7 - 1 © 1998 Prentice-Hall, Inc. Chapter 7 Inferences Based on a Single Sample: Estimation with Confidence Intervals

7 - 2 © 1998 Prentice-Hall, Inc. Learning Objectives 1.State what is estimated 2.Distinguish point & interval estimates 3.Explain interval estimates 4.Compute confidence interval estimates for population mean & proportion 5.Compute sample size

7 - 3 © 1998 Prentice-Hall, Inc. Thinking Challenge Suppose you’re interested in estimating the average amount of money that second-year business students (population) have on them. How would you find out? AloneGroupClass

7 - 4 © 1998 Prentice-Hall, Inc. Introduction to Estimation

7 - 5 © 1998 Prentice-Hall, Inc. Types of Statistical Applications

7 - 6 © 1998 Prentice-Hall, Inc. Estimation Process

7 - 7 © 1998 Prentice-Hall, Inc. Estimation Process Mean, , is unknown Population

7 - 8 © 1998 Prentice-Hall, Inc. Estimation Process Mean, , is unknown Population Random Sample Mean  X  = 50 Sample

7 - 9 © 1998 Prentice-Hall, Inc. Estimation Process Mean, , is unknown Population Random Sample I am 95% confident that  is between 42 & 58. Mean  X  = 50 Sample

© 1998 Prentice-Hall, Inc. Unknown Population Parameters Are Estimated

© 1998 Prentice-Hall, Inc. Unknown Population Parameters Are Estimated Estimate population parameter... with sample statistic

© 1998 Prentice-Hall, Inc. Unknown Population Parameters Are Estimated Estimate population parameter... with sample statistic Mean  x

© 1998 Prentice-Hall, Inc. Unknown Population Parameters Are Estimated Estimate population parameter... with sample statistic Mean  x Proportionp p ^

© 1998 Prentice-Hall, Inc. Unknown Population Parameters Are Estimated Estimate population parameter... with sample statistic Mean  x Proportionp p ^ Variance  2 s 2

© 1998 Prentice-Hall, Inc. Unknown Population Parameters Are Estimated Estimate population parameter... with sample statistic Mean  x Proportionp p ^ Variance  2 s 2 Differences  1 -  -  2  x 1 -  x 2

© 1998 Prentice-Hall, Inc. Estimation Methods

© 1998 Prentice-Hall, Inc. Estimation Methods Estimation

© 1998 Prentice-Hall, Inc. Estimation Methods Estimation Point Estimation

© 1998 Prentice-Hall, Inc. Estimation Methods Estimation Point Estimation Interval Estimation

© 1998 Prentice-Hall, Inc. Point Estimation

© 1998 Prentice-Hall, Inc. Estimation Methods Estimation Point Estimation Interval Estimation

© 1998 Prentice-Hall, Inc. Point Estimation 1.Provides single value Based on observations from 1 sample Based on observations from 1 sample 2.Gives no information about how close value is to the unknown population parameter 3.Example: Sample mean  x = 3 is point estimate of unknown population mean

© 1998 Prentice-Hall, Inc. Interval Estimation

© 1998 Prentice-Hall, Inc. Estimation Methods Estimation Point Estimation Interval Estimation

© 1998 Prentice-Hall, Inc. Interval Estimation 1.Provides range of values Based on observations from 1 sample Based on observations from 1 sample 2.Gives information about closeness to unknown population parameter Stated in terms of probability Stated in terms of probability Knowing exact closeness requires knowing unknown population parameter Knowing exact closeness requires knowing unknown population parameter 3.Example: unknown population mean lies between 50 & 70 with 95% confidence

© 1998 Prentice-Hall, Inc. Key Elements of Interval Estimation

© 1998 Prentice-Hall, Inc. Key Elements of Interval Estimation Sample statistic (point estimate)

© 1998 Prentice-Hall, Inc. Key Elements of Interval Estimation Confidence interval Sample statistic (point estimate)

© 1998 Prentice-Hall, Inc. Key Elements of Interval Estimation Confidence interval Sample statistic (point estimate) Confidence limit (lower) Confidence limit (upper)

© 1998 Prentice-Hall, Inc. Confidence interval Key Elements of Interval Estimation Sample statistic (point estimate) Confidence limit (lower) Confidence limit (upper) A probability that the population parameter falls somewhere within the interval.

© 1998 Prentice-Hall, Inc. Confidence Limits for Population Mean Parameter = Statistic ± Error © T/Maker Co.

© 1998 Prentice-Hall, Inc. Many Samples Have Same Interval

© 1998 Prentice-Hall, Inc. Many Samples Have Same Interval  x_ XXXX 

© 1998 Prentice-Hall, Inc. Many Samples Have Same Interval  x_ XXXX   X  =  ± Z   x

© 1998 Prentice-Hall, Inc. Many Samples Have Same Interval 90% Samples  x_ XXXX   X  =  ± Z   x    x    x

© 1998 Prentice-Hall, Inc. Many Samples Have Same Interval 90% Samples 95% Samples    x  x_ XXXX    x    x    x   X  =  ± Z   x

© 1998 Prentice-Hall, Inc. Many Samples Have Same Interval 90% Samples 95% Samples 99% Samples    x    x  x_ XXXX    x    x    x    x   X  =  ± Z   x

© 1998 Prentice-Hall, Inc. 1.Probability that the unknown population parameter falls within interval 2.Denoted (1 -   is probability that parameter is not within interval  is probability that parameter is not within interval 3.Typical values are 99%, 95%, 90% Confidence Level

© 1998 Prentice-Hall, Inc. Intervals & Confidence Level

© 1998 Prentice-Hall, Inc. Intervals & Confidence Level Sampling Distribution of Mean

© 1998 Prentice-Hall, Inc. Intervals & Confidence Level Sampling Distribution of Mean Confidence Interval

© 1998 Prentice-Hall, Inc. Intervals & Confidence Level Sampling Distribution of Mean Confidence Interval

© 1998 Prentice-Hall, Inc. Intervals & Confidence Level Sampling Distribution of Mean Intervals extend from  X - Z   X to  X + Z   X Confidence Interval

© 1998 Prentice-Hall, Inc. Intervals & Confidence Level Sampling Distribution of Mean Intervals extend from  X - Z   X to  X + Z   X Confidence Interval

© 1998 Prentice-Hall, Inc. Intervals & Confidence Level Sampling Distribution of Mean Intervals extend from  X - Z   X to  X + Z   X Confidence Interval

© 1998 Prentice-Hall, Inc. Intervals & Confidence Level Sampling Distribution of Mean Large number of intervals Intervals extend from  X - Z   X to  X + Z   X Confidence Interval

© 1998 Prentice-Hall, Inc. Intervals & Confidence Level Sampling Distribution of Mean Large number of intervals Intervals extend from  X - Z   X to  X + Z   X (1 -  )100 % of intervals contain .  % do not.

© 1998 Prentice-Hall, Inc. Factors Affecting Interval Width 1.Data dispersion Measured by  Measured by  2.Sample size   X =  /  n   X =  /  n 3.Level of confidence (1 -  ) Affects Z Affects Z Intervals extend from  X - Z   X to  X + Z   X © T/Maker Co.

© 1998 Prentice-Hall, Inc. Confidence Interval Estimates

© 1998 Prentice-Hall, Inc. Confidence Interval Estimates One Population

© 1998 Prentice-Hall, Inc. Confidence Interval Estimates One Population Mean

© 1998 Prentice-Hall, Inc. Confidence Interval Estimates One Population MeanProportion

© 1998 Prentice-Hall, Inc. Confidence Interval Estimates One Population Z Distribution Large Sample MeanProportion

© 1998 Prentice-Hall, Inc. Confidence Interval Estimates One Population Z Distribution t Large Sample MeanProportion Small Sample

© 1998 Prentice-Hall, Inc. Confidence Interval Estimates

© 1998 Prentice-Hall, Inc. Confidence Interval Estimate Mean (Large Sample)

© 1998 Prentice-Hall, Inc. Confidence Interval Estimates One Population Z Distribution t Large Sample Z Distribution MeanProportion Small Sample

© 1998 Prentice-Hall, Inc. Confidence Interval Mean (Large Sample) 1.Assumptions Sample size at least 30 (n  30) Sample size at least 30 (n  30) Random sample drawn Random sample drawn If population standard deviation unknown, use sample standard deviation If population standard deviation unknown, use sample standard deviation

© 1998 Prentice-Hall, Inc. Confidence Interval Mean (Large Sample) 1.Assumptions Sample size at least 30 (n  30) Sample size at least 30 (n  30) Random sample drawn Random sample drawn If population standard deviation unknown, use sample standard deviation If population standard deviation unknown, use sample standard deviation 2.Confidence interval estimate

© 1998 Prentice-Hall, Inc. Estimation Example Mean (Large Sample) The mean of a random sample of n = 36 is  X = 50. Set up a 95% confidence interval estimate for  if  = 12.

© 1998 Prentice-Hall, Inc. Estimation Example Mean (Large Sample) The mean of a random sample of n = 36 is  X = 50. Set up a 95% confidence interval estimate for  if  = 12.

© 1998 Prentice-Hall, Inc. Thinking Challenge You’re a Q/C inspector for Gallo. The  for 2-liter bottles is.05 liters. A random sample of 100 bottles showed  X = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles? 2 liter © T/Maker Co. 2 liter AloneGroupClass

© 1998 Prentice-Hall, Inc. Confidence Interval Solution*

© 1998 Prentice-Hall, Inc. Confidence Interval Estimate Mean (Small Sample)

© 1998 Prentice-Hall, Inc. Confidence Interval Estimates One Population Z Distribution t Large Sample Z Distribution MeanProportion Small Sample

© 1998 Prentice-Hall, Inc. Confidence Interval Mean (Small Sample) 1.Assumptions Sample size less than 30 (n < 30) Sample size less than 30 (n < 30) Population normally distributed Population normally distributed Population standard deviation unknown Population standard deviation unknown 2.Use Student’s t distribution

© 1998 Prentice-Hall, Inc. Confidence Interval Mean (Small Sample) 1.Assumptions Sample size less than 30 (n < 30) Sample size less than 30 (n < 30) Population normally distributed Population normally distributed Population standard deviation unknown Population standard deviation unknown 2.Use Student’s t distribution 3.Confidence interval estimate

© 1998 Prentice-Hall, Inc. Student’s t Distribution

© 1998 Prentice-Hall, Inc. Z Student’s t Distribution 0 Standard Normal

© 1998 Prentice-Hall, Inc. Z t Student’s t Distribution 0 Standard Normal t (df = 13) Bell-ShapedSymmetric ‘Fatter’ Tails

© 1998 Prentice-Hall, Inc. Z t Student’s t Distribution 0 t (df = 5) Standard Normal t (df = 13) Bell-ShapedSymmetric ‘Fatter’ Tails

© 1998 Prentice-Hall, Inc. Student’s t Table

© 1998 Prentice-Hall, Inc. Student’s t Table

© 1998 Prentice-Hall, Inc. Student’s t Table t values

© 1998 Prentice-Hall, Inc. Student’s t Table t values  / 2

© 1998 Prentice-Hall, Inc. Student’s t Table t values  / 2 Assume: n = 3 df= n - 1 = 2  =.10  /2 =.05

© 1998 Prentice-Hall, Inc. Student’s t Table t values  / 2 Assume: n = 3 df= n - 1 = 2  =.10  /2 =.05

© 1998 Prentice-Hall, Inc. Student’s t Table t values  / 2 Assume: n = 3 df= n - 1 = 2  =.10  /2 =.05.05

© 1998 Prentice-Hall, Inc. Student’s t Table Assume: n = 3 df= n - 1 = 2  =.10  /2 = t values  / 2.05

© 1998 Prentice-Hall, Inc. Degrees of Freedom ( df ) 1.Number of observations that are free to vary after sample statistic has been calculated

© 1998 Prentice-Hall, Inc. Degrees of Freedom ( df ) 1.Number of observations that are free to vary after sample statistic has been calculated 2.Example: Sum of 3 numbers is 6 X 1 = X 2 = X 3 = Sum = 6

© 1998 Prentice-Hall, Inc. Degrees of Freedom ( df ) 1.Number of observations that are free to vary after sample statistic has been calculated 2.Example: Sum of 3 numbers is 6 X 1 = 1 (Or any number) X 2 = X 3 = Sum = 6

© 1998 Prentice-Hall, Inc. Degrees of Freedom ( df ) 1.Number of observations that are free to vary after sample statistic has been calculated 2.Example: Sum of 3 numbers is 6 X 1 = 1 (Or any number) X 2 = 2 (Or any number) X 3 = Sum = 6

© 1998 Prentice-Hall, Inc. Degrees of Freedom ( df ) 1.Number of observations that are free to vary after sample statistic has been calculated 2.Example: Sum of 3 numbers is 6 X 1 = 1 (Or any number) X 2 = 2 (Or any number) X 3 = 3 (Cannot vary) Sum = 6

© 1998 Prentice-Hall, Inc. Degrees of Freedom ( df ) 1.Number of observations that are free to vary after sample statistic has been calculated 2.Example: Sum of 3 numbers is 6 X 1 = 1 (Or any number) X 2 = 2 (Or any number) X 3 = 3 (Cannot vary) Sum = 6 degrees of freedom = n -1 = 3 -1 = 2

© 1998 Prentice-Hall, Inc. Estimation Example Mean (Small Sample) A random sample of n = 25 has  x = 50 & s = 8. Set up a 95% confidence interval estimate for .

© 1998 Prentice-Hall, Inc. Estimation Example Mean (Small Sample) A random sample of n = 25 has  x = 50 & s = 8. Set up a 95% confidence interval estimate for .

© 1998 Prentice-Hall, Inc. Thinking Challenge You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time? AloneGroupClass

© 1998 Prentice-Hall, Inc. Confidence Interval Solution*  X = 3.7 S = S = n = 6, df = n - 1 = = 5 n = 6, df = n - 1 = = 5 S /  n = /  6 = S /  n = /  6 = t.05,5 = t.05,5 = (2.015)(1.592)  (2.015)(1.592) (2.015)(1.592)  (2.015)(1.592)   6.908

© 1998 Prentice-Hall, Inc. Confidence Interval Estimate of Proportion

© 1998 Prentice-Hall, Inc. Data Types

© 1998 Prentice-Hall, Inc. Qualitative Data 1.Qualitative random variables yield responses that classify e.g., gender (male, female) e.g., gender (male, female) 2.Measurement reflects # in category 3.Nominal or ordinal scale 4.Examples Do you own savings bonds? Do you own savings bonds? Do you live on-campus or off-campus? Do you live on-campus or off-campus?

© 1998 Prentice-Hall, Inc. Proportions 1.Involve qualitative variables 2.Fraction or % of population in a category 3.If two qualitative outcomes, binomial distribution Possess or don’t possess characteristic Possess or don’t possess characteristic

© 1998 Prentice-Hall, Inc. Proportions 1.Involve qualitative variables 2.Fraction or % of population in a category 3.If two qualitative outcomes, binomial distribution Possess or don’t possess characteristic Possess or don’t possess characteristic 4.Sample proportion (p) ^

© 1998 Prentice-Hall, Inc. Sampling Distribution of Proportion

© 1998 Prentice-Hall, Inc. Sampling Distribution of Proportion Sampling Distribution P ^ P(P ^ )

© 1998 Prentice-Hall, Inc. 1.Approximated by normal distribution excludes 0 or n excludes 0 or n Sampling Distribution of Proportion Sampling Distribution P ^ P(P ^ )

© 1998 Prentice-Hall, Inc. 1.Approximated by normal distribution excludes 0 or n excludes 0 or n 2.Mean Sampling Distribution of Proportion Sampling Distribution P ^ P(P ^ )

© 1998 Prentice-Hall, Inc. p a 1.Approximated by normal distribution excludes 0 or n excludes 0 or n 2.Mean 3.Standard error Sampling Distribution of Proportion Sampling Distribution  p ^ p n  1 f P ^ P(P ^ )

© 1998 Prentice-Hall, Inc. Confidence Interval Estimates One Population Z Distribution t Large Sample Z Distribution MeanProportion Small Sample

© 1998 Prentice-Hall, Inc. Confidence Interval Proportion

© 1998 Prentice-Hall, Inc. Confidence Interval Proportion 1.Assumptions Two categorical outcomes Two categorical outcomes Population follows binomial distribution Population follows binomial distribution Normal approximation can be used Normal approximation can be used does not include 0 or 1 does not include 0 or 1

© 1998 Prentice-Hall, Inc. Confidence Interval Proportion 1.Assumptions Two categorical outcomes Two categorical outcomes Population follows binomial distribution Population follows binomial distribution Normal approximation can be used Normal approximation can be used does not include 0 or 1 does not include 0 or 1 2.Confidence interval estimate

© 1998 Prentice-Hall, Inc. Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p.

© 1998 Prentice-Hall, Inc. Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p.

© 1998 Prentice-Hall, Inc. Thinking Challenge You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective? AloneGroupClass

© 1998 Prentice-Hall, Inc. Confidence Interval Solution*

© 1998 Prentice-Hall, Inc. Finding Sample Sizes

© 1998 Prentice-Hall, Inc. Finding Sample Sizes for Estimating 

© 1998 Prentice-Hall, Inc. Finding Sample Sizes for Estimating 

© 1998 Prentice-Hall, Inc. Finding Sample Sizes for Estimating 

© 1998 Prentice-Hall, Inc. Finding Sample Sizes for Estimating  Error is also called bound, B

© 1998 Prentice-Hall, Inc. Finding Sample Sizes for Estimating  I don’t want to sample too much or too little! Error is also called bound, B

© 1998 Prentice-Hall, Inc. Sample Size Example What sample size is needed to be 90% confident of being correct within  5? A pilot study suggested that the standard deviation is 45.

© 1998 Prentice-Hall, Inc. Sample Size Example What sample size is needed to be 90% confident of being correct within  5? A pilot study suggested that the standard deviation is 45.

© 1998 Prentice-Hall, Inc. Thinking Challenge You work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that  was about $400. What sample size do you use? AloneGroupClass

© 1998 Prentice-Hall, Inc. Sample Size Solution*

© 1998 Prentice-Hall, Inc. Conclusion 1.Stated what is estimated 2.Distinguished point & interval estimates 3.Explained interval estimates 4.Computed confidence interval estimates for population mean & proportion 5.Computed sample size

© 1998 Prentice-Hall, Inc. This Class... 1.What was the most important thing you learned in class today? 2.What do you still have questions about? 3.How can today’s class be improved? Please take a moment to answer the following questions in writing:

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