# Sample Size Determination-Confidence Intervals and Hypothesis Tests

| Home | | Advanced Mathematics |

## Chapter: Biostatistics for the Health Sciences: Inferences Regarding Proportions

Using the formulas for the normal approximation, sample sizes can be derived in a manner similar to that employed.

SAMPLE SIZE DETERMINATION—CONFIDENCE INTERVALS AND HYPOTHESIS TESTS

Using the formulas for the normal approximation, sample sizes can be derived in a manner similar to that employed in Chapters 8 and 9. Again, these calculations would be based on the width of the confidence interval or the power of a test at a specific alternative. The resulting formulas are slightly different from those for continuous variables. In the case of variance in the test of a single proportion, or calculating a confidence interval about a proportion, we guess at p to find the necessary standard deviation. We make this estimate because W is p(1 – p)/n, and we do not know p (the population parameter). We also can be conservative in deter-mining the confidence interval, because for all 0 p 1, p(1 – p) is largest at p = 1/2.

Therefore, the variance of W = p(1 – p)/n (1/2)(1/2)/n = 1/(4n). This upper bound, 1/(4n), on the variance of W can be used in the formulas to obtain a mini-mum sample size that will satisfy the condition for any value of p. We could not find such a bound for the unknown variance of a normal distribution.

Again, software packages such as the ones reviewed by Chernick and Liu (2002) provide solutions for all the cases (using both exact and approximate methods).

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The passage discusses the methods for determining sample sizes for confidence intervals and hypothesis tests regarding proportions, including the use of normal approximation and variance estimation.
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Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
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# Sample Size Determination-Confidence Intervals and Hypothesis Tests

| Home | | Advanced Mathematics |

## Chapter: Biostatistics for the Health Sciences: Inferences Regarding Proportions

Using the formulas for the normal approximation, sample sizes can be derived in a manner similar to that employed.

SAMPLE SIZE DETERMINATION—CONFIDENCE INTERVALS AND HYPOTHESIS TESTS

Using the formulas for the normal approximation, sample sizes can be derived in a manner similar to that employed in Chapters 8 and 9. Again, these calculations would be based on the width of the confidence interval or the power of a test at a specific alternative. The resulting formulas are slightly different from those for continuous variables. In the case of variance in the test of a single proportion, or calculating a confidence interval about a proportion, we guess at p to find the necessary standard deviation. We make this estimate because W is p(1 – p)/n, and we do not know p (the population parameter). We also can be conservative in deter-mining the confidence interval, because for all 0 p 1, p(1 – p) is largest at p = 1/2.

Therefore, the variance of W = p(1 – p)/n (1/2)(1/2)/n = 1/(4n). This upper bound, 1/(4n), on the variance of W can be used in the formulas to obtain a mini-mum sample size that will satisfy the condition for any value of p. We could not find such a bound for the unknown variance of a normal distribution.

Again, software packages such as the ones reviewed by Chernick and Liu (2002) provide solutions for all the cases (using both exact and approximate methods).

Related Topics
user
The passage discusses the methods for determining sample sizes for confidence intervals and hypothesis tests regarding proportions, including the use of normal approximation and variance estimation.

TYPE: NO_QUESTION