# Introduction to Sampling Distributions

Chapter 7 Introduction to Sampling Distributions Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania Terms, Statistics & Parameters

Terms: Population, Sample, Parameter, Statistics Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7|2 Why Sample? At times, wed like to know something about the population, but because our time, resources, and efforts are limited, we can take a sample to learn about the population.

Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7|3 Types of Inference 1) Estimation: We estimate the value of a population parameter. 2) Testing: We formulate a decision about a population parameter. 3) Regression: We make predictions about the value of a statistical variable.

Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7|4 Sampling Distributions To evaluate the reliability of our inference, we need to know about the probability distribution of the statistic we are using. Typically, we are interested in the sampling distributions for sample means and sample proportions.

Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7|5 The Central Limit Theorem (Normal) If x is a random variable with a normal distribution, mean = , and standard deviation = , then the following holds for any sample size: Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved.

7|6 The Standard Error The standard error is just another name for the standard deviation of the sampling distribution. Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7|7 The Central Limit Theorem (Any Distribution)

If a random variable has any distribution with mean = and standard deviation = , the sampling distribution of x will approach a normal distribution with mean = and standard deviation = n as n increases without limit. Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7|8 Sample Size Considerations For the Central Limit Theorem (CLT) to be

applicable: If the x distribution is symmetric or reasonably symmetric, n 30 should suffice. If the x distribution is highly skewed or unusual, even larger sample sizes will be required. If possible, make a graph to visualize how the sampling distribution is behaving. Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7|9

Critical Thinking Bias A sample statistic is unbiased if the mean of its sampling distribution equals the value of the parameter being estimated. Variability The spread of the sampling distribution indicates the variability of the statistic. Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7 | 10

r Sampling Distributions for Proportions p n If np > 5 and nq > 5, then p can be approximated by a normal variable with mean and standard deviation p p and pq

p n Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7 | 11 Continuity Corrections Since p is discrete, but x is continuous, we have to make a continuity correction.

For small n, the correction is meaningful. Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7 | 12 Control Charts for Proportions Used to examine an attribute or quality of an observation (rather than a measurement). We select a fixed sample size, n, at fixed time intervals, and determine the sample proportions at each interval.