Compute Standard Error Proportion
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Financial Calculator books AP calculator review Statistics AP study guides Probability Survey sampling standard error of difference between two proportions calculator Excel Graphing calculators Book reviews Glossary AP practice exam Problems and solutions Formulas Notation Share with Friends Confidence Interval: standard error two proportions calculator Proportion (Large Sample) This lesson describes how to construct a confidence interval for a sample proportion, p, when the sample size is large. Estimation Requirements The approach described in this lesson is valid https://onlinecourses.science.psu.edu/stat200/node/43 whenever the following conditions are met: The sampling method is simple random sampling. The sample is sufficiently large. As a rule of thumb, a sample is considered "sufficiently large" if it includes at least 10 successes and 10 failures. Note the implications of the second condition. If the population proportion were close to 0.5, the sample size required to produce at least 10 successes and at http://stattrek.com/estimation/confidence-interval-proportion.aspx?Tutorial=Stat least 10 failures would probably be close to 20. But if the population proportion were extreme (i.e., close to 0 or 1), a much larger sample would probably be needed to produce at least 10 successes and 10 failures. For example, imagine that the probability of success were 0.1, and the sample were selected using simple random sampling. In this situation, a sample size close to 100 might be needed to get 10 successes. The Variability of the Sample Proportion To construct a confidence interval for a sample proportion, we need to know the variability of the sample proportion. This means we need to know how to compute the standard deviation and/or the standard error of the sampling distribution. Suppose k possible samples of size n can be selected from the population. The standard deviation of the sampling distribution is the "average" deviation between the k sample proportions and the true population proportion, P. The standard deviation of the sample proportion σp is: σp = sqrt[ P * ( 1 - P ) / n ] * sqrt[ ( N - n ) / ( N - 1 ) ] where P is the population proportio
0 otherwise. The standard deviation of any variable involves the standard error expression . Let's suppose there are m 1s (and n-m 0s) among the n subjects. Then, and is equal to (1-m/n) for m observations and 0-m/n standard error of for (n-m) observations. When these results are combined, the final result is and the sample variance (square of the SD) of the 0/1 observations is The sample proportion is the mean of n of these observations, so the standard error of the proportion is calculated like the standard error of the mean, that is, the SD of one of them divided by the square root of the sample size or Copyright © 1998 Gerard E. Dallal
on the Mean Learning Objectives Estimate the population proportion from sample proportions Apply the correction for continuity Compute a confidence interval A candidate in a two-person election commissions a poll to determine who is ahead. The pollster randomly chooses 500 registered voters and determines that 260 out of the 500 favor the candidate. In other words, 0.52 of the sample favors the candidate. Although this point estimate of the proportion is informative, it is important to also compute a confidence interval. The confidence interval is computed based on the mean and standard deviation of the sampling distribution of a proportion. The formulas for these two parameters are shown below: μp = π Since we do not know the population parameter π, we use the sample proportion p as an estimate. The estimated standard error of p is therefore We start by taking our statistic (p) and creating an interval that ranges (Z.95)(sp) in both directions, where Z.95 is the number of standard deviations extending from the mean of a normal distribution required to contain 0.95 of the area (see the section on the confidence interval for the mean). The value of Z.95 is computed with the normal calculator and is equal to 1.96. We then make a slight adjustment to correct for the fact that the distribution is discrete rather than continuous.
Normal Distribution Calculator sp is calculated as shown below: To correct for the fact that we are approximating a discrete distribution with a continuous distribution (the normal distribution), we subtract 0.5/N from the lower limit and add 0.5/N to the upper limit of the interval. Therefore the confidence interval is Lower limit: 0.52 - (1.96)(0.0223) - 0.001 = 0.475 Upper limit: 0.52 + (1.96)(0.0223) + 0.001 = 0.565 0.475 ≤ π ≤ 0.565 Since the interval extends 0.045 in both directions, the margin of error is 0.045. In terms of percent, between 47.5% and 56.5% of the voters favor the candidate and the margin of error is 4.5%. Keep in mind that the margin of error of 4.5% is the margin of error for the percent favoring the candidate and not the margin of error for the difference between the percent favoring the candidate and the percent favoring the opponent. The margin of error for the difference is 9%, twice the margin of error for the individual percent. Keep this in mind when you hear reports in the media; the media often get this wrong. Please answer the questions: feedback