Statistics: Unlocking the Power of Data Lock 5 Section 6.4 Distribution of a Sample Mean.

Slides:

Advertisements

Similar presentations

Chapter 23: Inferences About Means

Advertisements

Gossets student-t model What happens to quantitative samples when n is small?

An “app” thought!. VC question: How much is this worth as a killer app?

Statistics: Unlocking the Power of Data Lock 5 Inference Using Formulas STAT 101 Dr. Kari Lock Morgan Chapter 6 t-distribution Formulas for standard errors.

Confidence Interval Estimation of Population Mean, μ, when σ is Unknown Chapter 9 Section 2.

Statistics: Unlocking the Power of Data Lock 5 STAT 101 Dr. Kari Lock Morgan 10/25/12 Sections , Single Mean t-distribution (6.4) Intervals.

Objective: To estimate population means with various confidence levels.

Single-Sample t-Test What is the Purpose of a Single-Sample t- Test? How is it Different from a z-Test?What Are the Assumptions?

The Normal Distribution. n = 20,290  =  = Population.

Chapter 11: Inference for Distributions

1 (Student’s) T Distribution. 2 Z vs. T Many applications involve making conclusions about an unknown mean . Because a second unknown, , is present,

Getting Started with Hypothesis Testing The Single Sample.

Review As sample size increases, the distribution of sample means A.Becomes wider and more normally shaped B.Becomes narrower and more normally shaped.

Normal Distribution Chapter 5 Normal distribution

Copyright © 2010 Pearson Education, Inc. Chapter 23 Inference About Means.

Copyright © 2009 Pearson Education, Inc. Chapter 23 Inferences About Means.

Slide 23-1 Copyright © 2004 Pearson Education, Inc.

Education 793 Class Notes T-tests 29 October 2003.

T-distribution & comparison of means Z as test statistic Use a Z-statistic only if you know the population standard deviation (σ). Z-statistic converts.

INFERENCE ABOUT MEANS Chapter 23. CLT!! If our data come from a simple random sample (SRS) and the sample size is sufficiently large, then we know the.

X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ μ.

Dan Piett STAT West Virginia University

Albert Morlan Caitrin Carroll Savannah Andrews Richard Saney.

Confidence Intervals for Means. point estimate – using a single value (or point) to approximate a population parameter. –the sample mean is the best point.

Student’s t-distributions. Student’s t-Model: Family of distributions similar to the Normal model but changes based on degrees-of- freedom. Degrees-of-freedom.

When σ is Unknown The One – Sample Interval For a Population Mean Target Goal: I can construct and interpret a CI for a population mean when σ is unknown.

Inferential Statistics 2 Maarten Buis January 11, 2006.

Inference for Quantitative Variables 3/12/12 Single Mean, µ t-distribution Intervals and tests Difference in means, µ 1 – µ 2 Distribution Matched pairs.

Statistics: Unlocking the Power of Data Lock 5 Normal Distribution STAT 101 Dr. Kari Lock Morgan 10/18/12 Chapter 5 Normal distribution Central limit theorem.

H1H1 H1H1 HoHo Z = 0 Two Tailed test. Z score where 2.5% of the distribution lies in the tail: Z = Critical value for a two tailed test.

AP Statistics Chapter 23 Notes

Chapter 10 – Sampling Distributions Math 22 Introductory Statistics.

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.

July, 2000Guang Jin Statistics in Applied Science and Technology Chapter 7 - Sampling Distribution of Means.

Section 8-5 Testing a Claim about a Mean: σ Not Known.

Lesson 9 - R Chapter 9 Review.

Statistics for Business and Economics 8 th Edition Chapter 7 Estimation: Single Population Copyright © 2013 Pearson Education, Inc. Publishing as Prentice.

Chapter 12 Confidence Intervals and Hypothesis Tests for Means © 2010 Pearson Education 1.

Monday, October 22 Hypothesis testing using the normal Z-distribution. Student’s t distribution. Confidence intervals.

Inferring the Mean and Standard Deviation of a Population.

Statistics: Unlocking the Power of Data Lock 5 Inference for Means STAT 250 Dr. Kari Lock Morgan Sections 6.4, 6.5, 6.6, 6.10, 6.11, 6.12, 6.13 t-distribution.

Confidence Intervals for a Population Mean, Standard Deviation Unknown.

_ z = X -  XX - Wow! We can use the z-distribution to test a hypothesis.

Statistical Inference Statistical inference is concerned with the use of sample data to make inferences about unknown population parameters. For example,

Chapter 18 Sampling Distribution Models *For Means.

Confidence Interval Estimation of Population Mean, μ, when σ is Unknown Chapter 9 Section 2.

AGENDA Review In-Class Group Problems Review. Homework #3 Due on Thursday Do the first problem correctly Difference between what should happen over the.

Statistics: Unlocking the Power of Data Lock 5 Inference for Means STAT 250 Dr. Kari Lock Morgan Sections 6.4, 6.5, 6.6, 6.10, 6.11, 6.12, 6.13 t-distribution.

Inference About Means Chapter 23. Getting Started Now that we know how to create confidence intervals and test hypotheses about proportions, it’d be nice.

Monday, October 21 Hypothesis testing using the normal Z-distribution. Student’s t distribution. Confidence intervals.

Inference for distributions: - Comparing two means.

Introduction to the t statistic. Steps to calculate the denominator for the t-test 1. Calculate variance or SD s 2 = SS/n-1 2. Calculate the standard.

GOSSET, William Sealy How shall I deal with these small batches of brew?

Statistics: Unlocking the Power of Data Lock 5 Section 6.11 Confidence Interval for a Difference in Means.

AP Statistics Chapter 24 Comparing Means. Objectives: Two-sample t methods Two-Sample t Interval for the Difference Between Means Two-Sample t Test for.

Chapters 22, 24, 25 Inference for Two-Samples. Confidence Intervals for 2 Proportions.

Student’s t Distribution

Differences between t-distribution and z-distribution

Chapter 6 Inferences Based on a Single Sample: Estimation with Confidence Intervals Slides for Optional Sections Section 7.5 Finite Population Correction.

Chapter 24 Comparing Means.

Math 4030 – 10b Inferences Concerning Variances: Hypothesis Testing

Basic t-test 10/6.

Review Guppies can swim an average of 8 mph, with a standard deviation of 2 mph You measure 100 guppies and compute their mean What is the standard error.

Problem: If I have a group of 100 applicants for a college summer program whose mean SAT-Verbal is 525, is this group of applicants “above national average”?

Monday, October 19 Hypothesis testing using the normal Z-distribution.

Chapter 22 Inference About Means.

Chapter 23 Inference About Means.

Inferences from Matched Pairs

Presentation transcript:

Statistics: Unlocking the Power of Data Lock 5 Section 6.4 Distribution of a Sample Mean

Statistics: Unlocking the Power of Data Lock 5 Outline Standard error for a sample mean CLT for sample means t-distribution Distribution of sample mean

Statistics: Unlocking the Power of Data Lock 5 The larger the sample size, the smaller the SE

Statistics: Unlocking the Power of Data Lock 5 The standard deviation of the population is Standard Deviation

Statistics: Unlocking the Power of Data Lock 5 The standard deviation of the sample is Standard Deviation

Statistics: Unlocking the Power of Data Lock 5 The standard deviation of the sample mean is Standard Deviation The standard error is the standard deviation of the statistic.

Statistics: Unlocking the Power of Data Lock 5 Olympic Marathon Times

Statistics: Unlocking the Power of Data Lock 5 Olympic Marathon Times

Statistics: Unlocking the Power of Data Lock 5 CLT for a Mean Population Distribution of Sample Data Distribution of Sample Means n = 10 n = 30 n = 50

Statistics: Unlocking the Power of Data Lock 5 A normal distribution is usually a good approximation as long as n ≥ 30 Smaller sample sizes may be sufficient for symmetric distributions, and 30 may not be sufficient for very skewed distributions or distributions with high outliers

Statistics: Unlocking the Power of Data Lock 5 Math SAT Scores

Statistics: Unlocking the Power of Data Lock 5 Usually, we don’t know the population standard deviation , so estimate it with the sample standard deviation, s Standard Error

Statistics: Unlocking the Power of Data Lock 5 Replacing  with s changes the distribution of the standardized test statistic from normal to t The t distribution is very similar to the standard normal, but with slightly fatter tails to reflect this added uncertainty t-distribution

Statistics: Unlocking the Power of Data Lock 5 The t-distribution is characterized by its degrees of freedom (df) Degrees of freedom are calculated based on the sample size The higher the degrees of freedom, the closer the t-distribution is to the standard normal Degrees of Freedom

Statistics: Unlocking the Power of Data Lock 5 t-distribution

Statistics: Unlocking the Power of Data Lock 5 Aside: William Sealy Gosset

Statistics: Unlocking the Power of Data Lock 5 t-distribution If a population with mean µ 0 is approximately normal or if n is large (n ≥ 30), the standardized statistic for a mean using the sample s follows a t-distribution with n – 1 degrees of freedom:

Statistics: Unlocking the Power of Data Lock 5 t-distribution To use the t-distribution, either n has to be large or the population has to be normally distributed. If these conditions are met, then t has a t-distribution when the null hypothesis is true.

Statistics: Unlocking the Power of Data Lock 5 Normality Assumption Using the t-distribution requires that the data comes from a normal distribution Note: this assumption is about the population data, not the distribution of the statistic For large sample sizes we do not need to worry about this, because s will be a very good estimate of , and t will be very close to N(0,1) For small sample sizes (n < 30), we can only use the t-distribution if the distribution of the data is approximately normal

Statistics: Unlocking the Power of Data Lock 5 Normality Assumption One small problem: for small sample sizes, it is very hard to tell if the data actually comes from a normal distribution! Population Sample Data, n = 10

Statistics: Unlocking the Power of Data Lock 5 Small Samples If sample sizes are small, only use the t- distribution if the data look reasonably symmetric and do not have any extreme outliers. Even then, remember that it is just an approximation! In practice/life, if sample sizes are small, you should just use simulation methods (bootstrapping and randomization)