Sampling distribution of sample proportion worksheet. seed(0) #define number of samples.

If a poll is taken from a random sample of 80 adults in the large city, which of the following properly describes the sampling distribution of the sample proportion of adults who support the stadium? A sampling distribution of a statistic is the distribution of values for the statistic for all possible samples of a given size from a given population. in. - [Instructor] What we're gonna do in this video is talk about the idea of a sampling distribution. The sample proportion values range from . For a sampling distribution for proportions, we will take the sample proportion from all possible samples of our given size and average those together to find the mean of our sampling distribution. 43 ( 1 − 0. Therefore, the probability that the average height of those women falls below 160 cm is about 31. Independence: Groups from different randomly selected states should be independent. This. 025. escribe the measurements in a normally distributed population. n = 100 3. Obtain the proportion the population of. The larger the sample, the better the approximation will be. The variance of all differences, , is the sum of the variances, . 0. 5, H a: x ¯ > 4. Then, 1. 3 9. In a large city, 46% of adults support the local football team building a new stadium. 3 will involve carrying out two more (HW for Mon). For example, it finds the probability that 10 randomly chosen students have a mean ACT score less than 18 is 0. b) The sampling distribution for the sample proportion represents the distribution of possible values for the sample proportion if the study were repeated many times. 0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform. 065) 65 10. 5, Ha: μ < 4. That is, the population proportion p of the satisfaction is 0. [Example 6. 2: Sampling Distribution of the Sample Proportion 1. Your result is ready. 373899. Note: Each sample gave a different answer, which did not always match the population value of p = 0. For each part, indicate one of the following four responses: i. ) 6. Be sure to use the same scale on both (see scale used in the Activity). ulation of students at this university, is. 2 Practice Ch 6. Obt. A sampling distribution shows every possible result a statistic can take in every possible sample from a population and how often each result happens - and can help us use samples to make predictions about the chance tht something will occur. R. 1 Definitions. For large samples, the sample proportion is approximately normally distributed, with mean μ P ^ = p and standard deviation σ P ^ = p q / n. 3) Test Ch 5. 10% condition: Reasonable that the population of both states is greater than 10,000. 08 to . 68-95-99. Example: Stratified sampling The company has 800 female employees and 200 male employees. It is useful in this situation because it allows us to make inferences about the population Video transcript. Actually it includes sampling distributions for any statistic. For our purposes, it will be simpler to sample with replacement. Dec 9, 2022 · This leads to a big discussion on the graphs of the population distribution, the sample data distribution, and the sampling distribution. Although we cannot determine whether one sample The sample proportion ^p = x n = 65 100 p ^ = x n = 65 100 provides a point estimate of p p, the proportion of female students at MacEwan. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion Mon, Mar 1, 2010 7 / 33 Applications If p = 0:60 and our sample size is n = 150, then p^ is normal with Sampling distributions from non-normal populations are approximately normal provided n is large. Construct a 99. Suppose that we are planning to take a random sample of n (or the sample size) observations. n = 1000 True or False? In Exercises 5-7, determine whether the statement is true or false. p ^ is the sample proportion. UNC-3. Assume σ = 3. If we add these variances we get the variance of the differences between sample proportions. 25 0. This worksheet provides instructions that allow you to carry out simulations using your calculator, to investigate the sampling distribution of sample proportions. This will help to reveal to students that the The probability distribution of x is x P(x) 0 1=3 = 0:3333 1 2 =3 0:6667. Koether Experiment Results Computing the Sampling Distribution of ^p PDFs for n = 1;2;3;:::;30 Observations The Central Limit Theorem for Proportions Why Surveys Work Assignment. Only P(A) is given. This standard deviation formula is exactly correct as long as we have: Independent observations between the two samples. Then you use random or systematic sampling to select a sample from each subgroup. 4 0. The probability distribution of a . 0 3. Treating Sampling without replacement as independent if one of the following are satisfied: a) Assume a very big population when population size is not given. n = 10000. A statistical population is a set or collection of all possible observations of some characteristic. The first 10 samples along with the values of x are shown in the table: Sample Values x-bar In hypothesis testing, we assume the null hypothesis is true. Sampling with replacement – independent events. Find the probability that the sample proportion computed from a sample of size \(900\) will be within \(5\) percentage points of the true population proportion. n = 50 2. If 9 9 students are randomly sampled from each school, what is the probability that: Sep 12, 2021 · The Sampling Distribution of the Sample Proportion. Theorem 1 The Central Limit Theorem (CLT for proportions) The pro-portion of a random sample has a sampling distributi. The sampling distribution of a statistic describes the values of the statistic in all possible samples of the same size from the same population. 2. 8. o. For simplicity, we have been using N = 2 N = 2. This simulates the sampling distribution of the sample proportion. The GPAs of both schools are normally distributed. With a large sample, the sampling distribution of a proportion will have an approximate normal Jan 18, 2024 · Input the population parameters in the sampling distribution calculator (μ = 161. Given simple random samples of size n from a given population with a measured characteristic such as mean X, proportion lowing items. Let's say it's a bunch of balls, each of them have a number written on it. 5 and n ( 1 − p) = 50 ( 1 − 0. Remember, we set up the null hypothesis as H 0: p = p 0. sampling distribution: a probability distribution of a statistic; it is a distribution of all possible samples (random samples) from a population and how often each outcome occurs in repeated sampling (of the same size n). H 0: x ¯ = 4. The red li. 4 9. " If \ (np_0 < 10\) or \ (n (1-p_0) < 10\) then the distribution of sample proportions follows a binomial distribution. 95% that X is within 2 standard deviations of mean. If 9 9 students are randomly sampled from each school, what is the probability that: Ch 2 Test (Normal Distribution of Data)  AP Practice Worksheet 1 Ch 2 Practice with a Graphing Calculator Practice - A Little Harder Ch 6. 75 hours with a sample standard deviation of 2. 6. The examples on pp. Sep 19, 2019 · Based on the overall proportions of the population, you calculate how many people should be sampled from each subgroup. 4 Answers will vary. We can find the sampling distribution of any sample statistic that would estimate a certain population parameter of interest. It varies based on the sample. 60. n p = 50 ( 0. On the graph, what is represented on the x-axis:y-axis:5. 25. 4%. When the employees' satisfaction level with the company is investigated and the satisfaction is expressed as 1, the complaint is 0 as follows. A ** **school district gives vision tests to 133 incoming kindergarten children. The sampling distribution of a sample mean x ¯ has: μ x ¯ = μ σ x ¯ = σ n. __. 3 Sample size Worksheet on CI of Proportion Quiz on CI of Proportion Jun 18, 2024 · a) The sample proportion of respondents who support the new system is 600/1000 = 0. In a separate random sample of 532 adult Internet users aged 30-49, 14% used Twitter. 11 only 111 only I and Il only 1 and 111 only 1, Il, and 111. 27. 3 0. Jan 8, 2024 · As we saw before, due to sampling variability, sample proportion in random samples of size 100 will take numerical values which vary according to the laws of chance: in other words, sample proportion is a random variable. All of these sample means make up the sampling distribution, which can be graphed as a histogram. 2. First, we should check our conditions for the sampling distribution of the sample proportion. #create empty vector of length n. What is the mean of the sampling distribution of ˆ p? b. for(i in 1:n){. Distribution of sample proportions - Student Worksheet 5 In this section, you will explore the viability of modelling the discrete sampling distribution of Ö P with a continuous normal distribution, 2 N,ÖÖ PP , where Ö E Ö P Pp and Ö 1 P pp n . 22: Sampling Experiment (Worksheet) is shared under a CC BY 4. 2 Review Ch 5 (+ 6. Shape When n 1 p 1, n 1 (1 p 1), n 2 p 2 and n 2 (1 p 2) are all at least 10, the sampling distribution We use this document to lead students to describe the shape, center, and variability of the sampling distribution of the difference of proportions. 36 or 36%. We may sample with or without replacement. 5, Ha: ˉx > 4. 끫뺂끫뤢 = 끫뤢 2. 20 to 0. Oct 2, 2021 · Verify that the sample proportion \(\hat{p}\) computed from samples of size \(900\) meets the condition that its sampling distribution be approximately normal. SAMPLING DISTRIBUTION OF THE POPULATION PROPORTION. So, as a first step in developing inference procedures for population proportions, we need to know something about the sampling distribution of the sample proportion, pˆ . It is also important to keep in mind that there is a sampling distribution for various sample sizes. • The mean of the sampling distribution is the same as the mean of the 2-Sample Proportions Confidence Intervals Advanced Inference Worksheet #1 1. For large samples, the sample proportion is approximately normally distributed, with mean \(μ_{\hat{P}}=p\) and standard deviation \(\sigma _{\hat{P}}=\sqrt{\frac{pq}{n}}\). Sampling without replacement – dependent events. The sampling distribution of the range for N = 3 N = 3 is shown in Figure 9. 43) = 21. The first will be the population distribution of heights and the second will be the sampling distribution of sample mean heights. The sample mean was 4. The formulas men. 43 and n = 50. Notes 4 and 5 6) You sample 16 students in your school, and they average 13 hours of TV a week. 2 Practice Mar 26, 2023 · Verify that the sample proportion \(\hat{p}\) computed from samples of size \(900\) meets the condition that its sampling distribution be approximately normal. 2: Sample Proportions. Looking at this li. Steps for Generating a Sampling Distribution. c. 2: Sample P ropor tions The first type of sampling distribution you will encounter is a sampling distribution for proportions used to estimate a population proportion. , taste tests) with different sample groups. Conduct a hypothesis test. 1. 끫뺐끫뤢 = This page titled 1. Independent observations within each sample*. 3: Distribution of ranges for N = 2 N = 2. Worksheet 6. Variability. Therefore, the hypothesis is regarding Beta1, the slope of the line. Sample Proportion Proportion of a sample of a population. Input the sample data (n = 7, X = 160). The standard deviation of the difference is: σ p ^ 1 − p ^ 2 = p 1 ( 1 − p 1) n 1 + p 2 ( 1 − p 2) n 2. Apr 23, 2022 · The mean GPA for students in School A School A is 3. Dec 6, 2020 · The variances of the sampling distributions of sample proportion are. The central limit theorem (CLT) states that when the sample size is suficiently large, sampling distribution of the mean of. The following code shows how to generate a sampling distribution in R: set. Repetition – Repeat steps 1 and 2 many times. Before we begin, let’s make sure we review the terms and notation associated with proportions: \ (p\) is the population proportion. The center is at around . I n this chapt er, w e wil l explore the beha vior of sample statistics in repeat ed sampling and lear n one of the most impor tant theorems in Statistics ! The Central Limit Theorem. Take the average xof each sample. Question: Math 133: 8. One hundred samples of size 2 were generated and the value of x computed for each. The size of the sample is represented by the l. b) Use 5% guideline for cumbersome The mean of the sampling distribution is always equal to the population proportion (p), and the standard deviation is calculated as sqrt (p (1 − p) / n), where n is the sample size. Sample proportions arise most often when we are interested in categorical variables. 1 Sampling Distribution of Sample Proportions. of how sampling information varies f rom sample to sample. It also 4. 1 Sampling distribution of means. 7: Sampling Experiment (Worksheet) is shared under a CC BY 4. Statsmedic unit 5 mc review. 0; the mean GPA for students in School B School B is 2. Recall that the standard normal distribution is also known as the z distribution. a. 7) You are testing chocolate chip cookies to estimate the mean number of chips per cookie. seed(0) #define number of samples. 8. 6. 1 Point Estimator 9. 5% confidence interval for the mean lengths of all studs cut by this machine. We are testing a non-directional or bi-directional claim that the relationship is significant. Thus, this is known as a "single sample proportion z test" or "one sample proportion z test. Therefore, if n p 0 and n ( 1 − p The distribution of all of these sample means is the sampling distribution of the sample mean. It's going to be the square root of 0. Center: The mean of the distribution is p. 43) 75 ≈ 0. A statistics Worksheet: The student will demonstrate the simple random, systematic, stratified, and cluster 2 AP Stats Notes, 1 Review, &amp; 1 Test to teach SAMPLING DISTRIBUTIONS OF SAMPLE PROPORTIONS &amp; SAMPLING DISTRIBUTIONS OF SAMPLE MEANS. 2 Confidence Interval for proportion 9. 1) Select left-tailed, in this case. Sampling and independent event. 28 or 28%. The Sampling Distribution of the Sample Proportion. Do younger people use Twitter more often than older people? In a random sample of 316 adult Internet users aged 18-29, 26% used Twitter. Example. For each of the following scenarios, indicate if you can assume that the sampling distribution of the sample statistic is approximately normally distributed. The common mistake here for students is to want to add (or worse to increases, the sampling distribution of the sample mean remains centered on the population mean, but becomes more compactly distributed around that population mean Normal population 0. That is, the difference in sample proportions is an unbiased estimator of the difference in population propotions. Bias means that the center (mean) of the sampling distribution is not equal to the true value of the parameter. Remember, the sample proportion, pˆ , is a statistic. Sampling Distribution of a Sample Proportion Robb T. 7 Rule for Sample Proportion. Objectives: Students will: Calculate the mean and standard deviation of the sampling distribution of a sample proportion and interpret the standard deviation Determine if the sampling distribution of is approximately Normal If appropriate, use a Normal distribution to calculate probabilities involving. 1 / 6. Now, just to make things a little bit concrete, let's imagine that we have a population of some kind. This result is known as the Central Limit Theorem (or CLT) Chapter 7: Section 6 - Sampling distribution of 끫뤢 Let the population proportion be p. It calculates the probability using the sample size (n), population proportion (p), and the specified proportions range (if you don't know the te the random variable below: Mean Number = ⸇柙 of Pets p( (or P( 26. Simply enter the appropriate values for a given Part 2: Find the mean and standard deviation of the sampling distribution. In this Lesson, we will focus on the sampling distributions for the sample mean, \(\bar{x}\), and the sample proportion The Sampling Distribution of the Difference Between Sample Proportions Center The mean of the sampling distribution is p 1 p 2. ** a) Sketch and clearly label the sampling distribution model for the sample proportion by naming the model and telling its mean and standard deviation. In SRS of every size n from any population with mean and finite standard deviation, when n is large, the sampling distribution of the sample mean is approximately Normal sampling distribution of a sample proportion and sample mean The first will be the sampling distribution of X (number of successes) and the second will be the sampling distribution of phat (proportion of successes). 5 - both are greater than 5. all possible samples taken from the population) will have a mean u p =p. Sample 4: X = 7, proportion with gene = 7/25 = 0. A random sample of n = 1000 judo matches is obtained, and it is determined that 510 of the matches are won by the athletes wearing a blue uniform. 1 0. n about the mean of the population from which the sample is drawn. pets of the. 1 Practice Ch 6. taken at random from a large population with underlying. Sampling distribution of a statistic is the probability Sep 2, 2009 · Sampling Distributions and The Central Limit Theorem Consider taking many (theoretically, all possible) samples of size n from a population. The first will be the sampling distribution of X (number of successes) and the second will be the sampling distribution of phat (proportion of successes). This calculator finds the probability of obtaining a certain value for a sample mean, based on a population mean, population standard deviation, and sample size. Sampling Distribution of a Sample Proportion What Makes this Sampling Distribution Different: When we do not have certain characteristics of a sampling distribution, it is still possible to still find the sampling distribution, but we must find it using the sample proportion. Suppose that a sample survey contacts an SRS of 1000 young adult Internet users and calculates the proportion ˆ p in this sample who watch online video. Aug 17, 2020 · This page titled 1. Feb 2, 2022 · The mean GPA for students in School A School A is 3. A sample is taken from this population and the data can be found in Example #2 of Data Sets- Sampling Distributions-Proportions. 1-2 walk you through one simulation (HW for Fri), and the exercises on p. 3. Apr 7, 2020 · A sampling distribution is a probability distribution of a certain statistic based on many random samples from a single population. 00065. 0. Three different teams for different ice cream manufacturers conduct experiments regarding this preference (i. Check Conditions. Summary Statistic – Compute a summary statistic. If we take another sample of 2,500 households, we will most likely get a different estimate for p. on. It should be 0. The student will demonstrate the simple random, systematic, stratified, and cluster sampling techniques. Check that the 10% condition is met. These measures are useful for understanding the distribution's center and spread, respectively, regardless of its shape. You want to ensure that the sample reflects the gender 4 of them: you obtained above appear in. (The sample can be found in the Sample #1 worksheet. Five trial cuts are made to check the machine’s calibration. 05717 . Sample 3: X = 10, proportion with gene = 10/25 = 0. 4 In Exercises 1-4, a population has a mean µ = 100 and a standard deviation σ = 15. Great AP Stats Exam review prep! Aligns with AP College Board's updated 2019-2020 objectives and standards for AP Statistics Unit 5: Sampling Distributions. (where n 1 and n 2 are the sizes of each sample). We will work out the sampling distribution for ^p for sample sizes of 1, 2, and 3. If you chose an SRS of size n from a population with a given proportion p, and compute the proportion p of the sample then the (A) sampling distribution of p is approximately normal Preview text. Sample 2: X = 9, proportion with gene = 9/25 = 0. Pets ̅’s: of times that each of the 4 different ̕配 p( 3 (or P(Etc. Find a 99% Confidence Interval and interpret. Random Sample – take a random sample of a fixed size n from the population (may be simulated). The mean length of the studs produced is 104. We can characterize this sampling distribution as follows: Center: The center of the distribution is = 0. the expected value and standard deviation of of the above random variable7. For sample proportions. Possible Answers: Correct answer: Explanation: This question is about a linear regression between time spent meditating and time spent studying. po. and n(1 p) 10. You should have noticed the sampling distribution has the following characteristics for shape, center, and spread: Shape : In some cases, the sampling distribution of p ˆ can be approximat ed by a Normal curve. 2 The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic. This unit covers how sample proportions and sample means behave in repeated samples. Notes 2 and 3. The sampling distribution of a statistic is the distribution of values of that statistic over all possible samples of a given size n from the population. 2 Worksheet #2 Name: Section: 8. 0 f(X) Sampling Distributionof the Sample Mean Sampling Distribution: n = 2 Sampling Distribution: n =16 Sampling Distribution: n = 4 Solution. Note: For this standard deviation formula to be accurate, our sample size needs to be 10 % or less of the population so we can assume independence. 004 inch. 43, Standard deviation p ( 1 − p) n = 0. To summarize the behavior of any random variable, we focus on three features of its distribution: the center, the spread 114 ACTIVITY 8: Sampling distribution of sample proportion p ^WhyWe have looked at the sampling distribution of the sample mean x because we want to be able to use the sample mean to give informati. The standard deviation in both schools is 0. Robb T. The standard deviation of the difference between our sample proportions is going to be just the square root of this. This is very important! This statement says that we are assuming the unknown population proportion, p, is equal to the value p 0. A statistic can be an unbiased estimator or a biased estimator of a parameter. Since this is true, then we can follow the same logic above. You sample 25 cookies and you find a sample mean of 10 chips per cookie. Question A (Part 2) About this unit. If you summarize the population of samples means you could get when averaging n If n ≥ 30 (or the sample size is large). Notice that the simulation mimicked a simple random sample of the population, which is a straightforward sampling strategy that helps The document provides solutions to probability problems involving sampling distributions and normal distributions. 20. Each random sample that is selected may have a different value assigned to the statistics being studied. 8 2. 1: Sampling Distributions 9. 880, which is the same as the parameter. Assume σ = 2. 40 (40%). A sample is large if the interval [p − 3 σ P ^, p + 3 σ P ^] lies wholly within the interval [0,1]. 2 - Sampling Distribution of the Sample Proportion. **1) It is generally believed that nearsightedness affects about 12% of children. The cafeteria claims that 96% of students are satisfied with the food and prices. Confirm that the average of the population of sample means, average number o. \ (n\) is the size of the random sample. 40 or 40%. The center is once again at around . 314039. 43) = 28. any useful statistics have their own versions of the Central Limit Theorem. This will help to reveal to students that the Part1A_MC_SamplingDistributions_ToCheck. Nov 23, 2020 · Generate a Sampling Distribution in R. Practice questions for the SAMPLING DISTRIBUTION FOR THE PROPORTION (9 questions) and the SAMPLING DISTRIBUTION FOR THE MEAN. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. If a random sample of n observations is taken from a binomial population with parameter p, the sampling distribution (i. sample_means = rep(NA, n) #fill empty vector with means. This could be thought of as the number of successes over the number Mean of Sampling Distribution of the Proportion. The null and alternative hypotheses are: H0: ˉx = 4. 2, we investigate the shape, center, and variability of the sampling distribution of a sample proportion. ), probability is. It calculates probabilities and finds unknown values using the normal distribution and properties of mean, standard deviation, and sample sizes. Before we begin, let’s make sure we review the terms and notation associated with proportions: p is the population proportion. Get a hint. They must connect it to the shape, center, and variability for the sampling distribution of a single proportion from Lesson 7. V. For this problem, we know p = 0. H. 998 inches with sample standard deviation 0. p, probability is. 99. 68% that X is within 1 standard deviation of mean. 1% chance to get a sample proportion of 50% or higher in a sample size of 75. The sampling distribution of the sample proportion is approximately Normal with Mean μ = 0. ioned in the previous worksheet, = and = /√ p㠱 hold for samples of any size n. 3 days ago · This sampling distribution of the sample proportion calculator finds the probability that your sample proportion lies within a specific range: P (p₁ < p̂ < p₂), P (p₁ > p̂), or P (p₁ < p̂). Be sure to use the same scale on both…so the number of successes goes from 10 to 30 and the proportion of successes goes from 0. Navigate to Page 5. Sampling Distribution of sample proportion Worksheet on Sampling Distribution of proportion 8. 05 to . 2 0. \ (\hat {p}\) is the sample proportion. These two sections of notes formally define a sampling distribution for a sample proportion and a sampling distribution for a difference in sample proportions. e. 1] Let's call 10 employees of a company a population. Distribution – display the distribution of the summary statistics. For the sampling distribution of all differences, the mean, , of all differences is the difference of the means . 4. Nov 24, 2020 · To find the mean and standard deviation of this sampling distribution of sample means, we can first find the mean of each sample by typing the following formula in cell U2 of our worksheet: =AVERAGE(A2:T2) We can then hover over the bottom right corner of the cell until a tiny + appears and double click to copy this formula to every other cell QTM 100 Worksheet 7: Sampling Distribution of means; Inferences for a single mean. 4 Review Review Questions on Sampling Distribution Chapter-9: Estimation using a single Sample 9. Assume lengths are normally distributed. Answer. The sample proportion ^p = x n = 510 1000 p ^ = x n = 510 Apr 30, 2024 · Sampling distribution of a statistic is the distribution of all possible values taken by the statistic when all possible samples of a fixed size n are taken from the population. A sample is a part or subset of the population. For samples of size 60, the distribution of the sample proportions appears roughly mound-shaped (with a slight leftward skew). 7% that X is within 3 standard deviations of mean. We will not be conducting this test by hand in this Sample Means – Comparing Population and Sampling Distributions 2 1. NEW for 2020! This resource now includes both a PDF for printing out a physical copy AND a fully editable Google Doc link for remote or distance learning! Assign the Google doc to Apr 23, 2022 · Figure 9. Answers are provided for both parts at the end of this document (pages 4). Distinguish between a sample and a populat. And, the sample proportions range from . Find the standard deviation of the sampling distribution of ˆ p. 30. A random sample of size is a sample that is chosen in such a way as to ensure that every sample of size has the same probability of being chosen. 1 is an introduction to sampling distributions, which includes sampling distributions for proportions and sampling distributions for means. Since the conditions are satisfied, p ^ will have a sampling distribution that is approximately normal Described by the spread of its sampling distribution; determined primarily by the size of the random sample. Confirm that the standard σσ σσ /√2. 5. Also, you will need to prepare two posterboards for dotplots. on the graph shows the distribution of masses in the population. n = 250 4. 18. 2 Quiz Ch 6. Suppose that 65% of all Americans prefer chocolate ice cream to vanilla ice cream. . H0: μ ≥ 4. Section 7. Success/Failure: n p ˆ 1000(. This will help to reveal to students that the variability of the sampling Section 7. Find the mean and the standard deviation of a sampling distribution of sample means with the given sample size n. It is a fixed value. 1. This seems to depend on both the sample size n and the population proportion p . 4. n whose shape can be approximated by a normal model if np 10. Randomization: Each sample was drawn randomly from its respective state. 3, σ = 7. Therefore, there is a 11. Sampling Distributions: 9. And that is approximately equal to, let's just take the square root, and we get this, 0. In the same way, we are interested in proportions. Sample means are used use quantitative variables we are interested in other statistics such as the median or mean or standard deviation of the variable. n is the size of the random sample. lm zp mz qn fr xc bi eq ru ps