confidence intervals about a population proportion section 8.3 alan craig 770-274-5242...
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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
3
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
6
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
7
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
13
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
14
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
15
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
17
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
18
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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