Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-1 Statistics for Managers Using Microsoft® Excel 5th Edition Chapter 8 Confidence Interval Estimation

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-2 Learning Objectives In this chapter, you will learn:  To construct and interpret confidence interval estimates for the mean and the proportion  How to determine the sample size necessary to develop a confidence interval for the mean or proportion  How to use confidence interval estimates in auditing

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-3 Chapter Outline 1. Confidence Intervals for the Population Mean, μ  when Population Standard Deviation σ is Known  when Population Standard Deviation σ is Unknown 2. Confidence Intervals for the Population Proportion, π 3. Determining the Required Sample Size

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-4 Point Estimates  A point estimate is a single number. For the population mean (and population standard deviation), a point estimate is the sample mean (and sample standard deviation).  A confidence interval provides additional information about variability Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-5 Confidence Interval Estimates  A confidence interval gives a range estimate of values:  Takes into consideration variation in sample statistics from sample to sample  Based on all the observations from 1 sample  Gives information about closeness to unknown population parameters  Stated in terms of level of confidence  Ex. 95% confidence, 99% confidence  Can never be 100% confident

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-6 Confidence Interval Estimates  The general formula for all confidence intervals is: Point Estimate ± (Critical Value) (Standard Error)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-7 Confidence Level  Confidence Level  Confidence in which the interval will contain the unknown population parameter  A percentage (less than 100%)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-8 Confidence Level  Suppose confidence level = 95%  Also written (1 -  ) =.95  A relative frequency interpretation:  In the long run, 95% of all the confidence intervals that can be constructed will contain the unknown true parameter  A specific interval either will contain or will not contain the true parameter

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-9 Confidence Interval for μ (σ Known) Assumptions  Population standard deviation σ is known  Population is normally distributed  If population is not normal, use large sample Confidence interval estimate: (where Z is the standardized normal distribution critical value for a probability of α/2 in each tail)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-10 Finding the Critical Value, Z Consider a 95% confidence interval: Z= -1.96Z= 1.96 Lower Confidence Limit Upper Confidence Limit Z units: X units: Point Estimate 0

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-11 Finding the Critical Value, Z Commonly used confidence levels are 90%, 95%, and 99% Confidence Level Confidence Coefficient Z value % 90% 95% 98% 99% 99.8% 99.9%

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-12 Intervals and Level of Confidence Confidence Intervals Intervals extend from to (1-  ) x 100% of intervals constructed contain μ; (  ) x 100% do not. Sampling Distribution of the Mean x x1x1 x2x2

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-13 Confidence Interval for μ (σ Known) Example  A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is.35 ohms.  Determine a 95% confidence interval for the true mean resistance of the population.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-14 Confidence Interval for μ (σ Known) Example  We are 95% confident that the true mean resistance is between and ohms  Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-15 Confidence Interval for μ (σ Unknown)  If the population standard deviation σ is unknown, we can substitute the sample standard deviation, S  This introduces extra uncertainty, since S is variable from sample to sample  So we use the t distribution instead of the normal distribution

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-16

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-17 If1) n < 30 2) The sample is a simple random sample. 3) The sample is from a normally distributed population. Case 1 (  is known): Largely unrealistic; Use z-scores Case 2 (  is unknown): Use Student t distribution Small Samples Assumptions

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-18 Student t Distribution If the distribution of a population is essentially normal, then the distribution of  is essentially a Student t Distribution for all samples of size n.  is used to find critical values denoted by t = x - µ s n t  / 2

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-19 Definition Degrees of Freedom ( df ) corresponds to the number of sample values that can vary after certain restrictions have imposed on all data values

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-20 Degrees of Freedom ( df ) corresponds to the number of sample values that can vary after certain restrictions have imposed on all data values df = n - 1 in this section

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-21 Degrees of freedom Large (z) (one tail).01 (two tails) (one tail).02 (two tails).025 (one tail).05 (two tails).05 (one tail).10 (two tails).10 (one tail).20 (two tails).25 (one tail).50 (two tails)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-22 Given a variable that has a t-distribution with the specified degrees of freedom, what percentage of the time will it be in the indicated region? a. 10 df, between and b. 10 df, between and c. 24 df, between and d. 24 df, between and e. 24 df, outside the interval from and f. 24 df, to the right of g. 10 df, to the left of % 95% 99 % 1 %.5 % 5 %

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-23 What are the appropriate t critical values for each of the confidence intervals? a. 95 % confidence, n = 17 b. 90 % confidence, n = 12 c. 99 % confidence, n = 14 d. 90 % confidence, n = 25 e. 90 % confidence, n = 13 f. 95 % confidence, n =

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-24 Important Properties of the Student t Distribution 1.The Student t distribution is different for different sample sizes (see Figure 6-5 for the cases n = 3 and n = 12). 2.The Student t distribution has the same general symmetric bell shape as the normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples. 3.The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4.The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a  = 1). 5.As the sample size n gets larger, the Student t distribution gets closer to the normal distribution. For values of n > 30, the differences are so small that we can use the critical z values instead of developing a much larger table of critical t values. (The values in the bottom row of Table A-3 are equal to the corresponding critical z values from the standard normal distribution.)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-25 Student t distribution with n = 3 Student t Distributions for n = 3 and n = 12 0 Student t distribution with n = 12 Standard normal distribution

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-26 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-27 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873  = 0.05  / 2 = 0.025

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-28 Degrees of freedom Large (z) (one tail).01 (two tails) (one tail).02 (two tails).025 (one tail).05 (two tails).05 (one tail).10 (two tails).10 (one tail).20 (two tails).25 (one tail).50 (two tails) Table A-3 t Distribution

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-29 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873  = 0.05  / 2 = t  / 2 = ME = t  2 s = (2.201)(15,873) = 10, n 12

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-30 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873  = 0.05  / 2 = t  / 2 = ME = t  2 s = (2.201)(15,873) = 10,085.3 n 12 x - ME < µ < x +ME

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-31 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873  = 0.05  / 2 = t  / 2 = E = t  2 s = (2.201)(15,873) = 10,085.3 n 12 26, ,085.3 < µ < 26, ,085.3 x - E < µ < x + E

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-32 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873  = 0.05  / 2 = t  / 2 = ME = t  2 s = (2.201)(15,873) = 10,085.3 n 12 26, ,085.3 < µ < 26, ,085.3 $16,141.7 < µ < $36,312.3 x - ME < µ < x + ME

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-33 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873  = 0.05  / 2 = t  / 2 = E = t  2 s = (2.201)(15,873) = 10,085.3 n 12 26, ,085.3 < µ < 26, ,085.3 x - E < µ < x + E We are 95% confident that this interval contains the average cost of repairing a Dodge Viper. $16,141.7 < µ < $36,312.3

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-34 Confidence Interval for μ (σ 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: (where t is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-35 Student’s t Distribution  The t value depends on degrees of freedom (d.f.)  Number of observations that are free to vary after sample mean has been calculated d.f. = n - 1

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-36 Degrees of Freedom If the mean of these three values is 8.0, then X 3 must be 9 (i.e., X 3 is not free to vary) Here, n = 3, so degrees of freedom = n – 1 = 3 – 1 = 2 (2 values can be any numbers, but the third is not free to vary for a given mean) Idea: Number of observations that are free to vary after sample mean has been calculated Example: Suppose the mean of 3 numbers is 8.0  Let X 1 = 7  Let X 2 = 8  What is X 3 ?

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-37 Student’s t Distribution t 0 t (df = 5) t (df = 13) t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal Standard Normal (t with df = ∞) Note: t Z as n increases

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-38 Student’s t Table Upper Tail Area df t The body of the table contains t values, not probabilities Let: n = 3 df = n - 1 = 2  =.10  /2 =.05  /2 =.05

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-39 Confidence Interval for μ (σ Unknown) Example A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for μ  d.f. = n – 1 = 24, so  The confidence interval is (46.698, )

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-40 Confidence Intervals for the Population Proportion, π  An interval estimate for the population proportion ( π ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p )

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-41 Confidence Intervals for the Population Proportion, π Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation We will estimate this with sample data:

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-42 Confidence Intervals for the Population Proportion, π Upper and lower confidence limits for the population proportion are calculated with the formula where  z is the standardized normal value for the level of confidence desired  p is the sample proportion  n is the sample size

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-43 Confidence Intervals for the Population Proportion, Example A random sample of 100 people shows that 25 have opened IRA’s this year. Form a 95% confidence interval for the true proportion of the population who have opened IRA’s.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-44 Confidence Intervals for the Population Proportion, Example  We are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49%. Although the interval from.1651 to.3349 may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-45 Determining Sample Size  The required sample size can be found to reach a desired margin of error (e) with a specified level of confidence (1 -  )  The margin of error is also called sampling error  the amount of imprecision in the estimate of the population parameter  the amount added and subtracted to the point estimate to form the confidence interval

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-46 Determining Sample Size  To determine the required sample size for the mean, you must know:  The desired level of confidence (1 -  ), which determines the critical Z value  The acceptable sampling error (margin of error), e  The standard deviation, σ Now solve for n to get

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-47 Determining Sample Size If  = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence? So the required sample size is n = 220

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-48 Determining Sample Size  If unknown, σ can be estimated when using the required sample size formula  Use a value for σ that is expected to be at least as large as the true σ  Select a pilot sample and estimate σ with the sample standard deviation, S

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-49 Determining Sample Size To determine the required sample size for the proportion, you must know:  The desired level of confidence (1 -  ), which determines the critical Z value  The acceptable sampling error (margin of error), e  The true proportion of “successes”, π  π can be estimated with a pilot sample, if necessary (or conservatively use π =.50) Now solve for n to get

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-50 Determining Sample Size  How large a sample would be necessary to estimate the true proportion defective in a large population within ±3%, with 95% confidence?  (Assume a pilot sample yields p =.12)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-51 Determining Sample Size Solution: For 95% confidence, use Z = 1.96 e =.03 p =.12, so use this to estimate π So use n = 451

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-52 Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using . Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used .

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-53 n = [z  /2 ] 2 p (1-p) ME 2 Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e- mail. Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used .

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-54 n = [z  /2 ] 2 p(1-p) ME 2 = [1.645] 2 (0.169)(0.831) Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using . Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used .

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-55 = [1.645] 2 (0.169)(0.831) n = [z  /2 ] 2 p (1-p) ME 2 = = 238 households Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using . Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used . To be 90% confident that our sample percentage is within four percentage points of the true percentage for all households, we should randomly select and survey 238 households

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-56 n = [z  /2 ] 2 (0.25) ME 2 = (1.645) 2 (0.25) Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using . Assuming that we want 90% confidence in our results, how many households must we survey? There is no prior information suggesting a possible value for the sample percentage. = = 423 households

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-57 Applications in Auditing  Six advantages of statistical sampling in auditing  Sample result is objective and defensible  Based on demonstrable statistical principles  Provides sample size estimation in advance on an objective basis  Provides an estimate of the sampling error

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-58 Applications in Auditing  Can provide more accurate conclusions on the population  Examination of the population can be time consuming and subject to more nonsampling error  Samples can be combined and evaluated by different auditors  Samples are based on scientific approach  Samples can be treated as if they have been done by a single auditor  Objective evaluation of the results is possible  Based on known sampling error

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-59 Population Total Amount  Point estimate:  Confidence interval estimate: (This is sampling without replacement, so use the finite population correction in the confidence interval formula)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-60 Population Total Amount  A firm has a population of 1000 accounts and wishes to estimate the total population value.  A sample of 80 accounts is selected with average balance of $87.6 and standard deviation of $22.3.  Find the 95% confidence interval estimate of the total balance.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-61 Population Total Amount The 95% confidence interval for the population total balance is $82, to $92,362.48

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-62 Confidence Interval for Total Difference  Point estimate:  Where the mean difference,, is:

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-63 Confidence Interval for Total Difference Confidence interval estimate: where

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-64 One Sided Confidence Intervals Application: find the upper bound for the proportion of items that do not conform with internal controls where  Z is the standardized normal value for the level of confidence desired  p is the sample proportion of items that do not conform  n is the sample size  N is the population size

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-65 Ethical Issues  A confidence interval (reflecting sampling error) should always be reported along with a point estimate  The level of confidence should always be reported  The sample size should be reported  An interpretation of the confidence interval estimate should also be provided

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-66 Chapter Summary  Introduced the concept of confidence intervals  Discussed point estimates  Developed confidence interval estimates  Created confidence interval estimates for the mean (σ known)  Determined confidence interval estimates for the mean (σ unknown)  Created confidence interval estimates for the proportion In this chapter, we have

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 8-67 Chapter Summary  Determined required sample size for mean and proportion settings  Developed applications of confidence interval estimation in auditing  Confidence interval estimation for population total  Confidence interval estimation for total difference in the population  One sided confidence intervals  Addressed confidence interval estimation and ethical issues In this chapter, we have