Formula For Margin Of Error For Proportions
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Margin Of Error Calculator
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How To Find Margin Of Error With Confidence Interval
to Calculate the Margin of Error for a Sample Proportion How to Calculate the Margin of 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
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formula for the margin of error for a sample proportion (if certain conditions are met) is where is the sample 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 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
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 confidence interval for a large sample proportion margin of error definition as Confidence Interval Margin of Error for a Population Proportion Example 1000 randomly
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selected Americans were asked if they believed the minimum wage should be raised. 600 said yes. Construct a 95% confidence interval for the proportion of Americans sampling error formula who believe that the minimum wage should be raised. Solution: We have p = 600/1000 = .6 zc = 1.96 and n = 1000 We calculate: Hence we can conclude that between 57 and 63 percent of http://www.dummies.com/education/math/statistics/how-to-calculate-the-margin-of-error-for-a-sample-proportion/ 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 with a margin of error of .5 for m knowing s = 10. How many people should we ask? Solution Solving https://www.ltcconline.net/greenl/courses/201/estimation/ciprop.htm 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 this proportion, with a margin of error of 4%, How many computers should you drop? Solution The formula states that Squaring both sides, we get that zc2 p(1 - p) E2 = n Multiplying by n, we get n
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: The margin of error E is half https://onlinecourses.science.psu.edu/stat500/node/31 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 https://en.wikipedia.org/wiki/Margin_of_error interval and thus the 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 margin of 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% confidence interval instead of a 95% confidence interval, the margin of error will also be smaller. However, when one reports margin of error 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% with 95% confidence. How many individuals should we sample? (In the last poll his approval rate was 72%). a. Educated Guess (use if it is relatively inexpensive to sample more elements when needed.) Z0.025 = 1.96, E = 0.01 Therefore, \(n=\frac
engineering, see Tolerance (engineering). For the eponymous movie, see Margin for error (film). The top portion charts probability density against actual percentage, showing the relative probability that the actual percentage is realised, based on the sampled percentage. In the bottom portion, each line segment shows the 95% confidence interval of a sampling (with the margin of error on the left, and unbiased samples on the right). Note the greater the unbiased samples, the smaller the margin of error. The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It asserts a likelihood (not a certainty) that the result from a sample is close to the number one would get if the whole population had been queried. The likelihood of a result being "within the margin of error" is itself a probability, commonly 95%, though other values are sometimes used. The larger the margin of error, the less confidence one should have that the poll's reported results are close to the true figures; that is, the figures for the whole population. Margin of error applies whenever a population is incompletely sampled. Margin of error is often used in non-survey contexts to indicate observational error in reporting measured quantities. In astronomy, for example, the convention is to report the margin of error as, for example, 4.2421(16) light-years (the distance to Proxima Centauri), with the number in parentheses indicating the expected range of values in the matching digits preceding; in this case, 4.2421(16) is equivalent to 4.2421 ± 0.0016.[1] The latter notation, with the "±", is more commonly seen in most other science and engineering fields. Contents 1 Explanation 2 Concept 2.1 Basic concept 2.2 Calculations assuming random sampling 2.3 Definition 2.4 Different confidence levels 2.5 Maximum and specific margins of error 2.6 Effect of population size 2.7 Ot