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**Confidence Intervals about a Population Proportion Section 8.3**

Alan Craig

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**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

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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.

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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.

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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.

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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.

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

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.

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Example: #8, (b), p.374 (b) Verify that the requirements for constructing a confidence interval about are satisfied. What do we need to do?

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Example: #8, (b), p.374 (b) Verify that the requirements for constructing a confidence interval about are satisfied. We must show that np(1-p) ≥ 10. 74 * * ( ) = > 10

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Example: #8, (c), p.374 (c) Construct a 99% confidence interval for the proportion of patients with ulcers receiving Pepcid who will have confirmed ulcer healing.

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Example: #8, (c), p.374 (c) Construct a 99% confidence interval for the proportion of patients with ulcers receiving Pepcid who will have confirmed ulcer healing.

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

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**Margin of Error Sample Size**

Solving margin of error to find sample size gives

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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

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**Sample Size for Estimating the Population Proportion p**

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

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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 (a) he uses a Census Bureau estimate of 67.5% from the 4th quarter of 2000? (b) he does not use any prior estimates?

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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?

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Example: # 16, p. 375 within 2 percentage points with 90% confidence if (b) he does not use any prior estimates?

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10.1 – Estimating with Confidence. Recall: The Law of Large Numbers says the sample mean from a large SRS will be close to the unknown population mean.

10.1 – Estimating with Confidence. Recall: The Law of Large Numbers says the sample mean from a large SRS will be close to the unknown population mean.

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