Section 8.3 Confidence Intervals for a Population Proportion 379 In practice, we don’t know the value of p, so we substitute ̂p instead to obtain the standard error we use for the confidence interval: Standard error = √ ̂p(1 − ̂p) n Since the point estimate ̂p is approximately normal with standard error √ ̂p(1 − ̂p)∕n, the appropriate margin of error is Margin of error = z��∕2 √ ̂p(1 − ̂p) n The confidence interval is Point estimate ± Margin of error ̂p ± z��∕2 √ ̂p(1 − ̂p) n The method we have just described requires certain assumptions, which we now state. Explain It Again Reasons for the assumptions: The population must be much larger than the sample (at least 20 times as large), so that the sampled items are independent. The assumption that there are at least 10 items in each category is an approximate check on the assumption that both np and n(1 − p) are at least 10, which ensures that the sampling distribution of p̂ is approximately normal. Assumptions for Constructing a Confidence Interval for p 1. We have a simple random sample. 2. The population is at least 20 times as large as the sample. 3. The items in the population are divided into two categories. 4. The sample must contain at least 10 individuals in each category. Following is a step-by-step description of the procedure for constructing a confidence interval for a population proportion p. Procedure for Constructing a Confidence Interval for p Check to be sure the assumptions are satisfied. If they are, then proceed with the following steps. Step 1: Compute the value of the point estimate ̂p. Step 2: Find the critical value z��∕2 corresponding to the desired confidence level, either from the last line of Table A.3, from Table A.2, or with technology. Step 3: Compute the standard error √ ̂p(1 − ̂p)∕n and multiply it by the critical value to obtain the margin of error z��∕2 √ ̂p(1 − ̂p)∕n. Step 4: Use the point estimate and the margin of error to construct the confidence interval: Point estimate ± Margin of error ̂p ± z��∕2 √ ̂p(1 − ̂p) n ̂p − z��∕2 √ ̂p(1 − ̂p) n < p < ̂p + z��∕2 √ ̂p(1 − ̂p) n Step 5: Interpret the result. Explain It Again Round-off rule: When constructing a confidence interval for a proportion, round the final result to three decimal places. Rounding off the final result When constructing confidence intervals for a proportion, we will round the final result to three decimal places. Note that you should round only the final result, and not the calculations you have made along the way.
navidi_monk_elementary_statistics_2e_ch7-9
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