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# Standard Error Of Sample Proportion Formula

repeatedly randomly drawn from a population, and the proportion of successes in each sample is recorded ($$\widehat{p}$$),the distribution of the sample proportions (i.e., the sampling distirbution) can be approximated by a normal distribution given that sampling distribution of p hat calculator both $$n \times p \geq 10$$ and $$n \times (1-p) \geq 10$$. This

## Standard Error Of Proportion Definition

is known as theRule of Sample Proportions. Note that some textbooks use a minimum of 15 instead of 10.The mean of population proportion the distribution of sample proportions is equal to the population proportion ($$p$$). The standard deviation of the distribution of sample proportions is symbolized by $$SE(\widehat{p})$$ and equals $$\sqrt{\frac {p(1-p)}{n}}$$; this is known as standard deviation of sample proportion thestandard error of $$\widehat{p}$$. The symbol $$\sigma _{\widehat p}$$ is also used to signify the standard deviation of the distirbution of sample proportions. Standard Error of the Sample Proportion$SE(\widehat{p})= \sqrt{\frac {p(1-p)}{n}}$If $$p$$ is unknown, estimate $$p$$ using $$\widehat{p}$$The box below summarizes the rule of sample proportions: Characteristics of the Distribution of Sample ProportionsGiven both $$n \times p \geq 10$$ and $$n \times (1-p) \geq 10$$, the distribution of

## Standard Error Of P Hat

sample proportions will be approximately normally distributed with a mean of $$\mu_{\widehat{p}}$$ and standard deviation of $$SE(\widehat{p})$$Mean $$\mu_{\widehat{p}}=p$$Standard Deviation ("Standard Error")$$SE(\widehat{p})= \sqrt{\frac {p(1-p)}{n}}$$ 6.2.1 - Marijuana Example 6.2.2 - Video: Pennsylvania Residency Example 6.2.3 - Military Example ‹ 6.1.2 - Video: Two-Tailed Example, StatKey up 6.2.1 - Marijuana Example › Printer-friendly version Navigation Start Here! Welcome to STAT 200! Search Course Materials Faculty login (PSU Access Account) Lessons Lesson 0: Statistics: The “Big Picture” Lesson 1: Gathering Data Lesson 2: Turning Data Into Information Lesson 3: Probability - 1 Variable Lesson 4: Probability - 2 Variables Lesson 5: Probability Distributions Lesson 6: Sampling Distributions6.1 - Simulation of a Sampling Distribution of a Proportion (Exact Method) 6.2 - Rule of Sample Proportions (Normal Approximation Method)6.2.1 - Marijuana Example 6.2.2 - Video: Pennsylvania Residency Example 6.2.3 - Military Example 6.3 - Simulating a Sampling Distribution of a Sample Mean 6.4 - Central Limit Theorem 6.5 - Probability of a Sample Mean Applications 6.6 - Introduction to the t Distribution 6.7 - Summary Lesson 7: Confidence Intervals Lesson 8: Hypothesis Testing Lesson 9: Comparing Two Groups Lesson 10: One-Way Analysis of Variance (ANOVA) Lesson 11: Association Between Categorical Variables Lesson 12: Inference About Regression Special Topic: Mult

test AP formulas FAQ AP study guides AP calculators Binomial Chi-square f Dist Hypergeometric Multinomial Negative binomial Normal Poisson t Dist Random numbers Probability Bayes rule Combinations/permutations Factorial Event counter Wizard Graphing Scientific Financial Calculator books AP calculator review Statistics AP study guides Probability Survey sampling Excel Graphing calculators Book reviews Glossary AP probability of sample proportion calculator practice exam Problems and solutions Formulas Notation Share with Friends Important Statistics Formulas This web page sample proportion symbol presents statistics formulas described in the Stat Trek tutorials. Each formula links to a web page that explains how to use the formula. Parameters Population mean

## What Is P Hat

= μ = ( Σ Xi ) / N Population standard deviation = σ = sqrt [ Σ ( Xi - μ )2 / N ] Population variance = σ2 = Σ ( Xi - μ )2 / N Variance of population proportion https://onlinecourses.science.psu.edu/stat200/node/43 = σP2 = PQ / n Standardized score = Z = (X - μ) / σ Population correlation coefficient = ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] } Statistics Unless otherwise noted, these formulas assume simple random sampling. Sample mean = x = ( Σ xi ) / n Sample standard deviation = s = sqrt [ Σ ( xi - x )2 / ( n - 1 ) http://stattrek.com/statistics/formulas.aspx ] Sample variance = s2 = Σ ( xi - x )2 / ( n - 1 ) Variance of sample proportion = sp2 = pq / (n - 1) Pooled sample proportion = p = (p1 * n1 + p2 * n2) / (n1 + n2) Pooled sample standard deviation = sp = sqrt [ (n1 - 1) * s12 + (n2 - 1) * s22 ] / (n1 + n2 - 2) ] Sample correlation coefficient = r = [ 1 / (n - 1) ] * Σ { [ (xi - x) / sx ] * [ (yi - y) / sy ] } Correlation Pearson product-moment correlation = r = Σ (xy) / sqrt [ ( Σ x2 ) * ( Σ y2 ) ] Linear correlation (sample data) = r = [ 1 / (n - 1) ] * Σ { [ (xi - x) / sx ] * [ (yi - y) / sy ] } Linear correlation (population data) = ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] } Simple Linear Regression Simple linear regression line: ŷ = b0 + b1x Regression coefficient = b1 = Σ [ (xi - x) (yi - y) ] / Σ [ (xi - x)2] Regression slope intercept = b0 = y - b1 * x Regression coefficient = b1 = r * (sy / sx) Standard error of regression slope = sb1 = sqrt [ Σ(yi

population parameters like p are typically unknown and estimated from the data. Consider estimating the proportion p of the current WMU graduating class who plan http://www.stat.wmich.edu/s216/book/node70.html to go to graduate school. Suppose we take a sample of 40 graduating http://onlinestatbook.com/2/estimation/proportion_ci.html students, and suppose that 6 out of the 40 are planning to go to graduate school. Then our estimate is of the graduating class plan to go to graduate school. Now is based on a sample, and unless we got really lucky, chances are the .15 estimate missed. By how sample proportion much? On the average, a random variable misses the mean by one SD. From the previous section, the SD of equals . It follows that the expected size of the miss is . This last term is called the standard error of estimation of the sample proportion, or simply standard error (SE) of the proportion . However, since we do not know p, standard error of we cannot calculate this SE. In a situation like this, statisticians replace p with when calculating the SE. The resulting quantity is called the estimated standard error of the sample proportion . In practice, however, the word estimated'' is dropped and the estimated SE is simply called the SE . Exercise 4. a. If 6 out of 40 students plan to go to graduate school, the proportion of all students who plan to go to graduate school is estimated as ________. The standard error of this estimate is ________. b. If 54 out of 360 students plan to go to graduate school, the proportion of all students who plan to go to graduate school is estimated as ________. The standard error of this estimate is ________. Exercise 4 shows the effect of of increasing the sample size on the SE of the sample proportion. Multiplying the sample size by a factor of 9 (from 40 to 360) makes the SE decrease by a factor of 3. In the formula for the SE of , the sample size appears (i) in the denominator, and (ii) inside a squareroot. Therefore

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

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Sample Proportion Standard Error p repeatedly randomly drawn from a population and the proportion of successes in each sample is recorded widehat p the distribution sample proportion formula of the sample proportions i e the sampling distirbution can be standard error of proportion formula approximated by a normal distribution given that both n times p geq and n times standard error of p hat -p geq This is known as theRule of Sample Proportions Note that some textbooks use a minimum of instead of The mean of the p Population Proportion p distribution of sample proportions is equal to the

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Sample Proportion Error p repeatedly randomly drawn from a population and the proportion of successes in each sample is recorded widehat p the distribution of the sample proportions i e the sampling distirbution can be approximated by a normal sample proportion formula distribution given that both n times p geq and n times -p p Standard Error Of Proportion Formula p geq This is known as theRule of Sample Proportions Note that some textbooks use a minimum of p Sampling Distribution Of P Hat Calculator p instead of The mean of the distribution of sample proportions is equal to the

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Standard Error For The Sampling Distribution Of Proportion p repeatedly randomly drawn from a population and the proportion of successes in each sample is p Standard Deviation Of Sample Proportion p recorded widehat p the distribution of the sample proportions i e the sampling sample proportion formula distirbution can be approximated by a normal distribution given that both n times p sample proportion definition geq and n times -p geq This is known as theRule of Sample Proportions Note that some textbooks use a minimum of p Sample Proportion Calculator p instead of The mean of the distribution of sample

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Standard Error For Sample Proportions p repeatedly randomly drawn from a population and the proportion of successes in each sample is recorded widehat p the distribution of the sample proportions i e the sampling distirbution can be approximated by a standard error of proportion formula normal distribution given that both n times p geq and n times sample proportion formula -p geq This is known as theRule of Sample Proportions Note that some textbooks use a minimum of standard error of p hat instead of The mean of the distribution of sample proportions is equal to the population proportion p

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Standard Error Of Sampling Distribution Of Sample Proportion p repeatedly randomly drawn from a population and the proportion of successes in each sample is recorded widehat p the distribution of the sample proportions i e the sampling standard deviation of sample proportion distirbution can be approximated by a normal distribution given that both n p Sample Proportion Formula p times p geq and n times -p geq This is known as theRule of Sample Proportions sample proportion definition Note that some textbooks use a minimum of instead of The mean of the distribution of sample proportions is equal to the

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Standard Error Of Proportion Example p repeatedly randomly drawn from a population and the proportion of successes in each sample sample proportion formula is recorded widehat p the distribution of the sample proportions i e the p Standard Error Of Proportion Formula p sampling distirbution can be approximated by a normal distribution given that both n times standard error of p hat p geq and n times -p geq This is known as theRule of Sample Proportions Note that some textbooks use a minimum p Sample Proportion Calculator p of instead of The mean of the distribution of sample proportions

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Standard Error Of The Sampling Distribution Of A Sample Proportion p repeatedly randomly drawn from a population and the proportion of successes in each sample is recorded widehat p the distribution of the sample proportions i e the sampling p Standard Deviation Of Sample Proportion p distirbution can be approximated by a normal distribution given that both n sample proportion formula times p geq and n times -p geq This is known as theRule of Sample Proportions sample proportion definition Note that some textbooks use a minimum of instead of The mean of the distribution of sample proportions is equal

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What Is The Standard Error Of The Sample Proportion p repeatedly randomly drawn from a population and the proportion of successes in each sample standard error of proportion formula is recorded widehat p the distribution of the sample proportions i e p Sample Proportion Formula p the sampling distirbution can be approximated by a normal distribution given that both n p Standard Error Of P Hat p times p geq and n times -p geq This is known as theRule of Sample Proportions Note that some textbooks use a p Sample Proportion Calculator p minimum of instead of The mean

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