Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Estimates and Sample Sizes Chapter 6 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 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 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 Estimating a Population Mean: Large Samples

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 volume (or point) used to approximate a population parameter

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 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 population mean µ. is the best point estimate of the

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 #

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8  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 9 Definition  Degree of Confidence (level of confidence or confidence coefficient) the probability 1 –  (often expressed as the equivalent percentage value) that the confidence interval contains the true value of the population parameter  usually 95% or 99% (  = 5%) (  = 1%)

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10 Confidence Intervals from Different Samples x This confidence interval does not contain µ µ = (but unknown to us) Figure 6-3

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 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 z  /2 is a critical value that is a z score with the property that it separates an area  / 2 in the right tail of the standard normal distribution. There is an area of 1 –  between the vertical borderlines at – z  /2 and z  /2.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12 If Degree of Confidence = 95%

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 If Degree of Confidence = 95% –z  2 z  2 95%  2 = 2.5% =.025  = 5%

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 If Degree of Confidence = 95% –z  2 z  2 95%  2 = 2.5% =.025  = 5% Critical Values

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 95% Degree of Confidence

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16 95% Degree of Confidence Use Table A-2 to find a z score of 1.96  =  = 0.05

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17 95% Degree of Confidence.025 – z  2 = 1.96   Use Table A-2 to find a z score of 1.96  =  = 0.05

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18 Definition Margin of Error

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.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 20 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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 21 x – E 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 < µ < x + E

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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 23 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 upper limitlower limit

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 24 Calculating the Margin of Error When  Is Unknown

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 25 Calculating the Margin of Error When  Is known  If n > 30, we can replace  in Formula 6-1 by the sample standard deviation s.  If n  30, the population must have a normal distribution and we must know  to use Formula 6-1. E = z  /2 Formula 6-1  n

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 26 Confidence Interval 1. If using original data, round to one more decimal place than used in data. Round off Rules

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 27 Confidence Interval 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. Round off Rules

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 28 Procedure for Constructing a Confidence Interval for µ ( based on a large sample: n > 30 )

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 z  2 that corresponds to the desired degree of confidence.

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 z  2 that corresponds to the desired degree of confidence. 2. Evaluate the margin of error E= z  2  n. If the population standard deviation  is unknown and n > 30, use the value of the sample standard deviation s.

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 z  2 that corresponds to the desired degree of confidence. 2. Evaluate the margin of error E = z  2  n. If the population standard deviation  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 32 Procedure for Constructing a Confidence Interval for µ ( based on a large sample: n > 30 ) 1. Find the critical value z  2 that corresponds to the desired degree of confidence. 2. Evaluate the margin of error E= z  2  n. If the population standard deviation  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 33 Confidence Intervals from Different Samples

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Determining Sample Size

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 35 Determining Sample Size

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 36 Determining Sample Size z  / 2 E =  n

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 37 Determining Sample Size z  / 2 E =  n (solve for n by algebra)

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 38 Determining Sample Size If n is not a whole number, round it up to the next higher whole number. z  / 2 E =  n (solve for n by algebra) z  / 2 E  n = 2 Formula 6-2

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 39 What happens when E is doubled ?

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 40 What happens when E is doubled ?  / 2 z 1  n = = 2 1  / 2 ( z )  2 E = 1 :

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 41 What happens when E is doubled ?  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.  / 2 z 1  n = = 2 1  / 2 ( z )  2  / 2 z 2  n = = 2 4  / 2 ( z )  2 E = 1 : E = 2 :

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 42 What if  is unknown ? 1. Use the range rule of thumb to estimate the standard deviation as follows:   range / 4 or 2. Calculate the sample standard deviation s and use it in place of . That value can be refined as more sample data are obtained.