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Confidence Intervals about a Population Proportion Section 8.3

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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Confidence Intervals about a Population Proportion Section 8.3

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    1. Confidence Intervals about a Population Proportion Section 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 ???????????????

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