Population Proportion Margin Of Error Calculator
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Calculate the Margin of Error for a Sample Proportion Related Book Statistics For Dummies, 2nd Edition By Deborah J. Rumsey When you margin of error sample size report the results of a statistical survey, you need to include the margin of error. The general formula for the margin of error for a sample proportion (if certain conditions are met) is where is the sample http://www.scor.qc.ca/en_calculez.html proportion, n is the sample size, and z* is the appropriate z*-value for your desired level of confidence (from the following table). z*-Values for Selected (Percentage) Confidence Levels Percentage Confidence z*-Value 80 1.28 90 1.645 95 1.96 98 2.33 99 2.58 Note that these values are taken from the standard normal (Z-) distribution. The area between each z* value and the negative of that z* value is the confidence percentage (approximately). For example, the area http://www.dummies.com/education/math/statistics/how-to-calculate-the-margin-of-error-for-a-sample-proportion/ between z*=1.28 and z=-1.28 is approximately 0.80. Hence this chart can be expanded to other confidence percentages as well. The chart shows only the confidence percentages most commonly used. Here are the steps for calculating the margin of error for a sample proportion: Find the sample size, n, and the sample proportion. The sample proportion is the number in the sample with the characteristic of interest, divided by n. Multiply the sample proportion by Divide the result by n. Take the square root of the calculated value. You now have the standard error, Multiply the result by the appropriate z*-value for the confidence level desired. Refer to the above table for the appropriate z*-value. If the confidence level is 95%, the z*-value is 1.96. Here's an example: Suppose that the Gallup Organization's latest poll sampled 1,000 people from the United States, and the results show that 520 people (52%) think the president is doing a good job, compared to 48% who don't think so. First, assume you want a 95% level of confidence, so z* = 1.96. The number of Americans in the sample who said they approve of the president was found to be 520. This means that the sample proportion, is 520 / 1,000 = 0.52. (The sample size, n, was 1,000.) The margin of error for this polling question
version Unit Summary Margin of Error Determining the Required Sample Size Cautions About Sample Size Calculations Reading AssignmentAn Introduction to Statistical Methods and Data Analysis, (See Course Schedule). Margin of Error Note: https://onlinecourses.science.psu.edu/stat500/node/31 The margin of error E is half of the width of the confidence interval. \[E=z_{\alpha/2}\sqrt{\frac{\hat{p}\cdot (1-\hat{p})}{n}}\] Confidence and precision (we call wider intervals as having poorer precision): Note that the higher the confidence level, the wider the width (or equivalently, half width) of the interval and thus the poorer the precision. One television poll stated that the recent approval rating of the president is 72%; the margin margin of of error of the poll is plus or minus 3%. [For most newspapers and magazine polls, it is understood that the margin of error is calculated for a 95% confidence interval (if not stated otherwise). A 3% margin of error is a popular choice.] If we want the margin of error smaller (i.e., narrower intervals), we can increase the sample size. Or, if you calculate a 90% margin of error confidence interval instead of a 95% confidence interval, the margin of error will also be smaller. However, when one reports it, remember to state that the confidence interval is only 90% because otherwise people will assume a 95% confidence. Determining the Required Sample Size If the desired margin of error E is specified and the desired confidence level is specified, the required sample size to meet the requirement can be calculated by two methods: a. Educated Guess \[n=\frac {(z_{\alpha/2})^2 \cdot \hat{p}_g \cdot (1-\hat{p}_g)}{E^2}\] Where \(\hat{p}_g\) is an educated guess for the parameter π. b. Conservative Method \[n=\frac {(z_{\alpha/2})^2 \cdot \frac{1}{2} \cdot \frac{1}{2}}{E^2}\] This formula can be obtained from part (a) using the fact that: For 0 ≤ p ≤ 1, p (1 - p) achieves its largest value at \(p=\frac{1}{2}\). The sample size obtained from using the educated guess is usually smaller than the one obtained using the conservative method. This smaller sample size means there is some risk that the resulting confidence interval may be wider than desired. Using the sample size by the conservative method has no such risk. For the next poll of the president's approval rating, we want to get a margin of error of 1% wi