( /, ) n ( ) (.0555 ).( 0.6 ) 8 0 [-.30, 4.0] ( 8)( 93. ) ( 0)( 5. ) /6/07 TA 3: Biostatistics Dr. Ahmed Jaradat, AGU 5
6 Figure 6.4.: Flowchart for use in deciding whether the reliability factor should be z, t, or when making inferences about the difference between two oulation means. (*Use a nonarametric rocedure), Page 8. 3/6/07 TA 3: Biostatistics Dr. Ahmed Jaradat, AGU 6
7 Examle: Exercise P84/85 In a study of factors thought to be resonsible for the adverse effects of smoking on human reroduction, cadmium level determinations (nanograms er gram) were made on lacenta tissue of a samle of 4 mothers who were smokers and an indeendent random samle of 8 nonsmoking mothers. The results were as follows: Nonsmokers: 0.0, 8.4,.8, 5.0,.8, 9.8,.5, 5.4, 3.5, 9.4, 5., 9.5, 5.5, 9.8, 7.5,.8,., 5.0 mokers: 30.0, 30., 5.0, 4., 30.5, 7.8, 6.8, 4.8, 3.4, 8.5, 7.5, 4.4,.5, 0.4 Construct 95% and 90% confidence intervals. Does it aear likely that the mean cadmium level is higher among smokers thaonsmokers? Why do you reach this conclusion? olution: Assume that the two oulations are normally distributed, the variances are unknown but equal. Nonsmoker: n =8, mean=4.7, =6.0 mokers n =4, mean=0.4, =6.8 =0.05 /=0.05, -/=0.975, t (0.975, 30) = %confidence interval for is given by: (X X ) t [-0.407, -0.99] n ( α/,n n ) n 90% C.I is given by: [-9.6, -.786] Yes, it does aear likely mean cadmium level is higher among smokers thaonsmokers because the 90 and 95% confident intervals do not contain 0. ( 8 )( 60. ) ( 4 )( 68. ) ( ) ( 043. ).( ) 8 4 3/6/07 TA 3: Biostatistics Dr. Ahmed Jaradat, AGU 7
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