When choosing a sample size, we must consider the following issues:• Objectives: What population parameters we want to estimate/test hypothesis• Sampling/research design is selected• Degree of accuracy required for the study• Spread/variation (variability) of the population• Response rate, practicality: how hard is it to collect data• Time and money available
1)Sample size for Simple Random SamplingTo estimate mean 2 2 Z N n = 2 2 2 Z ( N 1) E Z2 2 n E2
Sample size for Simple Random SamplingTo estimate proportion 2 Z NP (1 P ) n = 2 2 Z P (1 P ) ( N 1) E n 2P 1 P Z ( ) n E 2
1,628 (Pilot survey) z 2 NP (1 P) 5% z 2 P (1 P ) NE 2 % n = )2 (1.645 (1,6280.2)( .8) )( 0 (1.645 (0.2)( .8) (1,6280.052 )2 0 )( ) n = = 156.53 157 5%90%
? z 2 P (1 P ) 95% E 2 n = 2(P 1) 1 (1.96 )( ) P 2 2 P = ½=0 (0.052 )n= = 384.16 385
Equal sample L L N2S2 hhn = h1 L N2E2 NhS2 h1 hnh n L
Proportional Allocation L N NhS2 h h1 n = L N2E2 NhS2 h h1 Nh nh L Nh h1
2) Sample size determination forhypothesis testing2.1 Sample size determination for thetest of one proportion
Example In a particular province theproportion of pregnant women provided withprenatal care in the first trimester of pregnancyis estimated to be 40% by the provincialdepartment of health. Health officials inanother province are interested in comparingtheir success at providing prenatal care withthese figures. How many women should besampled to test the hypothesis that the coveragerate in the second province is % against thealternative that it is not %? The investigatorswish to detect a difference of % with thepower of the test equal at % and at
P : coverage rateHo: P = . Ha: P . ( . orMINITAB can be used to assist in thissample size determination byselectingStat > Power and sample size >proportion.
If alternative values of p is equal to.45, a sample size of 1022 would beneeded.If alternative values of p is equal to. , a sample size of would beneeded.We choose the large sample size, thus asample size of 1022 is needed for thestudy.
2.2 Sample size determination for thetest of two proportionsTwo-sided test (Z 2pq Z p2q2 p1q1)2 n = 2 (p2 p1)2
Example 5 It is believed that the proportionof patients who develop complications afterundergoing one type of surgery is % whilethe proportion of patients who developcomplications after a second type of surgeryis %. How large should the sample size bein each of the two groups of patients if aninvestigator wishes to detect, with a powerof %, whether the second procedure has acomplication rate significantly higher thanthe first at the % level of significance?
Use MINITAB, click Stat > Powerand sample size > proportion.You would complete the dialog box.You want to test one-sided test, clickon the options button and choose lessthan
Power and Sample SizeTest for Two ProportionsTesting proportion = proportion (versus <)Calculating power for proportionAlpha = Sample Target ActualProportion Size Power PowerA sample size of would be needed in eachgroup.
2.3 Sample size determination for the tesof one meanTwo-sided test 2 2 (Z Z ) n 2 ( 0 1 )2
Example Consider the cholesterolstudy. Suppose that the null mean is mg% /ml, the alternative mean is mg%/ml, the standard deviation is , and we wish to conduct asignificance test for one-sided test atthe % level with a power of %.How large should the sample size be?
MINITAB> click Stat > Power andsample size > sample Z.You want to test one-sided test, clickon the options button and choosegreater than
-Sample Z TestTesting mean = null (versus > nulCalculating power for mean = nullAlpha = . Sigma = Sample Target ActualDifference Size Power Power Thus, 96 people are needed. To achieve a power of 90% using a 5% significance level
2.4 Sample size determination for thetest of two meansTwo-sided test (Z Z )2( 1 2 2)2 2 2 n = ( 2 1)2
Example Consider the blood pressure studyfor drug A users and non-drug A users as apilot study conducted to obtain parameterestimates to plan for a larger study. We wishto test the hypothesis : = versus : .Determine the appropriate sample size forthe large study using a two–sided test with asignificance level of . and a power ofIn the pilot study, we obtained = . ,S = . = . ,S
In the pilot study, we obtained= . ,S = . = . ,Sn=( - We would require a sample size of 152 people in each group