In this chapter you will learn: Define a point estimator, a point estimate, and desirable properties of a point estimator such as unbiasedness, efficiency, and consistency; define an interval estimator and an interval estimate; define a confidence interval, confidence level, margin of error, and a confidence interval estimate;...
9 1 Estimation and Confidence Intervals Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 9 2 When you have completed this chapter, you will be able to: Define a point estimator, a point estimate, and desirable properties of a point estimator such as unbiasedness, efficiency, and consistency. Define an interval estimator and an interval estimate Define a confidence interval, confidence level, margin of error, and a confidence interval estimate Construct a confidence interval for the population mean when the population standard deviation is known Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 9 3 Construct a confidence interval for the population variance when the population is normally distributed Construct a confidence interval for the population mean when the population is normally distributed and the population standard deviation is unknown Construct a confidence interval for a population proportion Determine the sample size for attribute and variable sampling Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Terminology 9 4 Point Estimate …is a single value (statistic) used to estimate a population value (parameter) Interval Estimate …states the range within which a population parameter probably lies Confidence Interval …is a range of values within which the population parameter isexpectedtooccur Copyrightâ2004byTheMcGrawưHillCompanies,Inc.Allrightsreserved. Desirablepropertiesofapointestimator 9ư5 efficient efficient possiblevaluesareconcentrated closetothevalueoftheparameter consistent consistent valuesaredistributedevenlyon bothsidesofthevalueofthe parameter unbiased unbiased unbiasedwhentheexpectedvalueequalsthevalue ofthepopulationparameterbeingestimated. Otherwise,itisbiased! Copyrightâ2004byTheMcGrawưHillCompanies,Inc.Allrightsreserved. Terminology Standard error of the sample mean Standard error of the sample mean 9 6 …is the standard deviation of the sampling distribution of the sample means x It is computed by n …is the symbol for the standard error of the sample mean x …is the standard deviation of the population n …is the size of the sample Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Standard Error Standard Error of the Means of the Means is not known and n > 30, IfIf is not known and n > 30, the standard deviation of the the standard deviation of the sample(ss) is used ) is used sample( to approximate the population standard to approximate the population standard deviation deviation Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. x Computed by… s s n 9 7 9 8 …that determine the width of a confidence interval are: 1 2 The sample size, n The sample size, n The variability in the population, The variability in the population, usually estimated by ss usually estimated by The desired level of confidence The desired level of confidence Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Constructing Constructing Confidence Intervals Confidence Intervals 9 9 IN GENERAL, A confidence interval for a mean is computed by: zα/2 s n x Interpreting… Interpreting… Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Interpreting Interpreting Confidence Intervals Confidence Intervals 9 10 The Globe Suppose that you read that “…the average selling price “…the average selling price of a family home in of a family home in York Region is York Region is $200 000 +/ $15000 $200 000 +/ $15000 at 95% confidence!” at 95% confidence!” This means…what? This means…what? Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Student’s tdistribution Student’s tdistribution 9 31 $4.45 $4.05 $4.95 $3.25 $4.68 $5.75 $6.01 $3.99 $5.25 $2.95 Step 1 Step 1 Determine the sample mean and standard deviation Determine the sample mean and standard deviation = $4.53 s = $1.00 X Step 2 Step 2 Enter the key data into the appropriate formula Enter the key data into the appropriate formula n = 10 x Formula Formula df = 10 – 1 = 9 = 199% = .01 1.00 = 4.53 3.25 10 α/2 n = $4.53 +/ $1.03 t s We are 99% confident that the mean amount spent We are 99% confident that the mean amount spent per customer is between $3.50 and per customer is between $3.50 and Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. $5.56 Constructing Confidence Intervals for Population Proportions A confidence interval for a population proportion is estimated by: Formula Formula p z p (1 p ) n p …is the symbol for the sample proportion Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 9 32 Constructing Confidence Intervals for Population Proportions A sample of 500 executives who own their own home revealed 175 planned to sell their homes and retire to Victoria. Develop a 98% confidence interval for the proportion of executives that plan to sell and move to Victoria Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 9 33 Constructing Confidence Intervals for Population Proportions A sample of 500 A sample of 500 executives who own executives who own their own home their own home revealed 175 revealed 175 planned to sell their planned to sell their homes and retire to homes and retire to Victoria. Victoria. Develop a 98% Develop a 98% confidence interval confidence interval for the proportion for the proportion of executives… of executives… Formula Formula ˆp 9 34 p(1 − p) n zα /2 n = p = z = 500 2.33 175/500 = .35 n = p = z = 35 33 98% CL = Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 35 ( 35 ) 500 35 0497 FinitePopulation FinitePopulation Correction Factor Correction Factor 9 35 Used when n/N is 0.05 or more Used when n/N is 0.05 or more Formula Formula x n N n N Correction Correction Factor Factor The attendance at the college hockey game last night The attendance at the college hockey game last night was 2700 2700. A random A random sample of 250 sample of 250 of those in of those in was attendance revealed that the average average number of number of attendance revealed that the drinks consumed per person was 1.8 1.8 drinks consumed per person was with a standard deviation of 0.40 standard deviation of 0.40 with a Develop a 90% confidence interval estimate for the mean number of drinks consumed per person Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. FinitePopulation FinitePopulation Correction Factor Correction Factor X Formula Formula Zα/2 9 36 s N n N n The attendance at the The attendance at the college hockey game college hockey game last night was 2700. last night was 2700. N = n = x = 2700 250 1.8 N = n = x = A sample of 250 of A sample of 250 of 0.40 s = /2 = 0.05 those in attendance s = /2 = those in attendance revealed that the revealed that the average number of average number of drinks consumed per Since 250/2700 >.05, use the correction factor drinks consumed per person was 1.8 with a person was 1.8 with a standard deviation of 2700 250 standard deviation of 0.40. 1.8 1.645 ( )( ) 0.40. Develop a 90% 2700 Develop a 90% 250 confidence interval confidence interval estimate.… estimate.… 90% CL = Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 04 9 37 Selecting the Sample Size Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Factors …that determine the sample size are: 1 2 3 The degree of confidence selected The degree of confidence selected The maximum allowable error The maximum allowable error The variation in the population The variation in the population Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 9 38 Selecting the Sample Size Formula Formula n = zα/2 s E 9 39 E … is the allowable error Z …is the zscore for the chosen level of confidence S …is the sample deviation of the pilot survey Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Selecting the Sample Size A consumer group would like to estimate the A consumer group would like to estimate the mean monthly electricity charge for a single mean monthly electricity charge for a single family house in July (within $5) $5) using a using a family house in July (within 99 percent level of confidence. 99 percent level of confidence. Based on similar studies the Based on similar studies the sstandard tandard ddeviation is eviation is estimated to be $20.00. estimated to be $20.00. How large a sample is required? How large a sample is required? Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 9 40 Selecting the Sample Size A consumer group A consumer group would like to estimate the would like to estimate the mean monthly electricity mean monthly electricity charge for a single family charge for a single family house in July (within $5) house in July (within $5) using a 99 percent level using a 99 percent level of confidence. of confidence. Based on Based on similar studies the similar studies the standard deviation is standard deviation is estimated to be $20.00. estimated to be $20.00. Formula Formula zα/2 s E 9 41 2.58 20 5.00 = (10.32)2 = 106.5 A minimum of 90% CL = A minimum of 107 homes 107 homes must be sampled must be sampled Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Selecting the Sample Size 9 42 The Kennel Club wants to estimate the proportion of children that have a dog as a pet. Assume a 95% level of confidence and that the club estimates that 30% of the children have a dog as a pet. If the club wants the estimate to be within 3% of the population proportion, how many children would they need to contact? Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Selecting the Sample Size New The Kennel Club The Kennel Club wants to estimate the wants to estimate the proportion of children proportion of children that have a dog as a that have a dog as a pet. pet. Assume a Assume a 95% level of 95% level of confidence and that and that confidence Formula Formula n (1 9 43 p (1 Z p) E 96 3) 03 ( 21 ) 65 33 n = 896.4 the club estimates that A minimum of 897 children the club estimates that A minimum of 897 children 30% of the children of the children must be sampled 30% must be sampled have a dog have a dog Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 2 Test your learning… … Test your learning … … n o n o k ilcick CCl www.mcgrawhill.ca/college/lind Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s EStat data …and much more! Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 9 44 9 45 This completes Chapter 9 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. ... Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Interpreting Interpreting Confidence Intervals Confidence Intervals 9 14 90% Confidence Interval 90% Confidence Interval … 10% chance of falling outside this interval... Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Constructing Constructing Confidence Intervals Confidence Intervals 9 9 IN GENERAL, A confidence interval for a mean is computed by: zα/2 s n x Interpreting…... Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Interpreting Interpreting Confidence Intervals Confidence Intervals You select a You select a random sample random sample