Lesson Central Limit Theorem for Sample Means

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STATWAY STUDENT HANDOUT STUDENT NAME DATE INTRODUCTION Exploring Sampling Distributions In lesson 10.1.1 you found random samples from a population of 400 real acorn weights.

1 STATWAY STUDENT HANDOUT STUDENT NAME DATE INTRODUCTION Exploring Sampling Distributions In lesson you found random samples from a population of 400 real acorn weights. In this lesson you will use computer simulations to further your understanding of sampling distributions of sample means. You will pay special attention to the shape of the population distribution and how it compares to the shape of the sampling distribution for different sample sizes. This lesson uses the applet from your homework in Lesson : TRY THESE Sampling from a Uniformly Distributed Population To begin, we initially sample from a population that is uniformly distributed. We will simulate 1000 random samples using three different sample sizes. Set the population to uniform by adjusting the top input box. The population distribution is the top graph. Set the Population type to Uniform, and then click the Reset button. 1 Describe the shape of this population. What are the possible values in this population?

2 STATWAY STUDENT HANDOUT 2 2 Does it appear that some of the (discrete) population values are more likely to occur than others? Below the population curve is a plot of sample data. On the side there is a Sample button, and two input boxes. Set the sample size, n, to 2 Set the number of samples, N, to 1000 Click Sample In the Sample data plot, you see the results of a single sample of size 2. Below this, in the Sample means plot, you see the sample means from 1000 random samples of size 2. 3 What is the range of the sample means in this simulated sampling distribution? 4 What is the shape of the simulated sampling distribution? 5 Sketch the simulated sampling distribution, when n = 2, in the box below. Run the simulation for sample sizes n = 10, and n = 30. Sketch your results in the boxes below. For each simulation, set N = 1000 so 1000 random samples are created. Uniform Population n = 2 n = 10 n = 30

3 STATWAY STUDENT HANDOUT 3 6 What happens to the center and variability of the simulated sampling distribution when the sample size increases to n = 30? 7 Identify the mean and standard deviation (or error) of the simulated sampling distribution when n = 30. How do they compare to the mean and standard deviation of the population? Sampling from a Skewed Right Population We will now sample from a population whose distribution is skewed right. We will again simulate 1000 random samples using three different sample sizes. Set the Population type to Skewed, and then click the Reset button. If you want to make the population distribution more skewed, drag your mouse pointer along the population graph and adjust the shape. Set the sample size, n, to 2 and set the number of samples, N, to 1000 Click Sample 8 What is the range of the sample means in this simulated sampling distribution? 9 What is the shape of the simulated sampling distribution?

4 STATWAY STUDENT HANDOUT 4 10 Sketch the simulated sampling distribution, when n = 2, in the box below. Run the simulation for sample sizes n = 10, and n = 30. Sketch your results in the boxes below. For each simulation, set N = 1000 so 1000 random samples are created. Skewed Right Population n = 2 n = 10 n = What happens to the center and variability of the sampling distribution when the sample size increases to n = 30? 12 Identify the mean and standard deviation (or error) of the sampling distribution when n = 30. How do they compare to the mean and standard deviation of the population? Sampling from Any Population We now sample from a population with an unusual distribution. Try to create a population distribution so weird, so wild, that even when we increase the sample size, the sampling distribution will not become bell shaped. We will again simulate 1000 random samples using three different sample sizes. Set the Population type to Custom, and then click the Reset button. Click on the population graph to draw the wildest population you can imagine. Set the sample size, n, to 2 and set the number of samples, N, to Click Sample.

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6 STATWAY STUDENT HANDOUT 6 13 Sketch your population in the box below. 14 What is the range of the sample means in this simulated sampling distribution? 15 What is the shape of the sampling distribution? 16 Sketch the sampling distribution, when n = 2, in the box below. Run the simulation for sample sizes n = 10, and n = 30. Sketch your results in the boxes below. For each simulation, set N = 1000 so 1000 random samples are created. The Wild Population n =2 n =10 n = What happens to the center and variability of the sampling distribution when the sample size increases to n = 50?

7 STATWAY STUDENT HANDOUT 7 18 Identify the mean and standard deviation (or error) of the sampling distribution when n = 50. How do they compare to the mean and standard deviation of the population? Sampling from a Normal Population For the last example, we sample from a normal population. Set the Population type to Bell shaped, and then click the Reset button. Set the sample size, n, to 2 and set the number of samples, N, to Click Sample 19 Does the distribution of sample means appear normal? 20 Experiment with different values of n, up to 100, clicking Sample for each. Are you able to construct any distribution of sample means that does not appear normal for any sample size? 21 When sampling from a normal population, is any restriction upon sample size necessary in order for the distribution of sample means to be normal?

8 STATWAY STUDENT HANDOUT 8 YOU NEED TO KNOW The Given any population with mean µ and standard deviation σ, the sampling distribution of sample means sampled with replacement from random samples of size n will have a distribution that approaches normality with increasing sample size. The mean and standard error of the sampling distribution are: The criteria for the approximate normality of a sampling distribution are that either the population from which we are sampling should be normal, or the sample size should be greater than 30. Very non-normal populations may require samples substantially larger than 30. TAKE IT HOME 1 A possible population distribution is one that is triangular. Use the applet to construct a triangular population distribution. A Use the Central Limit Theorem to calculate the mean and standard error of the sampling distribution when the sample size becomes very large. Use the population mean and standard deviation. B How does the shape of the sampling distribution change as the sample size increases?

9 STATWAY STUDENT HANDOUT 9 C How close are the values from (a) to the mean and standard error of your sampling distribution? 2 Yet another possible population distribution is one that is bimodal. For example, suppose you took samples of the length of human hair in your school. It is probable (but not certain!) that young women tend to have long hair and young men have short hair. Use the applet to construct a bimodal population distribution. A Use the Central Limit Theorem to calculate the mean and standard error of the sampling distribution when the sample size becomes very large. Use the population mean and standard deviation. B How does the shape of the sampling distribution change as the sample size increases? C How close are the values from (a) to the mean and standard error of your sampling distribution?

10 STATWAY STUDENT HANDOUT 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 carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/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.

Lesson Least Squares Regression Line as Line of Best Fit

Lesson Least Squares Regression Line as Line of Best Fit STATWAY STUDENT HANDOUT STUDENT NAME DATE INTRODUCTION Comparing Lines for Predicting Textbook Costs In the previous lesson, you predicted the value of the response variable knowing the value of the explanatory