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STATWAY? STUDENT HANDOUTLesson 10.1.1 Sampling Distributions of Sample MeansSTUDENT NAMEDATEIntroduction In Modules 7–9, you used sampling distributions of sample proportions to estimate or test a claim about a population proportion. In this module, you will focus on sample means from random samples. The first part of this lesson asks you to recall what you know about sampling distributions of sample proportions. You will use these ideas to make a connection to the sampling distribution of sample means.Try These – Part 1Polling Data – ProportionsThis dotplot below shows a sampling distribution of sample proportions, formed from a simulation. Two hundred samples of size 50 were simulated. Sample proportions, representing the proportion of voters in the sample who voted for candidate A, were calculated. And these sample proportions are all represented in the dotplot below.16929101403351What does a dot represent in this sampling distribution?2Based on the dotplot, give a single number estimate for the population proportion of voters in the city who support Candidate A. Explain your answer. The simulation was conducted two more times, using different sample sizes. Each time, 200 samples were collected. One simulation used samples of size 25, while the other used samples of size 100. The results are below.3Which of the graphs do you believe represents samples of size 25? Which graph represents samples of size 100? Explain your answer. Polling Data – MeansImagine that you want to examine the amount of time it takes someone to vote from the time that person reaches the voting station until the vote is cast. In fact, you would like to know the average amount of time it takes a person to vote. To explore this idea further we have created a simulation.Two hundred random samples were simulated. Each sample consisted of 50 people with various voting times. The mean (or average) voting time (in minutes) was calculated for each sample. These sample means are plotted below.4What does a dot represent in the dotplot of the sampling distribution? 5Give a single number to estimate the mean (or average) amount of time it takes an individual to vote. Explain your answer. 6Suppose you were to run the simulation again using a sample size of 100. How would the sampling distribution from samples of size 100 differ from that of samples of size 50, which is shown above? 7Suppose that you plotted the voting times of all 50 people in one of your samples (instead of plotting the average voting times from all samples). In this case, each dot would represent a single person. How would the distribution of this dotplot differ from that of the sampling distribution dotplot? Try These – Part 2You will now continue your investigation of the sampling distribution of sample means, but this time you will use real data collected by a group of students in Austin, Texas. Students in Austin collected and measured the mass of acorns from live oak trees. Botanists use acorn weight as a factor in predicting future oak tree germination in a region. A decline in the growth of new oak trees could have a severe impact on the ecosystem in this part of the country because many birds and small mammals eat acorns. Collecting a Sample1Start by randomly selecting nine acorns from the Acorn Mass Table. Use a table of random digits or a random number generator to select the nine acorns. Write down the mass for the nine acorns in your random sample. The mass is presented in grams. 2Calculate the mean mass for your sample of nine acorns. 3Ten random samples of 9 acorns were collected, and the mean mass of each sample was calculated. These ten means are plotted below. Add your sample mean to the dotplot. 4As time permits, repeat the process of randomly selecting nine acorns several times. Calculate the mean mass for the sample, and plot the sample mean on the distribution above. Write the value of each new sample mean below. A dotplot and histogram of the mean masses created from 500 random samples, each with nine acorns, are shown below. The values are shown in grams.5In the dotplot of the sampling distribution, what does a dot represent? 6How might you use the distribution of sample means to estimate the true mean mass of all individual acorns measured by the students? 7What is your estimate of the mean mass of individual acorns? 8In the context of a distribution of estimates, such as sample means, recall that any deviation from the population mean is error. This is why we refer to the standard deviation of all estimates as the standard error. Estimate the standard error of the sampling distribution above. 9Do you think the standard deviation of the individual acorn measurements for the collection of 400 acorns is larger, smaller, or about the same as the standard error? Explain your answer. The simulation was conducted two more times, using different sample sizes. Each time, 500 samples were collected. One simulation used samples of size 25, while the other used samples of size 100. The results are below.10Which of the graphs do you believe represents samples of size 25? Which graph represents samples of size 100? Explain your answer. Finding the Standard ErrorThe following table gives the standard error of sample means for different sample sizes. Sample SizeStandard Error (rounded to hundredths)11.0240.5190.34250.201000.104000.0510,0000.0111What relationship do you notice between sample size and the value of the standard error? 12The standard deviation of the population of acorns is 1.02 grams. The population can be considered the collection of samples of size n = 1. The formula for standard error depends on sample size. Study the patterns in the table and see if you can figure out the formula for the standard error based on the sample size. Take It HomeBrowse to the website: . This applet simulates random samples from a population with a bell shaped distribution with mean 25 and standard deviation 8.22. It plots the sample means of these simulated samples.To use the applet:Set the population type to Bell Shaped.Set the number of samples in the simulation, N, to 100.Set the sample size, n, as indicated in the table below.Click Sample after changing the sample size.The screen shot below shows the simulation results for N = 100 samples of size n = 2.The bell shaped curve on the top shows the population distribution. The population has mean 25 and standard deviation 8.22.The histogram in the middle shows the 2 values obtained in a single sample. These two values have a mean of 24.75 and a standard deviation of 10.15.The histogram on the bottom shows the sample mean calculated from each of the 100 samples of size 2. Note that these means have a mean of 24.78 and a standard deviation of 6.51. Recall, the standard deviation of a set of means is called the standard error. The standard error of the sample means in this case is 6.51.1For the various sample sizes in the table below, run the simulation, and give the means and standard errors of the distribution of sample means (corresponding to the third histogram in the simulation).Sample Size2491625Mean of Sample MeansStandard Error2As the sample size increases, how does the standard error change?3As the sample size increases, what happens to the mean of the sample means?+++++This lesson is part of STATWAY?, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway? was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit . For the most recent version of instructional materials, visit kernel.+++++STATWAY? and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation. ................
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