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5 - 1 1997 Prentice-Hall, Inc. Importance of Normal Distribution n Describes many random processes or continuous phenomena n Can be used to approximate discrete probability distributions l Binomial l Poisson n Basis for classical statistical inference

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5 - 2 1997 Prentice-Hall, Inc. Normal Distribution n Bell-shaped & symmetrical n Mean, median, mode are equal Middle spread is 1.33 Middle spread is 1.33 n Random variable has infinite range Mean Median Mode

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5 - 3 1997 Prentice-Hall, Inc. Standardize the Normal Distribution One table! Normal Distribution Standardized Normal Distribution

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5 - 4 1997 Prentice-Hall, Inc. Standardizing Example Normal Distribution

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5 - 5 1997 Prentice-Hall, Inc. Standardizing Example Normal Distribution Standardized Normal Distribution

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5 - 6 1997 Prentice-Hall, Inc. Obtaining the Probability.0478.0478.02 0.1.0478 Standardized Normal Probability Table (Portion) ProbabilitiesProbabilities Shaded area exaggerated

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5 - 7 1997 Prentice-Hall, Inc. Example P(3.8 X 5)

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5 - 8 1997 Prentice-Hall, Inc. Example P(3.8 X 5) Normal Distribution.0478 Standardized Normal Distribution Shaded area exaggerated

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5 - 9 1997 Prentice-Hall, Inc. Example P(2.9 X 7.1)

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5 - 10 1997 Prentice-Hall, Inc. Example P(2.9 X 7.1) Normal Distribution.1664.1664.0832.0832 Standardized Normal Distribution Shaded area exaggerated

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5 - 11 1997 Prentice-Hall, Inc. Example P(X 8)

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5 - 12 1997 Prentice-Hall, Inc. Example P(X 8) Normal Distribution Standardized Normal Distribution.1179.1179.5000.3821.3821 Shaded area exaggerated

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5 - 13 1997 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough ( 30)... sampling distribution becomes almost normal.

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5 - 14 1997 Prentice-Hall, Inc. Introduction to Estimation

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5 - 15 1997 Prentice-Hall, Inc. Statistical Methods

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5 - 16 1997 Prentice-Hall, Inc. Estimation Process Mean, , is unknown Population Random Sample I am 95% confident that is between 40 & 60. Mean X = 50 Sample

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5 - 17 1997 Prentice-Hall, Inc. Population Parameters Are Estimated

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5 - 18 1997 Prentice-Hall, Inc. Point Estimation n Provides single value l Based on observations from 1 sample n Gives no information about how close value is to the unknown population parameter Example: Sample mean X = 3 is point estimate of unknown population mean Example: Sample mean X = 3 is point estimate of unknown population mean

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5 - 19 1997 Prentice-Hall, Inc. Interval Estimation n Provides range of values l Based on observations from 1 sample n Gives information about closeness to unknown population parameter l Stated in terms of probability n Example: Unknown population mean lies between 50 & 70 with 95% confidence

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5 - 20 1997 Prentice-Hall, Inc. Key Elements of Interval Estimation Confidence interval Sample statistic (point estimate) Confidence limit (lower) Confidence limit (upper) A probability that the population parameter falls somewhere within the interval.

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5 - 21 1997 Prentice-Hall, Inc. Confidence Limits for Population Mean Parameter = Statistic Error 1984-1994 T/Maker Co.

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5 - 22 1997 Prentice-Hall, Inc. Many Samples Have Same Interval 90% Samples x_ XXXX X = Z x +1.65 x -1.65 x

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5 - 23 1997 Prentice-Hall, Inc. Many Samples Have Same Interval 90% Samples 95% Samples +1.65 x x_ XXXX +1.96 x -1.65 x -1.96 x X = Z x

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5 - 24 1997 Prentice-Hall, Inc. Many Samples Have Same Interval 90% Samples 95% Samples 99% Samples +1.65 x +2.58 x x_ XXXX +1.96 x -2.58 x -1.65 x -1.96 x X = Z x

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5 - 25 1997 Prentice-Hall, Inc. n Probability that the unknown population parameter falls within interval Denoted (1 - Denoted (1 - is probability that parameter is not within interval is probability that parameter is not within interval n Typical values are 99%, 95%, 90% Level of Confidence

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5 - 26 1997 Prentice-Hall, Inc. Intervals & Level of Confidence Sampling Distribution of Mean Large number of intervals Intervals extend from X - Z X to X + Z X (1 - ) % of intervals contain . % do not.

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5 - 27 1997 Prentice-Hall, Inc. Factors Affecting Interval Width n Data dispersion Measured by Measured by n Sample size X = / n X = / n Level of confidence (1 - ) Level of confidence (1 - ) l Affects Z Intervals extend from X - Z X to X + Z X 1984-1994 T/Maker Co.

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5 - 28 1997 Prentice-Hall, Inc. Confidence Interval Estimates

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5 - 29 1997 Prentice-Hall, Inc. Confidence Interval Estimate Mean ( Known)

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5 - 30 1997 Prentice-Hall, Inc. Confidence Interval Estimates

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5 - 31 1997 Prentice-Hall, Inc. Confidence Interval Mean ( Known) n Assumptions l Population standard deviation is known l Population is normally distributed If not normal, can be approximated by normal distribution (n 30) If not normal, can be approximated by normal distribution (n 30)

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5 - 32 1997 Prentice-Hall, Inc. Confidence Interval Mean ( Known) n Assumptions l Population standard deviation is known l Population is normally distributed If not normal, can be approximated by normal distribution (n 30) If not normal, can be approximated by normal distribution (n 30) n Confidence interval estimate

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5 - 33 1997 Prentice-Hall, Inc. Estimation Example Mean ( Known) The mean of a random sample of n = 25 is X = 50. Set up a 95% confidence interval estimate for if = 10.

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5 - 34 1997 Prentice-Hall, Inc. Estimation Example Mean ( Known) The mean of a random sample of n = 25 is X = 50. Set up a 95% confidence interval estimate for if = 10.

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5 - 35 1997 Prentice-Hall, Inc. Confidence Interval Solution*

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5 - 36 1997 Prentice-Hall, Inc. Confidence Interval Estimate Mean ( Unknown)

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5 - 37 1997 Prentice-Hall, Inc. Confidence Interval Estimates

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5 - 38 1997 Prentice-Hall, Inc. Confidence Interval Mean ( Unknown) n Assumptions l Population standard deviation is unknown l Population must be normally distributed n Use Students t distribution n Confidence interval estimate

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5 - 39 1997 Prentice-Hall, Inc. Students t Distribution 0 t (df = 5) Standard normal t (df = 13) Bell- shaped Symmetric Fatter tails

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5 - 40 1997 Prentice-Hall, Inc. Students t Table t values / 2 Assume: n = 3 df= n - 1 = 2 =.10 /2 =.05.05

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5 - 41 1997 Prentice-Hall, Inc. Students t Table Assume: n = 3 df= n - 1 = 2 =.10 /2 =.05 2.920 t values / 2.05

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5 - 42 1997 Prentice-Hall, Inc. Estimation Example Mean ( Unknown) A random sample of n = 25 has X = 50 & S = 8. Set up a 95% confidence interval estimate for .

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5 - 43 1997 Prentice-Hall, Inc. Thinking Challenge Youre a time study analyst in manufacturing. Youve 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

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5 - 44 1997 Prentice-Hall, Inc. Confidence Interval Solution* X = 3.7 S = 3.8987 S = 3.8987 n = 6, df = n - 1 = 6 - 1 = 5 n = 6, df = n - 1 = 6 - 1 = 5 S / n = 3.8987 / 6 = 1.592 S / n = 3.8987 / 6 = 1.592 t.05,5 = 2.0150 t.05,5 = 2.0150 3.7 - (2.015)(1.592) 3.7 + (2.015)(1.592) 3.7 - (2.015)(1.592) 3.7 + (2.015)(1.592) 0.492 6.908 0.492 6.908

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5 - 45 1997 Prentice-Hall, Inc. Estimation of Mean for Finite Populations

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5 - 46 1997 Prentice-Hall, Inc. Confidence Interval Estimates

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5 - 47 1997 Prentice-Hall, Inc. Estimation for Finite Populations n Assumptions l Sample is large relative to population s n / N >.05 n Use finite population correction factor Confidence interval (mean, unknown) Confidence interval (mean, unknown)

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5 - 48 1997 Prentice-Hall, Inc. Confidence Interval Estimate of Proportion

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5 - 49 1997 Prentice-Hall, Inc. Confidence Interval Estimates

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5 - 50 1997 Prentice-Hall, Inc. Confidence Interval Proportion n Assumptions l Two categorical outcomes l Population follows binomial distribution l Normal approximation can be used np 5 & n(1 - p) 5 n Confidence interval estimate

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5 - 51 1997 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.

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5 - 52 1997 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.

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5 - 53 1997 Prentice-Hall, Inc. Thinking Challenge Youre 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

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5 - 54 1997 Prentice-Hall, Inc. Confidence Interval Solution* np 5 n(1 - p) 5

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5 - 55 1997 Prentice-Hall, Inc. This Class... n What was the most important thing you learned in class today? n What do you still have questions about? n How can todays class be improved? Please take a moment to answer the following questions in writing: