STAT 200 LAB 3 1 2 SAMPLING DISTRIBUTIONS STAT 200 Lab Activity for Section 3 1 2 Sampling Distributions Learning Objectives Recognize that a sampling distribution shows how sample statistics tend to Recognize that statistics from random samples tend to be centered at the vary population parameter Estimate the standard error of a statistic from its sampling distribution Explain how sample size affects a sampling distribution Activity 1 Sleep Tracking Class Data Creating a sampling distribution revisited In the last lab you calculated two sample proportions pn 30 for n 30 and pn 60 for n 60 from our sleep tracking data set Next we ll compare the sampling distributions and see the actual proportion of yes responses the parameter we were estimating The situation We ve gained access to the original survey data Among all students across all sections of STAT 200 UP True proportion of the class who said yes to using a sleep tracker p 0 11 You and your classmates random sample statistics built two sampling distributions of the sample proportion One from random samples of size 30 and one from random samples of size 60 View the provided sampling distributions of the sample proportion 1 Describe the shape of each sampling distribution of the sample proportion e g symmetric skewed bell shaped 2 Estimate the center of each distribution How close is each to the true proportion p Why might they differ slightly 3 Which sampling distribution of the sample proportion has more variability spread How can you tell 12 15 25 University 2025 The Pennsylvania State STAT 200 LAB 3 1 2 SAMPLING DISTRIBUTIONS 4 What causes the difference in variability between the two sampling distributions of the sample proportion 5 Since we have a bell shaped distribution we can apply what we know about the 95 rule to estimate the range where most sample means are expected to fall within about 2 standard errors of the population mean a Look at the sampling distributions for n 30 and n 60 Estimate the lower and upper bounds where about 95 percent of the sample proportions fall Express each as an interval n 30 n 60 b How do these 95 ranges compare How does the sample size affect the width of this range 6 If you needed a more precise estimate one that varies less from sample to sample of the true proportion would you prefer a sample of n 30 or n 60 Why 7 Predict how a sampling distribution of the sample proportion would look if you took random samples of 100 students from the survey data How would its spread and center compare to the n 30 and n 60 distributions 8 In this lab you saw nearly 2000 random samples But in real research we usually only take one How does seeing a full sampling distribution help you judge whether that one sample result is typical or unusual Activity 2 Band Size The Rock Performers 3e dataset in StatKey contains the number of members in a population of rock bands You ll analyze the mean number of members to explore the variability and reliability of estimates and to see how sampling distributions provide insight into real world inference especially when you only get one sample 12 15 25 University 2025 The Pennsylvania State STAT 200 LAB 3 1 2 SAMPLING DISTRIBUTIONS Part 1 Population Distribution In StatKey the population distribution is shown by default when you load the dataset Go to Sampling Distributions Mean Choose the built in dataset Rock Performers 3e People in Band Before generating any samples the graph in the upper right shows the full population distribution of the number of band members 1 In this population dotplot what does each dot represent 2 What variable is being measured and what type of variable is it 3 In the table below complete the row for the population distribution use proper notation along with the values in the table Part 2 Single Sample Distribution 4 Set sample size n 10 select Generate 1 Sample and view the sample distribution a What does each dot represent in the sample distribution b What is the variable 5 In the table below complete the row for the sample distribution use proper notation along with the values in the table Part 3 Sampling Distribution n 10 6 Keep n 10 and generate about 5000 samples to simulate a sampling distribution of sample means 12 15 25 University 2025 The Pennsylvania State STAT 200 LAB 3 1 2 SAMPLING DISTRIBUTIONS a What does each dot represent in the sampling distribution of sample means b What is the variable 7 In the table below complete the row for the sampling n 10 distribution of the sample mean just the values for the center and spread Part 4 Larger Sample Size n 40 Select Reset Plot to refresh the StatKey app Change the sample size to n 40 and generate 5000 samples again to create a new sampling distribution of the sample mean 8 In the table below complete the row for the sampling n 40 distribution of the sample mean just the values for the center and spread Distribution of values Shape Population Sample Sampling n 10 Sampling n 40 Use exact values from the output do not round Part 5 Making Connections Center mean Spread sd or se SE SE 9 Which distributions have the same or nearly the same center mean Why does that make sense 10 Based on the activities as sample size increases the variability of the sampling distribution answer1 and the sample statistics tend to fall answer2 to the true value of the answer3 Options for answer1 increases decreases stays the same Options for answer2 farther closer equally close Oputions for answer3 sample statistic population parameter 11 How does the shape of the sampling distributions of the sample means compare to the shape of the population Why might they differ 12 15 25 University 2025 The Pennsylvania State STAT 200 LAB 3 1 2 SAMPLING DISTRIBUTIONS 12 A sampling distribution is not the same as a sample distribution because a dot in the sampling distribution represents a answer1 while a dot in a single sample distribution represents a answer2 Options a Sample mean of a random sample b Average number of band members in the population c Number of band members in one band d Number of bands in the population 13 Suppose you take a random sample of 40 bands from the population and find the average number of members is 4 5 Based on the sampling distribution of sample means for n 40 would this sample mean be considered typical or unusual 12 15 25 University 2025 The Pennsylvania State