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)