Stat 528: Sampling Distribution & Confidence Intervals, Study notes of Statistics

Download Stat 528: Sampling Distribution & Confidence Intervals and more Study notes Statistics in PDF only on Docsity! 1 Christopher Holloman, The Ohio State University, Summer 2006 Statistics 528 Data Analysis I Lecture #7 July 13, 2006 Christopher Holloman, The Ohio State University, Summer 2006 Overview of Today’s Lecture
IPS Sections 5.2 – 6.1  Sampling Distribution of a Sample Mean  Statistical Confidence Christopher Holloman, The Ohio State University, Summer 2006 )( 1 21 n XXX n x +++= L The Sampling Distribution of the Sample Mean
Imagine that we have an SRS of size n from a population and measure a variable X on each individual in the sample.
Each Xi is a measurement from the population and therefore has the distribution of the population  µXi = µ and σXi = σ.
The sample mean of an SRS of size n is Christopher Holloman, The Ohio State University, Summer 2006 Question: What are the mean and standard deviation of ? The mean of the distribution of the sample mean:  is an unbiased estimate of µ. x )( 1 21 nxxxx n µµµµ +++= L )( 1 µµµ +++= L n µ= x 2 Christopher Holloman, The Ohio State University, Summer 2006 The standard deviation of the distribution of the sample mean: The observations are independent, so we can use the addition rule for variances - So, )( 1 222 2 2 21 nxxxx n σσσσ +++      = L )( 1 222 2 σσσ +++= L n 22 2 11 σσ n n n =      = n x σ σ = Christopher Holloman, The Ohio State University, Summer 2006
Example: The height X of a single randomly chosen young woman varies according to the N(64.5, 2.5) distribution. Suppose we randomly sample 100 young women. What is mean and variance of the distribution of ?x Christopher Holloman, The Ohio State University, Summer 2006
We know the mean and variance, but that doesn’t tell us everything we need to know about the distribution of the sample mean.
First, let’s examine one special case: the normal distribution. Christopher Holloman, The Ohio State University, Summer 2006 Sampling Distribution of the Sample Mean If a population is distributed , then the sample mean of n independent observations has the distribution.  The sampling distribution of depends on the sample size (n) - the distribution is more spread out (larger variance) the smaller the sample size. ),( σµN x )/,( nN σµ x 5 Christopher Holloman, The Ohio State University, Summer 2006 Example: Assume that the sampling distribution of is N(µ, 4.5). lies within +/- 9 of µ in 95% of all samples, so µ also lines within +/- 9 of in those samples. x x x Christopher Holloman, The Ohio State University, Summer 2006  In 95% of samples, - 9 < µ < + 9. We say that ( - 9, + 9) is a 95% confidence interval for µ. Requirements of a Confidence Interval for an Unknown Parameter: 1. an interval of the form (a, b), where a and b are numbers computed from the data 2. a confidence level that gives the probability that an interval computed this way covers the parameter. Usually, confidence levels are 90% or 95%. x x x x Christopher Holloman, The Ohio State University, Summer 2006 Definition of a Confidence Interval: A level C confidence interval for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter. Note: The following statement is INCORRECT – The probability that the unknown parameter is contained within a level C confidence interval is C. Why is this wrong? Christopher Holloman, The Ohio State University, Summer 2006 6 Christopher Holloman, The Ohio State University, Summer 2006 Confidence Intervals for the Population Mean Recall: is approximately by the Central Limit Theorem. To construct a level C confidence interval for µ (assuming we know σ) :
Let z* be the point such that the area under the N(0,1) curve between z* and -z* is C. x )/,( nN σµ Christopher Holloman, The Ohio State University, Summer 2006 Christopher Holloman, The Ohio State University, Summer 2006
Notice that any normal curve has probability C between the points z* standard deviations below the mean and z* standard deviations above the mean. Why?
So there is probability C that lies between and x n z σ µ *− n z σ µ *+ Christopher Holloman, The Ohio State University, Summer 2006
This is the same as saying that 95% percent of the time (in repeated sampling from the population with mean µ and standard deviation σ) µ will lie between and  This is our level C confidence interval for µ, i.e. our estimate of µ is and our margin of error is n zx σ *− n zx σ *+ x nz /*σ 7 Christopher Holloman, The Ohio State University, Summer 2006 C 90% 95% 99% z* 1.645 1.96 2.576 The most commonly used confidence levels are z* for other confidence levels can be found similarly from the Normal Table (Table A) , from the bottom row of Table D (t distribution critical values) or using Minitab. Christopher Holloman, The Ohio State University, Summer 2006 Example: Scores on a test of quantitative skills range from 0 to 500. A simple random sample of 840 men aged 21 to 25 took the exam. Their average score was =272. Suppose we know that the population standard deviation for this test, σ, is equal to 60. What can we say about the population mean score, µ, of all 9.5 million men in this age group? a) Find a 90% confidence interval for the mean test score. x Christopher Holloman, The Ohio State University, Summer 2006 b) Find a 99% confidence interval for the mean test score. c) Find an 80% confidence interval for the mean test score. Christopher Holloman, The Ohio State University, Summer 2006 Meaning of Confidence Note: We don’t know if any of the above confidence intervals contain µ or not! Then what do we mean by confidence? The meaning of “Confidence”: When we say “95% confident,” we mean that if you use 95% confidence intervals often, in the long run 95% of your intervals will contain the true value of µ.