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2. Objectives 8.3. Obtain a point estimate for the population proportionObtain and interpret a confidence interval for the population proportionDetermine the sample size for estimating a population proportion. 3. Point Estimate of a Population Proportion. Suppose a simple random sample of size n i
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1. Confidence Intervals about a Population ProportionSection 8.3 Alan Craig
770-274-5242
acraig@gpc.edu
2. 2 Objectives 8.3 Obtain a point estimate for the population proportion
Obtain and interpret a confidence interval for the population proportion
Determine the sample size for estimating a population proportion
3. 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.
4. 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. 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.
6. 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.
7. 7 For a simple random sample of size n, a
(1-a) 100% confidence interval for p is given by
provided that np(1-p) = 10. Constructing a (1-a) 100% Confidence Interval for a Population Proportion
8. 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?
9. 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
10. 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
11. 11 (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
12. 12 Construct a 99% confidence interval for the proportion of patients with ulcers receiving Pepcid who will have confirmed ulcer healing.
Using Calculator: STAT?TESTS?A: 1-PropZInt
Enter 58 for x, 74 for n, and .99 for C-Level
Example: #8, (c), p.374
13. 13 Margin of Error ? Sample Size Solving margin of error to find sample size gives
14. 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 vertexthat is its maximum value
Alternatively, we can use Calculus to find the maximum value.
In either case = 0.25, so
15. 15 The sample of size needed for a (1-a) 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
16. 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
17. 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?
18. 18 Example: # 16, p. 375 within 2 percentage points with 90% confidence if
(b) he does not use any prior estimates?
19. 19 Questions ???????????????