Confidence Intervals Chapter 8 Confidence Intervals for numerical

Confidence Intervals (Chapter 8) • Confidence Intervals for numerical data: – Standard deviation known – Standard deviation unknown • Confidence Intervals for categorical data

Estimation Process: Example • We are interested in knowing the average household income in a certain county. • A sample with 144 observations yields a sample mean X=$72, 000. • It is also “known” that in this county, =$24, 000 • How can we get a “good” estimate for the true average household income ? Or: • How far away (“how bad”) can X be as an estimate for ?

Estimation Process Population Random Sample Mean, , is unknown Mean X = 50 Sample I am 95% confident that is between 40 & 60.

Point Estimates Estimate Population Parameters … Mean Proportion Variance Difference with Sample Statistics

Interval Estimates • Provides range of values – Take into consideration variation in sample statistics from sample to sample – Based on observation from 1 sample – Give information about closeness to unknown population parameters – Stated in terms of level of confidence • Never 100% sure

Confidence Interval Estimates Confidence Intervals Mean Known Proportion Unknown

Confidence Interval for ( Known) • Assumptions – Population standard deviation is known – Population is normally distributed – If population is not normal, use large sample • Confidence interval estimate

General Formula The general formula for a confidence interval is: Point Estimate ± Margin of Error Point Estimate ± (Critical Value)(Standard Error) Where: • Point Estimate is the sample statistic estimating the population parameter of interest • Critical Value is a table value based on the sampling distribution of the point estimate and the desired confidence level • Standard Error is the standard deviation of the point estimate

Elements of Confidence Interval Estimation • Level of confidence – Confidence in which the interval will contain the unknown population parameter • Precision (range) – Closeness to the unknown parameter • Cost – Cost required to obtain a sample of size n

Level of Confidence • Denoted by • A relative frequency interpretation – In the long run, of all the confidence intervals that can be constructed will contain the unknown parameter • A specific interval will either contain or not contain the parameter – No probability involved in a specific interval

Interval and Level of Confidence Sampling Distribution of the _ Mean Intervals extend from of intervals constructed contain ; to Confidence Intervals not. do

Factors Affecting Margin of error (Precision) • Data variation – Measured by • Sample size – Intervals Extend from X - Z x to X + Z x • Level of confidence – © 1984 -1994 T/Maker Co.

Determining Sample Size (Cost) Too Big: Too small: • Requires too much resources • Won’t do the job

Determining Sample Size for Mean What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard deviation is 45. Round Up

Do You Ever Truly Know σ? • Probably not! • In virtually all real world business situations, σ is not known. • If there is a situation where σ is known then µ is also known (since to calculate σ you need to know µ. ) • If you truly know µ there would be no need to gather a sample to estimate it.

Confidence Interval for ( Unknown) • Assumptions – Population standard deviation is unknown – Population is normally distributed – If population is not normal, use large sample • Use Student’s t Distribution • Confidence Interval Estimate –

Student’s t Distribution Standard Normal Bell-Shaped Symmetric ‘Fatter’ Tails t (df = 13) t (df = 5) 0 Z t

Example

Confidence Interval Estimate for Proportion • Assumptions – Two categorical outcomes – Population follows binomial distribution – Normal approximation can be used if and – Confidence interval estimate –

Example A random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95% confidence interval estimate for p.

Determining Sample Size for Proportion Out of a population of 1, 000, we randomly selected 100 of which 30 were defective. What sample size is needed to be within ± 5% with 90% confidence? Round Up

Excel Tutorial Constructing Confidence Intervals using Excel: • Tutorial • Excel spreadsheet