Confidence Intervals about a Population Proportion

Section 8.3

Alan Craig770-274-5242

acraig@gpc.edu

2

Objectives 8.3

1. Obtain a point estimate for the population proportion

2. Obtain and interpret a confidence interval for the population proportion

3. Determine the sample size for estimating a population proportion

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Point Estimate of a Population Proportion

Suppose a simple random sample of size n is obtained from a population in which each individual either does or does not have a certain characteristic. The best point estimate of p, denoted , the proportion of the population with a certain characteristic, is given by

where x is the number of individuals in the sample with the specified characteristic.

n

xp ˆ

p̂

4

Example: #8 (a), p. 374

A study of 74 patients with ulcers was conducted in which they were prescribed 40 mg of Pepcid. After 8 weeks, 58 reported confirmed ulcer healing.

(a) Obtain a point estimate for the proportion of patients with ulcers receiving Pepcid who will have confirmed ulcer healing.

5

Example: #8 (a), p. 374

A study of 74 patients with ulcers was conducted in which they were prescribed 40 mg of Pepcid. After 8 weeks, 58 reported confirmed ulcer healing.

(a) Obtain a point estimate for the proportion of patients with ulcers receiving Pepcid who will have confirmed ulcer healing.

784.74

58ˆ n

xp

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Sampling Distribution of

For a simple random sample of size n such that n ≤ .05N (i.e., sample size is no more than 5% of the population), the sampling distribution of is approximately normal with

mean

and standard deviation

provided that np(1-p) ≥ 10.

n

ppp

)1(ˆ

p̂

p̂

pp ˆˆ

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For a simple random sample of size n, a(1-) ·100% confidence interval for p is given by

provided that np(1-p) ≥ 10.

Constructing a (1-) ·100% Confidence Interval for a

Population Proportion

n

ppzp

n

ppzp

)ˆ1(ˆˆ :boundUpper

)ˆ1(ˆˆ :boundLower

2/

2/

8

Example: #8, (b), p.374

(b) Verify that the requirements for constructing a confidence interval about are satisfied.

What do we need to do?

p̂

9

(b) Verify that the requirements for constructing a confidence interval about are satisfied.

We must show that np(1-p) ≥ 10.

74 * 0.784 * (1 - 0.784) = 12.53 > 10

Example: #8, (b), p.374

p̂

10

(c) Construct a 99% confidence interval for the proportion of patients with ulcers receiving Pepcid who will have confirmed ulcer healing.

Example: #8, (c), p.374

5). Slide (from 784.0ˆ that Recall p

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(c) Construct a 99% confidence interval for the proportion of patients with ulcers receiving Pepcid who will have confirmed ulcer healing.

Example: #8, (c), p.374

907.074

)784.01(784.0575.2784.0

)ˆ1(ˆˆ :boundUpper

661.074

)784.01(784.0575.2784.0

)ˆ1(ˆˆ :boundLower

2/

2/

n

ppzp

n

ppzp

12

(c) Construct a 99% confidence interval for the proportion of patients with ulcers receiving Pepcid who will have confirmed ulcer healing.

Using Calculator: STATTESTSA: 1-PropZInt

Enter 58 for x, 74 for n, and .99 for C-Level

Example: #8, (c), p.374

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Margin of Error Sample Size

• Solving margin of error to find sample size gives

for! solving are what weis which involves This . ˆBut

)ˆ1(ˆ

)ˆ1(ˆ :Error ofMargin

2

2/

2/

nn

xp

E

zppn

n

ppzE

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Margin of Error Sample Size

• So we can use a prior estimate for p, • or we can find the largest value of .

• Using the fact that this is a parabola that opens down (see Figure 17 p. 373), we can find the y-coordinate of the vertex—that is its maximum value

• Alternatively, we can use Calculus to find the maximum value.

• In either case ≤ 0.25, so

)ˆ1(ˆ pp p̂

)ˆ1(ˆ pp

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The sample of size needed for a (1-) ·100% confidence interval for p with a margin of error E is given by

(rounded up to next integer) where is a prior estimate of p. If a prior estimate of p is unavailable, the sample size required is

Sample Size for Estimating the Population Proportion p

2

2/

2

2/

25.0

)ˆ1(ˆ

E

zn

E

zppn

p̂

16

(a) he uses a Census Bureau estimate of 67.5% from the 4th quarter of 2000?

(b) he does not use any prior estimates?

Example: # 16, p. 375

An urban economist wishes to estimate the percentage of Americans who own their house. What size sample should be obtained if he wishes the estimate to be within 2 percentage points with 90% confidence if

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Example: # 16, p. 375

within 2 percentage points with 90% confidence if

(a) he uses a Census Bureau estimate of 67.5% from the 4th quarter of 2000?

1485 UP)(round 086.1484

02.

645.1)675.01(675.0)ˆ1(ˆ

675.ˆ02.0

22

2/

05.2/

E

zppn

pzzE

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1692 UP)(round 27.1691

02.

645.125.025.0

02.0

22

2/

05.2/

E

zn

zzE

Example: # 16, p. 375

within 2 percentage points with 90% confidence if

(b) he does not use any prior estimates?

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Questions

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