Now, consider the mean-difference $\bar{X}_n - \bar{X}_N$ measuring the difference between the sample mean and population mean. It is also learned that the population standard deviation is 10.37 pounds. Thats the nature of the uncertainty still left in the 80% not sampled. 7.4 Finite Population Correction Factor - OpenStax $$\mathrm{Cov}(X_i, X_i) = \mathrm{Var}(X_i) = \sigma^2$$ Math. As n increases, the adjusted sample size estimate decreases. It is appropriate when more than 5% of the population is being sampled and the population has a known population size. \sum_{k=1}^m v_k^2 n_k = N(\mu^2 + \sigma^2) \tag{6.2} \label{sum of squares} As you can see, within this framework the correction term pops out fairly simply in the course of attempting to estimate the mean of the finite population. How do barrel adjusters for v-brakes work? \sigma^2 &= \mathrm{E}[X_i^2] - \mu^2 \\ The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Download Free PDF \\[6pt] Introductory Business Statistics (OpenStax), { "7.00:_Introduction_to_the_Central_Limit_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.01:_The_Central_Limit_Theorem_for_Sample_Means" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.02:_Using_the_Central_Limit_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.03:_The_Central_Limit_Theorem_for_Proportions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.04:_Finite_Population_Correction_Factor" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.05:_Chapter_Formula_Review" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.06:_Chapter_Homework" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.07:_Chapter_Key_Terms" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.08:_Chapter_Practice" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.09:_Chapter_References" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.10:_Chapter_Review" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.11:_Chapter_Solution_(Practice__Homework)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Sampling_and_Data" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Descriptive_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Probability_Topics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Discrete_Random_Variables" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Continuous_Random_Variables" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_The_Normal_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_The_Central_Limit_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Confidence_Intervals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Hypothesis_Testing_with_One_Sample" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Hypothesis_Testing_with_Two_Samples" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_The_Chi-Square_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_F_Distribution_and_One-Way_ANOVA" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Linear_Regression_and_Correlation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "14:_Apppendices" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "authorname:openstax", "showtoc:no", "license:ccby", "Finite Population Correction Factor", "program:openstax", "licenseversion:40", "source@https://openstax.org/details/books/introductory-business-statistics" ], https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FApplied_Statistics%2FIntroductory_Business_Statistics_(OpenStax)%2F07%253A_The_Central_Limit_Theorem%2F7.04%253A_Finite_Population_Correction_Factor, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 7.3: The Central Limit Theorem for Proportions, source@https://openstax.org/details/books/introductory-business-statistics. Look for the words HTML or . Accessibility StatementFor more information contact us atinfo@libretexts.org. Past records indicate that the probability of customers exceeding their credit limit is .06. In other words, when $n/N\leq 0.05$ (i.e., $n$ is not 'too large' compared to $N$), the FPC can safely be ignored; it is easy to see how the correction factor evolves with varying $n$ for a fixed $N$: with $N=10,000$, we have $\text{FPC}=.9995$ when $n=10$ while $\text{FPC}=.3162$ when $n=9,000$. &= \frac{1}{n^2} \Bigg[ \Big( \frac{N-n}{N} \Big)^2 n \sigma^2 + \Big( \frac{n}{N} \Big)^2 (N-n) \sigma^2 \Bigg] \\[6pt] $\overline{X}=\frac{\sum_{i=1}^n X_i}{n}$, $$ \mathrm{Cov}(X_i, X_j) = 0, \quad i \ne j $$, $$\mathrm{Cov}(X_i, X_i) = \mathrm{Var}(X_i) = \sigma^2$$, \begin{align*} &= \sum_{k=1}^m \sum_{k=l} v_k^2 P(X_i=v_k, X_j=v_k) + \sum_{k=1}^m \sum_{k \ne l} v_k v_l P(X_i=v_k, X_j=v_l) \\ \begin{align*} It is learned that the population of White German Shepherds in the USA is 4,000 dogs, and the mean weight for German Shepherds is 75.45 pounds. Required fields are marked *. Solved We can use the finite population correction factor - Chegg ", Similar quotes to "Eat the fish, spit the bones". Statistics and Probability questions and answers. These populations dont comprise millions of people like the U.S. electorate, active Facebook users, or Netflix subscribers. The largest gap in coverage, like the SUS data, is when the sample is a substantial portion of the population (n = 100 and n = 200), where the confidence intervals produced without correction are much too conservative. Next, we need to consider how large our sample is relative to our population to determine if we need to use a finite population correction. Legal. =10.37, \mathrm{E}[X_i^2] &= \mu^2 + \sigma^2 We have assumed that the population is extremely large and that we have sampled a small part of the population. Learn to Use the Finite Population Correction (FPC) in Stata With Data From the IRCBP Public Services Baseline Survey (2005), London: SAGE Publications, Ltd.. \frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n \mathrm{Cov}(X_i, X_j) &= \frac{1}{n^2} \left( \sum_{i=1}^n\sum_{j=i} \mathrm{Cov}(X_i, X_j) + \sum_{i=1}^n\sum_{j \ne i} \mathrm{Cov}(X_i, X_j) \right) \\ I've added another paragraph to address the issue. Hence: Government agencies in the MEPS-IC include all state governments including the District of Columbia, as well as a sample of local governments. However, with this form, you will notice that we use a finite population correction term that is different to your expression. This would imply that we could discard all terms where $i \ne j$. As an Amazon Associate we earn from qualifying purchases. Example 7.4. In this case we have the equivalent expression: $$\text{CI}_N(1-\alpha) = \Bigg[ \bar{X}_n \pm \sqrt{\frac{N-n}{N-1}} \cdot \frac{t_{n-1, \alpha/2}}{\sqrt{n}} \cdot S_{N*} \Bigg].$$. For consistency, we use N1 in the denominator rather than N. When n = 1, the fraction is 0 so there is no effect on the standard sample size estimate. If n=N, the nite population correction factor equals zero, and so does x. If the sample size is 100 dogs, then find the probability that a sample will have a mean that differs from the true probability mean by less than 2 pounds. Mariano Ruiz Espejo a , Housila P. Singh b , Miguel Delgado Pineda c and Saralees Nadarajah d , Corresponding Author Contact . Add this content to your learning management system or webpage by copying the code below into the HTML editor on the page. \mathrm{Var}(\overline{X}) = \mathrm{Var} \left( \frac{\sum_{i=1}^n X_i}{n} \right) = \frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n \mathrm{Cov}(X_i, X_j) = 0.56 +/- 1.96*(.56(1-.56) / 100) * (1300-100) / (1300-1)= [0.4665, 0.6535]. &= \frac{n \sigma^2}{n^2} \\ This means that the apparently intractable sum above is just: \[\sigma_{\mathrm{p}^{\prime}}=\sqrt{\frac{p(1-p)}{n}} \times \sqrt{\frac{N-n}{N-1}}=\sqrt{\frac{0.06(1-0.06)}{360}} \times \sqrt{\frac{3000-360}{3000-1}}=0.0117\nonumber\], \[p_{1}=\frac{10}{360}=0.0278, \quad p_{2}=\frac{20}{360}=0.0556\nonumber\], \[Z=\frac{p^{\prime}-p}{\sqrt{\frac{p(1-p)}{n}} \cdot \sqrt{\frac{N-n}{N-1}}}=\frac{0.0278-0.06}{0.011744}=-2.74\nonumber\], \[p\left(\frac{0.0278-0.06}{0.011744}<\frac{0.0556-0.06}{0.011744}\right)\]. In Sage Research Methods Datasets Part 2. Create lists of favorite content with your personal profile for your reference or to share. Does teleporting off of a mount count as "dismounting" the mount? If we wish to allow a general notion of a "variance" of a set of numbers (as opposed to a variance of a probability distribution), it is .
Blue Heron, Lake Harmony, Pa, Articles W