INFERENCE: CONFIDENCE INTERVALS

Slides:



Advertisements
Similar presentations
Proportions. A question has 4 possible answers, only one which is correct. You randomly guess the correct answer. With 20 such questions, the distribution.
Advertisements

Chapter 7 Statistical Inference: Confidence Intervals
Confidence Intervals Chapter 7. Rate your confidence Guess my mom’s age within 10 years? –within 5 years? –within 1 year? Shooting a basketball.
5. Statistical Inference: Estimation
Chapter 11- Confidence Intervals for Univariate Data Math 22 Introductory Statistics.
© 2013 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems Introductory Statistics: Exploring the World through.
Lecture 5 Outline – Tues., Jan. 27 Miscellanea from Lecture 4 Case Study Chapter 2.2 –Probability model for random sampling (see also chapter 1.4.1)
Parameter Estimation Chapter 8 Homework: 1-7, 9, 10.
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 7-1 Introduction to Statistics: Chapter 8 Estimation.
Chapter 8 Estimation: Single Population
Chapter Topics Confidence Interval Estimation for the Mean (s Known)
Fall 2006 – Fundamentals of Business Statistics 1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter 7 Estimating Population Values.
Chapter 11: Inference for Distributions
Chapter 7 Estimating Population Values
Confidence Intervals Chapter 10. Rate your confidence Name my age within 10 years? within 5 years? within 1 year? Shooting a basketball at a wading.
Confidence Intervals for the Mean (σ Unknown) (Small Samples)
Estimation Goal: Use sample data to make predictions regarding unknown population parameters Point Estimate - Single value that is best guess of true parameter.
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 7-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter.
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved Estimation and Confidence Intervals.
+ DO NOW What conditions do you need to check before constructing a confidence interval for the population proportion? (hint: there are three)
Chapter 6 Confidence Intervals.
Chapter 8: Statistical Inference: Confidence Intervals
1 Math 10 Part 5 Slides Confidence Intervals © Maurice Geraghty, 2009.
Confidence Intervals (Chapter 8) Confidence Intervals for numerical data: –Standard deviation known –Standard deviation unknown Confidence Intervals for.
Chapter 7 Statistical Inference: Confidence Intervals
© 2003 Prentice-Hall, Inc.Chap 6-1 Business Statistics: A First Course (3 rd Edition) Chapter 6 Sampling Distributions and Confidence Interval Estimation.
Confidence Intervals Chapter 6. § 6.1 Confidence Intervals for the Mean (Large Samples)
Confidence Intervals about a Population Proportion
Estimation Bias, Standard Error and Sampling Distribution Estimation Bias, Standard Error and Sampling Distribution Topic 9.
Review of Chapters 1- 5 We review some important themes from the first 5 chapters 1.Introduction Statistics- Set of methods for collecting/analyzing data.
University of Ottawa - Bio 4118 – Applied Biostatistics © Antoine Morin and Scott Findlay 08/10/ :23 PM 1 Some basic statistical concepts, statistics.
Statistical Sampling & Analysis of Sample Data
STA291 Statistical Methods Lecture 18. Last time… Confidence intervals for proportions. Suppose we survey likely voters and ask if they plan to vote for.
Copyright © 2012 Pearson Education. All rights reserved © 2010 Pearson Education Copyright © 2012 Pearson Education. All rights reserved. Chapter.
Chap 7-1 A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc. A Course In Business Statistics 4 th Edition Chapter 7 Estimating Population Values.
Lecture 7 Dustin Lueker. 2  Point Estimate ◦ A single number that is the best guess for the parameter  Sample mean is usually at good guess for the.
Agresti/Franklin Statistics, 1 of 87  Section 7.2 How Can We Construct a Confidence Interval to Estimate a Population Proportion?
Lesson 9 - R Chapter 9 Review.
Confidence Intervals for the Mean (Small Samples) 1 Larson/Farber 4th ed.
Chap 7-1 A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc. A Course In Business Statistics 4 th Edition Chapter 7 Estimating Population Values.
CONFIDENCE INTERVALS.
Chapter 8: Confidence Intervals based on a Single Sample
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 7-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter.
Confidence Intervals for a Population Mean, Standard Deviation Unknown.
+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 8: Estimating with Confidence Section 8.3 Estimating a Population Mean.
Confidence Intervals for a Population Proportion Excel.
Ch 8 Estimating with Confidence 8.1: Confidence Intervals.
Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.2 Confidence Intervals for the Mean (  Unknown)
Section 6.2 Confidence Intervals for the Mean (Small Samples) Larson/Farber 4th ed.
Chapter 23 The t-distribution, Confidence Intervals, and Significance Testing for Quantitative Data.
Ex St 801 Statistical Methods Inference about a Single Population Mean (CI)
Review Confidence Intervals Sample Size. Estimator and Point Estimate An estimator is a “sample statistic” (such as the sample mean, or sample standard.
SECTION 7.2 Estimating a Population Proportion. Where Have We Been?  In Chapters 2 and 3 we used “descriptive statistics”.  We summarized data using.
Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 8 Statistical Inference: Confidence Intervals Section 8.1 Point and Interval Estimates.
Confidence Intervals Dr. Amjad El-Shanti MD, PMH,Dr PH University of Palestine 2016.
In your groups, go through all four steps of a confidence interval: The “Country Taste” bread making company wants to estimate the actual weight of their.
Section 6.2 Confidence Intervals for the Mean (Small Samples) © 2012 Pearson Education, Inc. All rights reserved. 1 of 83.
© 2001 Prentice-Hall, Inc.Chap 8-1 BA 201 Lecture 12 Confidence Interval Estimation.
 Normal Curves  The family of normal curves  The rule of  The Central Limit Theorem  Confidence Intervals  Around a Mean  Around a Proportion.
Topic 12 Sampling Distributions. Sample Proportions is determined by: = successes / size of sample = X/n If you take as SRS with size n with population.
Chapter 8 Confidence Intervals Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin.
Chapter Eight Estimation.
Estimation and Confidence Intervals
FINAL EXAMINATION STUDY MATERIAL II
Chapter 6 Inferences Based on a Single Sample: Estimation with Confidence Intervals Slides for Optional Sections Section 7.5 Finite Population Correction.
Sampling Distributions and Estimation
Chapter 6 Confidence Intervals.
Introduction to Inference
Chapter 6 Confidence Intervals.
Confidence Intervals: The Basics
Presentation transcript:

INFERENCE: CONFIDENCE INTERVALS Math 1530 - Chapter 8 INFERENCE: CONFIDENCE INTERVALS Chapter 8

8.1 What are Point and Interval Estimates of Population Parameters?

Point Estimate and Interval Estimate A point estimate is a single number that is our “best guess” for the parameter An interval estimate is an interval of numbers within which the parameter value is believed to fall.

Point Estimate vs. Interval Estimate A point estimate doesn’t tell us how close the estimate is likely to be to the parameter An interval estimate is more useful It incorporates a margin of error which helps us to gauge the accuracy of the point estimate aiaccess.net

Properties of Point Estimators Property 1: Good estimator has sampling distribution centered at the parameter An estimator with this property is unbiased Sample mean [proportion] is an unbiased estimator of population mean [proportion]

Properties of Point Estimators Property 2: Good estimator has smaller standard error than other estimators Falls closer than other estimates to parameter Sample mean has a smaller standard error than sample median montenasoft.com

Confidence Interval An interval containing the most believable values for a parameter The probability that the interval contains the parameter is the confidence level Close to 1, most commonly 0.95 comfsm.fm

Logic of Confidence Intervals  

Logic of Confidence Intervals   gmatclub.com

Margin of Error Measures accuracy of point estimate in estimating a parameter Multiple of standard error 1.96 standard errors for a 95% confidence interval of p

CI for a Proportion 19% of 1823 respondents agreed, “It is more important for a wife to help her husband’s career than to have one herself.” Assuming standard error is 0.01, calculate a 95% CI for p who agreed. Margin of error = 1.96*se=1.96*0.01=0.02 95% CI = 0.19±0.02 or (0.17 to 0.21) We predict that the population proportion who agreed is somewhere between 0.17 and 0.21. 2.bp.blogspot.com

8.2 How Can We Construct a Confidence Interval to Estimate a Population Proportion?

Finding the 95% CI for p  

Protecting the Environment “Are you willing to pay much higher prices in order to protect the environment?” 1154 respondents, 518 willing Find and interpret a 95% confidence interval

Needed Sample Size for CI of P   eaststaffsbc.gov.uk

Confidence Levels Other than 95% 95% confidence means a 95% chance that contains p With probability 0.05, CI misses p To increase chance of a correct inference, use larger confidence level, such as 0.99

Confidence Levels Other than 95% We must compromise between desired margin of error and desired confidence of a correct inference thefreemarketeers.files.wordpress.com

CI for a Proportion Of 598 respondents, 366 said yes, “If the wife in a family wants children, but the husband does not, is it all right for the husband to refuse to have children?” Assuming standard error is 0.01, calculate a 99% CI for p who said yes. Use calculator: 1-PropZInt vitaminsandnutrition.files.wordpress.com

Error Probability for CI Method?  

Summary   pirate.shu.edu

Effects of Confidence Level and Sample Size on Margin of Error The margin of error for a confidence interval: Increases as confidence level increases Decreases as sample size increases

Interpretation of Confidence Level For 95% confidence intervals, in the long run about 95% of those intervals would give correct results, containing the population proportion, p

8.3 How Can We Construct a Confidence Interval to Estimate a Population Mean?

Construct CI for μ  

Properties of the t Distribution Bell-shaped and symmetric about 0 Probabilities depend on degrees of freedom, df = n-1 Has thicker tails (more spread out) than standard normal distribution

t-Distribution

95% Confidence Interval for μ boost.org

t-CI for Other Confidence Levels?  

Table B

If the Population is Not Normal, is the Method Robust? Basic assumption is normal population distribution Many distributions far from normal Abnormal distributions are okay for large n because of the CLT CIs using t-scores usually work well: except for extreme outliers, the method is robust urbanbacon.com

Standard Normal Distribution is t-Distribution with df = ∞

8.4 How Do We Choose the Sample Size for a Study?

Sample Size for Estimating p  

Sample Size For Exit Poll   n = 0.58(1-0.58)*1.96^2/0.04^2 = 584.9 Even if we didn’t have a guess for p-hat, we could do the worst-case scenario: p-hat = 0.50. n = 0.50(1-0.50)*1.96^2/0.04^2 = 600.25

Sample Size for Estimating μ   statcan.gc.ca

Education of Black South Africans   Crude estimate of =range/6=18/6=3 n = 1.96^2*3^2/1^2 = 35

Factors Affecting Choice of Sample Size? Desired precision, as measured by the margin of error, m Confidence level Variability in data - If subjects have little variation (σis small), we need fewer data than with substantial variation Financial schoolloans.org

Using a Small n The t methods for μ are valid for any n, but look for extreme outliers or great departures from normal For a population proportion, p, the method works poorly for small samples because the CLT no longer holds stat.psu.edu

CI for p with Small Samples   businessandlearning.com