29. 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
30. 1)
Sample size for Simple Random Sampling
To estimate mean
2 2
Z N
n =
2 2 2
Z ( N 1) E
Z2 2
n
E2
31. Sample size for Simple Random Sampling
To 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
32. 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%
33. ? 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
37. 2) Sample size determination for
hypothesis testing
2.1 Sample size determination for the
test of one proportion
38. Example In a particular province the
proportion of pregnant women provided with
prenatal care in the first trimester of pregnancy
is estimated to be 40% by the provincial
department of health. Health officials in
another province are interested in comparing
their success at providing prenatal care with
these figures. How many women should be
sampled to test the hypothesis that the coverage
rate in the second province is % against the
alternative that it is not %? The investigators
wish to detect a difference of % with the
power of the test equal at % and at
39. P : coverage rate
Ho: P = . Ha: P . ( . or
MINITAB can be used to assist in this
sample size determination by
selecting
Stat > Power and sample size >
proportion.
40.
41. If alternative values of p is equal to
.45, a sample size of 1022 would be
needed.
If alternative values of p is equal to
. , a sample size of would be
needed.
We choose the large sample size, thus a
sample size of 1022 is needed for the
study.
42. 2.2 Sample size determination for the
test of two proportions
Two-sided test
(Z 2pq Z p2q2 p1q1)2
n = 2
(p2 p1)2
43. Example 5 It is believed that the proportion
of patients who develop complications after
undergoing one type of surgery is % while
the proportion of patients who develop
complications after a second type of surgery
is %. How large should the sample size be
in each of the two groups of patients if an
investigator wishes to detect, with a power
of %, whether the second procedure has a
complication rate significantly higher than
the first at the % level of significance?
44. Use MINITAB, click Stat > Power
and sample size > proportion.
You would complete the dialog box.
You want to test one-sided test, click
on the options button and choose less
than
45.
46. Power and Sample Size
Test for Two Proportions
Testing proportion = proportion (versus <)
Calculating power for proportion
Alpha =
Sample Target Actual
Proportion Size Power Power
A sample size of would be needed in each
group.
47. 2.3 Sample size determination for the tes
of one mean
Two-sided test
2 2
(Z Z )
n 2
( 0 1 )2
48. Example Consider the cholesterol
study. Suppose that the null mean is
mg% /ml, the alternative mean is
mg%/ml, the standard deviation is
, and we wish to conduct a
significance test for one-sided test at
the % level with a power of %.
How large should the sample size be?
49. MINITAB> click Stat > Power and
sample size > sample Z.
You want to test one-sided test, click
on the options button and choose
greater than
50.
51. -Sample Z Test
Testing mean = null (versus > nul
Calculating power for mean = null
Alpha = . Sigma =
Sample Target Actual
Difference Size Power Power
Thus, 96 people are needed.
To achieve a power of 90%
using a 5% significance level
52. 2.4 Sample size determination for the
test of two means
Two-sided test
(Z Z )2( 1
2 2)2
2
2
n =
( 2 1)2
53. Example Consider the blood pressure study
for drug A users and non-drug A users as a
pilot study conducted to obtain parameter
estimates to plan for a larger study. We wish
to test the hypothesis : = versus : .
Determine the appropriate sample size for
the large study using a two–sided test with a
significance level of . and a power of
In the pilot study, we obtained = . ,
S = . = . ,S
54. In the pilot study, we obtained
= . ,S = . =
. ,S
n=( -
We would require a sample size of 152 people
in each group
