How To Calculate Margin Of Error For A Proportion
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Error for a Sample Proportion Related Book Statistics For Dummies, 2nd Edition By Deborah J. Rumsey When you 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 proportion, n is the margin of error definition 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 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 conf
version Unit Summary Margin of Error Determining the Required Sample Size Cautions About Sample Size Calculations Reading AssignmentAn Introduction to Statistical Methods and Data
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Analysis, (See Course Schedule). Margin of Error Note: The margin how to find margin of error on ti 84 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 how to find margin of error with confidence interval 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 http://www.dummies.com/education/math/statistics/how-to-calculate-the-margin-of-error-for-a-sample-proportion/ poorer the precision. One television poll stated that the recent approval rating of the president is 72%; the margin 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 https://onlinecourses.science.psu.edu/stat500/node/31 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% 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
zc s x We can make a similar construction for a confidence interval for a population proportion. Instead of x, we can use p and instead of s, we use , hence, we can write the https://www.ltcconline.net/greenl/courses/201/estimation/ciprop.htm confidence interval for a large sample proportion as Confidence Interval Margin of Error for a Population Proportion Example 1000 randomly selected Americans were asked if they believed the minimum wage should be raised. 600 http://www.statisticshowto.com/how-to-calculate-margin-of-error/ said yes. Construct a 95% confidence interval for the proportion of Americans who believe that the minimum wage should be raised. Solution: We have p = 600/1000 = .6 zc = 1.96 and n = 1000 margin of We calculate: Hence we can conclude that between 57 and 63 percent of all Americans agree with the proposal. In other words, with a margin of error of .03 , 60% agree. Calculating n for Estimating a Mean Example Suppose that you were interested in the average number of units that students take at a two year college to get an AA degree. Suppose you wanted to find a 95% confidence interval margin of error with a margin of error of .5 for m knowing s = 10. How many people should we ask? Solution Solving for n in Margin of Error = E = zc s/ we have E = zcs zc s = E Squaring both sides, we get We use the formula: Example A Subaru dealer wants to find out the age of their customers (for advertising purposes). They want the margin of error to be 3 years old. If they want a 90% confidence interval, how many people do they need to know about? Solution: We have E = 3, zc = 1.65 but there is no way of finding sigma exactly. They use the following reasoning: most car customers are between 16 and 68 years old hence the range is Range = 68 - 16 = 52 The range covers about four standard deviations hence one standard deviation is about s @ 52/4 = 13 We can now calculate n: Hence the dealer should survey at least 52 people. Finding n to Estimate a Proportion Example Suppose that you are in charge to see if dropping a computer will damage it. You want to find the proportion of computers that break. If you want a 90% confidence interval for
Curve) Z-table (Right of Curve) Probability and Statistics Statistics Basics Probability Regression Analysis Critical Values, Z-Tables & Hypothesis Testing Normal Distributions: Definition, Word Problems T-Distribution Non Normal Distribution Chi Square Design of Experiments Multivariate Analysis Sampling in Statistics Famous Mathematicians and Statisticians Calculators Variance and Standard Deviation Calculator Tdist Calculator Permutation Calculator / Combination Calculator Interquartile Range Calculator Linear Regression Calculator Expected Value Calculator Binomial Distribution Calculator Statistics Blog Calculus Matrices Practically Cheating Statistics Handbook Navigation How to Calculate Margin of Error in Easy Steps Probability and Statistics > Critical Values, Z-Tables & Hypothesis Testing > How to Calculate Margin of Error Contents (click to skip to that section): What is a Margin of Error? How to Calculate Margin of Error (video) What is a Margin of Error? The margin of error is the range of values below and above the sample statistic in a confidence interval. The confidence interval is a way to show what the uncertainty is with a certain statistic (i.e. from a poll or survey). For example, a poll might state that there is a 98% confidence interval of 4.88 and 5.26. That means if the poll is repeated using the same techniques, 98% of the time the true population parameter (parameter vs. statistic) will fall within the interval estimates (i.e. 4.88 and 5.26) 98% of the time. What is a Margin of Error Percentage? A margin of error tells you how many percentage points your results will differ from the real population value. For example, a 95% confidence interval with a 4 percent margin of error means that your statistic will be within 4 percentage points of the real population value 95% of the time. The Margin of Error can be calculated in two ways: Margin of error = Critical value x Standard deviation Margin of error = Critical value x Standard error of the statistic Statistics Aren't Always Right! The idea behind confidence levels and margins of error is that any survey or poll will differ from the true population by a certain amount. However, confidence intervals and margins of error reflect the fact that there is room for error, so although 95% or 98% confidence with a 2 percent Margin of Error might sound like a very good statistic, room for error is built in, which means sometimes statistics are wrong. For example, a