Introduction to Sampling Distributions

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.

We then use the normal approximation of the sample proportion to determine the control limits. Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7 | 13 P-Chart Example Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7 | 14

Recently Viewed Presentations

  • Woolworths closes!

    Woolworths closes!

    This time there was an area of high pressure causing north westerly winds which blew everything in our direction. Ask the children to show you which direction is NW and also the other compass points. ... You could look at...
  • National Health Programmes 1. INTRODUCTION

    National Health Programmes 1. INTRODUCTION

    National Health Programs-India INTRODUCTION Dr. KANUPRIYA CHATURVEDI Lesson Objectives To know about the National Health Programs ( NHP) in India Understand the relevance of the NHP Understand difference between vertical and horizontal programs.
  • Organic halides - KSU

    Organic halides - KSU

    Organic Halides and their uses. Organic Halides are a large class of natural and synthetic chemicals that contain one or more halogens (fluorine, chlorine, bromine, or iodine) combined with carbon and other elements.
  • Discrepancies between API-510 7.4.2 and API-579 Part 5 Level 1

    Discrepancies between API-510 7.4.2 and API-579 Part 5 Level 1

    The API-510 paragraph 7.4.2 Evaluation of locally thinned corroded areas is not consistent with the API-579 Part 4 Level 1 Point Thickness Reading (PTR) method. The differences lie on the lack of a limit for metal loss uniformity for thickness...
  • AAAA Chapter's Update

    AAAA Chapter's Update

    Centric Grouping AAAA Chapter Grouping by Centric Category ARNG/Reserve Active CONUS Active OCONUS Industry/Blended Student 37 14 8 10 3. Chapter Membership Tracking and Transfers. CURRENT PROCEDURE. ... Mid-Atlantic, Flying Tigers, Bavarian. Pre-2015 Procedure.
  • Project Management 3e. - Gray and Larson

    Project Management 3e. - Gray and Larson

    A project is an interrelated set of activities with a definite starting and ending point, which result in a unique outcome for a specific allocation of resources. Project Defined: A complex, non-routine, one-time effort limited by time, budget, resources, and...
  • begin - Coe College

    begin - Coe College

    Postsynaptic Potentials Lesson 10 Postsynaptic Potentials Signaling in Dendrites & Soma Vm changes dendrites & soma Excitatory: + Inhibitory: - ~ Postsynaptic Potentials - PSPs Chemically-gated ion channels Graded Summation Duration 10 - 100 milliseconds Velocity Fast approximately speed of...
  • Dyspnea in a Heart Failure Patient - Dartmouth-Hitchcock

    Dyspnea in a Heart Failure Patient - Dartmouth-Hitchcock

    High Risk Groups for SCD Sudden Cardiac Death: Management Strategies Salvage and treat (too few are salvaged). Predict and prevent (pathophysiology complex, positive predictive value of risk stratifiers relatively low). * This slide demonstrates the reduction in mortality for patients...