Statistics Confidence Intervals

Slides:



Advertisements
Similar presentations
Introduction to Confidence Intervals using Population Parameters Chapter 10.1 & 10.3.
Advertisements

Chapter 19 Confidence Intervals for Proportions.
Confidence Intervals for Proportions
© 2010 Pearson Prentice Hall. All rights reserved Confidence Intervals for the Population Mean When the Population Standard Deviation is Unknown.
1 A heart fills with loving kindness is a likeable person indeed.
PSY 307 – Statistics for the Behavioral Sciences
Standard error of estimate & Confidence interval.
Estimation Goal: Use sample data to make predictions regarding unknown population parameters Point Estimate - Single value that is best guess of true parameter.
Chapter 7 Confidence Intervals and Sample Sizes
1/2555 สมศักดิ์ ศิวดำรงพงศ์
Chapter 11: Estimation Estimation Defined Confidence Levels
© 2002 Thomson / South-Western Slide 8-1 Chapter 8 Estimation with Single Samples.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc. Chap 8-1 Confidence Interval Estimation.
Population All members of a set which have a given characteristic. Population Data Data associated with a certain population. Population Parameter A measure.
Estimation in Sampling!? Chapter 7 – Statistical Problem Solving in Geography.
Ch 8 Estimating with Confidence. Today’s Objectives ✓ I can interpret a confidence level. ✓ I can interpret a confidence interval in context. ✓ I can.
Introduction to Inference Confidence Intervals for Proportions.
Determination of Sample Size: A Review of Statistical Theory
1 Section 10.1 Estimating with Confidence AP Statistics January 2013.
6.1 Inference for a Single Proportion  Statistical confidence  Confidence intervals  How confidence intervals behave.
Sampling distributions rule of thumb…. Some important points about sample distributions… If we obtain a sample that meets the rules of thumb, then…
Confidence Intervals Target Goal: I can use normal calculations to construct confidence intervals. I can interpret a confidence interval in context. 8.1b.
Introduction to Confidence Intervals using Population Parameters Chapter 10.1 & 10.3.
Confidence Intervals Population Mean σ 2 Unknown Confidence Intervals Population Proportion σ 2 Known Copyright © 2013 Pearson Education, Inc. Publishing.
Statistics 300: Elementary Statistics Sections 7-2, 7-3, 7-4, 7-5.
© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 8. Parameter Estimation Using Confidence Intervals.
Chapter 11: Estimation of Population Means. We’ll examine two types of estimates: point estimates and interval estimates.
+ The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE Chapter 8: Estimating with Confidence Section 8.1 Confidence Intervals: The.
Confidence Intervals for a Population Proportion Excel.
Ch 8 Estimating with Confidence 8.1: Confidence Intervals.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-1 Business Statistics, 3e by Ken Black Chapter.
INFERENCE Farrokh Alemi Ph.D.. Point Estimates Point Estimates Vary.
10.1 – Estimating with Confidence. Recall: The Law of Large Numbers says the sample mean from a large SRS will be close to the unknown population mean.
ESTIMATION OF THE MEAN. 2 INTRO :: ESTIMATION Definition The assignment of plausible value(s) to a population parameter based on a value of a sample statistic.
Ex St 801 Statistical Methods Inference about a Single Population Mean (CI)
Lab Chapter 9: Confidence Interval E370 Spring 2013.
SECTION 7.2 Estimating a Population Proportion. Where Have We Been?  In Chapters 2 and 3 we used “descriptive statistics”.  We summarized data using.
Chapter Seven Point Estimation and Confidence Intervals.
© 2001 Prentice-Hall, Inc.Chap 8-1 BA 201 Lecture 12 Confidence Interval Estimation.
Welcome to Week 07 College Statistics
Inference: Conclusion with Confidence
CHAPTER 8 Estimating with Confidence
CHAPTER 8 Estimating with Confidence
Estimation and Confidence Intervals
Chapter 6 Confidence Intervals.
More on Inference.
CHAPTER 8 Estimating with Confidence
Chapter 6 Confidence Intervals.
Inferences Based on a Single Sample
Welcome to . Week 08 Thurs . MAT135 Statistics.
Week 10 Chapter 16. Confidence Intervals for Proportions
Chapter 8: Estimating with Confidence
Confidence Intervals for Proportions
More on Inference.
Confidence Interval Estimation and Statistical Inference
Introduction to Inference
CHAPTER 22: Inference about a Population Proportion

Essential Statistics Introduction to Inference
Estimation Goal: Use sample data to make predictions regarding unknown population parameters Point Estimate - Single value that is best guess of true parameter.
Confidence Intervals: The Basics
Chapter 6 Confidence Intervals.
Confidence Intervals with Proportions
CHAPTER 8 Estimating with Confidence
Chapter 8: Estimating with Confidence
Chapter 8: Estimating With Confidence
Chapter 8: Estimating with Confidence
Chapter 8: Estimating with Confidence
Chapter 8: Estimating with Confidence
STA 291 Summer 2008 Lecture 14 Dustin Lueker.
Presentation transcript:

Statistics Confidence Intervals https://www.123rf.com/photo_6622261_statistics-and-analysis-of-data-as-background.html

Confidence Intervals We will use the sample mean 𝒙 to estimate the unknown population mean µ

Confidence Intervals Using the sample mean 𝒙 to estimate the unknown population mean µ is called “making inferences”

Confidence Intervals If you can assume the distribution of the sample means is normal, you can use the normal distribution probabilities for making probability statements about µ

as “n” increases, variability (spread) also decreases Confidence Intervals as “n” increases, variability (spread) also decreases

Confidence Intervals We use: s/ n for the measure of variability in the new population of 𝒙 s

Confidence Intervals The standard deviation of the 𝒙 s: s/ n is called the “standard error” abbreviated “se”

Confidence Intervals 𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se So our normal curve for the true value of the population mean µ is: 𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se

Confidence Intervals About 95% of the possible values for μ will be within 2 SE of 𝒙

Inferences About μ Although our normal curve graphs the 𝒙 s, we will use it to make inferences about what value μ actually has

This allows us to create a “confidence interval” for values of μ Confidence Intervals This allows us to create a “confidence interval” for values of μ

Confidence Intervals Confidence interval formula: 𝒙 - 2s/ n ≤ μ ≤ 𝒙 + 2s/ n or 𝒙 - 2se ≤ μ ≤ 𝒙 + 2se With a confidence level of 95%

The “2” in the equations is called the “critical value” Confidence Intervals The “2” in the equations is called the “critical value”

It comes from the normal curve, which gives us the 95% Confidence Intervals It comes from the normal curve, which gives us the 95%

2s/ n or 2se is called the “margin of error” Confidence Intervals 2s/ n or 2se is called the “margin of error”

What if we wanted a confidence level of 99% CONFIDENCE INTERVALS PROJECT QUESTION What if we wanted a confidence level of 99%

CONFIDENCE INTERVALS PROJECT QUESTION What if we wanted a confidence level of 99% We’d use a value of “3” rather than 2

Confidence Intervals For most scientific purposes, 95% is “good-enuff” In the law, 98% is required for a criminal case In medicine, 99% is required

Confidence Intervals For a 95% confidence interval, 95% of the values of μ will be within 2se of 𝒙

Confidence Intervals If we use the confidence interval to estimate a likely range for true values of μ, we will be right 95% of the time

For a 95% confidence interval, we will be WRONG 5% of the time Confidence Intervals For a 95% confidence interval, we will be WRONG 5% of the time

For a 99% confidence interval, how much of the time will we be wrong? CONFIDENCE INTERVALS PROJECT QUESTION For a 99% confidence interval, how much of the time will we be wrong?

CONFIDENCE INTERVALS PROJECT QUESTION For a 99% confidence interval, how much of the time will we be wrong? we will be wrong 1% of the time

Confidence Intervals The percent of time we are willing to be wrong is called “α” (“alpha”) or “the α-level”

Confidence Intervals Everyday use of confidence intervals: You will frequently hear that a poll has a candidate ahead by 10 points with a margin of error of 3 points

Confidence Intervals This means: 10-3 ≤ true difference ≤ 10+3 Or, the true difference is between 7 and 13 points (with 95% likelihood)

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 Can we assume normality?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 Can we assume normality? yes, because n>20

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the α-level?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the α-level? 5%

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the critical value?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the critical value? 2, because we want a 95% confidence interval

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the standard error?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the standard error? s/ n = 5/ 25 = 5/5 = 1

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the margin of error?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the margin of error? 2se = 2(1) = 2

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the confidence interval?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the confidence interval? 𝒙 - 2s/ n ≤ μ ≤ 𝒙 + 2s/ n 7 – 2 ≤ μ ≤ 7 + 2 5 ≤ μ ≤ 9 with 95% confidence

CONFIDENCE INTERVALS PROJECT QUESTION Interpreting confidence intervals: If the 95% confidence interval is: 5 ≤ µ ≤ 9 Is it likely that µ = 10?

CONFIDENCE INTERVALS PROJECT QUESTION No, because it’s outside of the interval That would only happen 5% of the time

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 Can we assume normality?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 Can we assume normality? no, because n<20-30

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 Can we assume normality? no, because n<20-30 But if the data is normal-ish, we can!

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the α-level?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the α-level? 5%

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the critical value?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the critical value? 2, because we want a 95% confidence interval

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the standard error?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the standard error? s/ n = 5/ 16 = 5/4 = 1.25

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the margin of error?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the margin of error? 2se = 2(1.25) = 2.5

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the confidence interval?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the confidence interval? 𝒙 - 2s/ n ≤ μ ≤ 𝒙 + 2s/ n 7 – 2.5 ≤ μ ≤ 7 + 2.5 4.5 ≤ μ ≤ 9.5 with 95% confidence

Confidence Intervals PROJECT QUESTION Interpreting confidence intervals: If the 95% confidence interval is: 4.5 ≤ µ ≤ 9.5 Is it likely that µ = 10?

Confidence Intervals PROJECT QUESTION No, because it’s outside of the interval That would only happen 5% of the time

Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = 49 CONFIDENCE INTERVALS PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = 49

CONFIDENCE INTERVALS PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = 49 49 ≤ µ ≤ 57

Can you say with 95% confidence that µ ≠ 55? CONFIDENCE INTERVALS PROJECT QUESTION Can you say with 95% confidence that µ ≠ 55?

CONFIDENCE INTERVALS PROJECT QUESTION Can you say with 95% confidence that µ ≠ 55? Nope… it’s in the interval It IS a likely value for µ

Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = 121 CONFIDENCE INTERVALS PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = 121

CONFIDENCE INTERVALS PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = 121 453 ≤ µ ≤ 509

Can you say with 95% confidence that µ might be 450? CONFIDENCE INTERVALS PROJECT QUESTION Can you say with 95% confidence that µ might be 450?

CONFIDENCE INTERVALS PROJECT QUESTION Can you say with 95% confidence that µ might be 450? µ is unlikely to be 450 – that value is outside of the confidence interval and would only happen 5% of the time

You will have a smaller interval if you have a larger value for n Confidence Intervals You will have a smaller interval if you have a larger value for n

So you want to take the LARGEST sample you can Confidence Intervals So you want to take the LARGEST sample you can

This is called the “LAW OF LARGE NUMBERS” Confidence Intervals This is called the “LAW OF LARGE NUMBERS”

What if you have a sample size smaller than 20??? Confidence Intervals What if you have a sample size smaller than 20???

Confidence Intervals What if you have a sample size smaller than 20??? You must use a different (bigger) critical value W.S. Gosset 1908

Questions?

Confidence Intervals We’ve done confidence intervals for measurement data with a mean μ based on the sample mean 𝒙 and margin of error 2se

CI for Proportions What about count data and proportions?

CI for Proportions We’ll use the normal curve for proportions: p -3 pq n p -2 pq n p - pq n p p + pq n p +2 pq n p +3 pq n

If p = .4 and n = 30 find the 95% CI for p CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p

If p = .4 and n = 30 find the 95% CI for p se = CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p se =

If p = .4 and n = 30 find the 95% CI for p se = pq n = .4x.6 30 ≈ .089 CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p se = pq n = .4x.6 30 ≈ .089

If p = .4 and n = 30 find the 95% CI for p me = CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p me =

If p = .4 and n = 30 find the 95% CI for p me = 2 × .089 = .178 CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p me = 2 × .089 = .178

If p = .4 and n = 30 find the 95% CI for p CI: CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p CI:

CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p CI: .4 - .178 ≤ p ≤ .4 + .178 .222 ≤ p ≤ .578

Questions? http://i.imgur.com/aliTlT3.jpg