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Estimates and Sample Sizes Chapter 6 M A R I O F. T R I O L A Copyright ©Copyright 1998, Triola, Elementary Statistics © 1998, Triola, Elementary Statistics Addison Wesley Wesley Longman Longman Addison 1 Chapter 6 Estimate and Sample Sizes 6-1 Overview 6-2 Estimating a Population Mean: Large Samples 6-3 Estimating a Population Mean: Small Samples 6-4 Determining Sample Size 6-5 Estimating a Population Proportion 6-6 Estimating a Population Variance Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 6-1 Overview This chapter presents: methods for estimating population means, proportions, and variances methods for determining sample sizes necessary to estimate the above parameters. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 6-2 Estimating a Population Mean: Large Samples Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 Definitions Estimator a sample statistic used to approximate a population parameter Estimate a specific value or range of values used to approximate some population parameter Point Estimate a single volume (or point) used to approximate a population parameter Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 Definitions Estimator a sample statistic used to approximate a population parameter Estimate a specific value or range of values used to approximate some population parameter Point Estimate a single volue (or point) used to approximate a popular parameter The sample mean x is the best point estimate of the population mean µ. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 Confidence Interval (or Interval Estimate) a range (or an interval) of values likely to contain the true value of the population parameter Lower # < population parameter < Upper # Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 Confidence Interval (or Interval Estimate) a range (or an interval) of values likely to contain the true value of the population parameter Lower # < population parameter < Upper # As an example Lower # < µ < Upper # Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 Definition Degree of Confidence (level of confidence or confidence coefficient) the probability 1 – a (often expressed as the equivalent percentage value) that the confidence interval contains the true value of the population parameter usually 95% or 99% (a = 5%) (a = 1%) Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9 Confidence Intervals from Different Samples µ = 98.25 (but unknown to us) 98.00 x • 98.08 • Figure 6-3 • • 98.50 • 98.32 • • Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman • This confidence interval does not contain µ • • 10 Definition Critical Value the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. The number value that is a z za/2 is a critical score with the property that it separates an area a/2 in the right tail of the standard normal distribution. There is an area of 1 – a between the vertical borderlines at –za/2 and Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman za/2. 11 If Degree of Confidence = 95% Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12 If Degree of Confidence = 95% 95% a = 5% a/2 = 2.5% = .025 .95 .025 –za/2 .025 za/2 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 If Degree of Confidence = 95% 95% a = 5% a/2 = 2.5% = .025 .95 .025 .025 –za/2 za/2 Critical Values Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 95% Degree of Confidence Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 95% Degree of Confidence a = 0.05 a/2 = 0.025 .4750 .025 Use Table A-2 to find a z score of 1.96 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16 95% Degree of Confidence a = 0.05 a/2 = 0.025 .4750 .025 Use Table A-2 to find a z score of 1.96 za/2 = +- 1.96 .025 –1.96 .025 1.96 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17 Definition Margin of Error Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18 Definition Margin of Error is the maximum likely difference between the observed sample mean, x, and true population mean µ. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 19 Definition Margin of Error is the maximum likely difference between the observed sample mean, x, and true population mean µ. denoted by E x –E µ Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman x +E 20 Definition Margin of Error is the maximum likely difference between the observed sample mean, x, and true population mean µ. denoted by E x –E µ x +E x–E<µ<x+E Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 21 Definition Margin of Error is the maximum likely difference between the observed sample mean, x, and true population mean µ. denoted by E µ x –E x +E x–E<µ<x+E lower limit Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 22 Definition Margin of Error is the maximum likely difference between the observed sample mean, x, and true population mean µ. denoted by E µ x –E x +E x – E < µ < x +E lower limit upper limit Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 23 Calculating the Margin of Error When s Is Unknown Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 24 Calculating the Margin of Error When s Is known E = za/2 • s n Formula 6-1 If n > 30, we can replace s in Formula 6-1 by the sample standard deviation s. If n 30, the population must have a normal distribution and we must know s to use Formula 6-1. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 25 Confidence Interval Round off Rules • 1. If using original data, round to one more decimal place than used in data. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 26 Confidence Interval Round off Rules • 1. If using original data, round to one more decimal place than used in data. • 2. If given summary statistics (n, x, s), round to same number of decimal places as in x. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 27 Procedure for Constructing a Confidence Interval for µ ( based on a large sample: Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman n > 30 ) 28 Procedure for Constructing a Confidence Interval for µ ( based on a large sample: n > 30 ) 1. Find the critical value za/2 that corresponds to the desired degree of confidence. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 29 Procedure for Constructing a Confidence Interval for µ ( based on a large sample: n > 30 ) 1. Find the critical value za/2 that corresponds to the desired degree of confidence. 2. Evaluate the margin of error E= za/2 • s / n . If the population standard deviation s is unknown and n > 30, use the value of the sample standard deviation s. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 30 Procedure for Constructing a Confidence Interval for µ ( based on a large sample: n > 30 ) 1. Find the critical value za/2 that corresponds to the desired degree of confidence. 2. Evaluate the margin of error E = za/2 • s / n . If the population standard deviation s is unknown and n > 30, use the value of the sample standard deviation s. 3. Find the values of x – E and x + E. Substitute those values in the general format of the confidence interval: x –E <µ< x +E Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 31 Procedure for Constructing a Confidence Interval for µ ( based on a large sample: n > 30 ) 1. Find the critical value za/2 that corresponds to the desired degree of confidence. 2. Evaluate the margin of error E= za/2 • s / n . If the population standard deviation s is unknown and n > 30, use the value of the sample standard deviation s. 3. Find the values of x – E and x + E. Substitute those values in the general format of the confidence interval: x –E <µ< x +E 4. Round using the confidence intervals round off rules. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 32 Confidence Intervals from Different Samples Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 33 6-3 Determining Sample Size Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 34 Determining Sample Size Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 35 Determining Sample Size s E = za / 2 • n Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 36 Determining Sample Size s E = za / 2 • n (solve for n by algebra) Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 37 Determining Sample Size s E = za / 2 • n (solve for n by algebra) n= za / 2 s E 2 Formula 6-2 If n is not a whole number, round it up to the next higher whole number. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 38 What happens when E is doubled ? Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 39 What happens when E is doubled ? (za / 2s ) z a / 2s n= 1 = 1 2 E=1: Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 40 What happens when E is doubled ? E=1: (za / 2s ) z a / 2s n= 1 = 1 E=2: (za / 2s ) z a / 2s n= = 4 2 2 2 2 2 Sample size n is decreased to 1/4 of its original value if E is doubled. Larger errors allow smaller samples. Smaller errors require larger samples. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 41 What if s is unknown ? 1. Use the range rule of thumb to estimate the standard deviation as follows: s range / 4 or 2. Calculate the sample standard deviation s and use it in place of s. That value can be refined as more sample data are obtained. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 42