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5.3 Determining Sample Size to Estimate p
To Estimate a Population Proportion p
• If you desire a C% confidence interval for a population proportion p with an accuracy specified by you, how large does the sample size need to be?
• We will denote the accuracy by ME, which stands for Margin of Error.
Required Sample Size n to Estimate a Population Proportion p
*
*
*
ˆ
ˆ
2
2
p ME
pqCI for p : p z
n
pqset z ME and solve for n :
n
z pqn = ;
(ME)
p1.96
pqp
n 1.96
pqp
n
.95
Confidence levelSampling distribution of
ME ME
2
2
set 1.96 and solve for
1.96
pqME n
n
pqn
ME
p̂
What About p and q=1-p?*2
2
z pqn = ; we don't know p or q;
(ME)
TWO METHODS :
1: if prior information is available concerning
the value of p, use that value of p to calculate
n;
2 : if no prior information about p is available,
to o
btain a conservative estimate of the
1required sample size, use p q
2
Example: Sample Size to Estimate a Population Proportion p
• The U. S. Crime Commission wants to estimate p = the proportion of crimes in which firearms are used to within .02 with 90% confidence. Data from previous years shows that p is about .6
Example: Sample Size to Estimate a Population Proportion p (cont.)
*
*
2
2
2
2
z pqn = ; ME .02; p is estimated to
(ME)
be about .6 from previous years' data;
90% z 1.645
(1.645) (.6)(.4)n 1,623.6; n 1624
(.02)
Example: Sample Size to Estimate a Population Proportion p
The Curdle Dairy Co. wants to estimate the proportion p of customers that will purchase its new broccoli-flavored ice cream.
Curdle wants to be 90% confident that they have estimated p to within .03. How many customers should they sample?
Example: Sample Size to Estimate a Population Proportion p (cont.)
• The desired Margin of Error is ME = .03• Curdle wants to be 90% confident, so
z*=1.645; the required sample size is
• Since the sample has not yet been taken, the sample proportion p is still unknown.
• We proceed using either one of the following two methods:
*2
2
z pqn =
(ME)
2
2
(1.645) pqn
(.03)
Example: Sample Size to Estimate a Population Proportion p (cont.)• Method 1:
– There is no knowledge about the value of p
– Let p = .5. This results in the largest possible n needed for
a 90% confidence interval of the form – If the proportion does not equal .5, the actual ME will be
narrower than .03 with the n obtained by the formula below.
03.p̂
2
2
1.645 .5 .5751.67 752
.03n
2
2
1.645 .2 .8481.07 482
.03n
• Method 2:– There is some idea about the value of p (say p ~ .2)– Use the value of p to calculate the sample size
*2
2
z pqn =
(ME)