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Statistics for Business and Economics Chapter 5 Inferences Based on a Single Sample: Estimation with Confidence Intervals

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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 6.Discuss Finite Population Correction Factor

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Thinking Challenge Suppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out?

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Introduction to Estimation

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Statistical Methods Estimation Hypothesis Testing Inferential Statistics Descriptive Statistics

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Estimation Process Mean, , is unknown Population Sample Mean X = 50 Random Sample I am 95% confident that is between 40 & 60.

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Unknown Population Parameters Are Estimated Estimate Population Parameter... with Sample Statistic Differences 1 - 2 x 1 - x 2 Variance 2 s 2 ^ Proportionp p Mean x

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Estimation Methods Estimation Interval Estimation Point Estimation

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Estimation Methods Estimation Interval Estimation Point Estimation

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1.Provides a single value Based on observations from one sample 2.Gives no information about how close the value is to the unknown population parameter 3.Example: Sample mean x = 3 is point estimate of unknown population mean

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Interval Estimation

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Estimation Methods Estimation Interval Estimation Point Estimation

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Interval Estimation 1.Provides a range of values Based on observations from one sample 2.Gives information about closeness to unknown population parameter Stated in terms of probability –Knowing exact closeness requires knowing unknown population parameter 3.Example: Unknown population mean lies between 50 and 70 with 95% confidence

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Key Elements of Interval Estimation Sample statistic (point estimate) Confidence interval Confidence limit (lower) Confidence limit (upper) A probability that the population parameter falls somewhere within the interval.

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Confidence Limits for Population Mean Parameter = Statistic ± Error © 1984-1994 T/Maker Co.

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Many Samples Have Same Interval x _ XX X = ± Z x 90% Samples +1.65 x -1.65 x 95% Samples +1.96 x -1.96 x 99% Samples -2.58 x +2.58 x

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1.Probability that the unknown population parameter falls within interval 2.Denoted (1 – is probability that parameter is not within interval 3.Typical values are 99%, 95%, 90% Confidence Level

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Intervals & Confidence Level x = 1 - /2 X _ x _ Sampling Distribution of Sample Mean Large number of intervals (1 – α)% of intervals contain μ α% do not Intervals extend from X – Zσ X to X + Zσ X

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Factors Affecting Interval Width 1.Data dispersion Measured by Intervals extend from X – Z X to X + Z X © 1984-1994 T/Maker Co. 3.Level of confidence (1 – ) Affects Z 2.Sample size

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Confidence Interval Estimates Confidence Intervals MeanProportion σ Known σ Unknown

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Confidence Interval Estimate Mean ( Known)

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Confidence Interval Estimates Confidence Intervals MeanProportion σ Known σ Unknown

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Confidence Interval Mean ( Known) 1.Assumptions Population standard deviation is known Population is normally distributed If not normal, can be approximated by normal distribution (n 30) 2.Confidence interval estimate

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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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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 © 1984-1994 T/Maker Co. 2 liter

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Confidence Interval Solution*

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Confidence Interval Estimate Mean ( Unknown)

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Confidence Interval Estimates Confidence Intervals MeanProportion σ Known σ Unknown

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Confidence Interval Mean ( Unknown) 1.Assumptions Population standard deviation is unknown Population must be normally distributed 2.Use Student’s t–distribution

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Z t Student’s t Distribution 0 t (df = 5) Standard Normal t (df = 13) Bell-Shaped Symmetric ‘Fatter’ Tails

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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

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v t.10 t.05 t.025 13.0786.314 12.706 21.8862.9204.303 31.6382.3533.182 Student’s t Table t values Assume: n = 3 df= n - 1 = 2 =.10 /2 =.05 t 0 / 2 t 2.920

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Confidence Interval Mean ( Unknown)

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Estimation Example Mean ( Unknown) A random sample of n = 25 has x = 50 and s = 8. Set up a 95% confidence interval estimate for .

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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?

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Confidence Interval Solution* x = 3.7 s = 3.8987 n = 6, df = n - 1 = 6 - 1 = 5 t.05 = 2.015

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Confidence Interval Estimate of Proportion

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Confidence Interval Estimates Confidence Intervals MeanProportion σ Known σ Unknown

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1.Assumptions Random sample selected Normal approximation can be used if Confidence Interval Proportion 2.Confidence interval estimate

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Estimation Example Proportion A random sample of 400 graduates showed 32 went to graduate school. Set up a 95% confidence interval estimate for p.

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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?

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Confidence Interval Solution*

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Finding Sample Sizes

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Finding Sample Sizes for Estimating SE = Sampling Error I don’t want to sample too much or too little!

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Sample Size Example What sample size is needed to be 90% confident the mean is within 5? A pilot study suggested that the standard deviation is 45.

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Finding Sample Sizes for Estimating p SE = Sampling Error If no estimate of p is available, use p = q =.5

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Sample Size Example What sample size is needed to estimate p with 90% confidence and a width of.03?

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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?

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Sample Size Solution*

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Finite Population Correction Factor

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Use when n, the sample size, is relatively large compared to N, the size of the population If n/N >.05 use the finite population correction factor Finite population correction factor:

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Finite Population Correction Factor Approximate 95% confidence interval for μ: Approximate 95% confidence interval for p:

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Finite Population Correction Factor Example You want to estimate a population mean, μ, where x =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for μ. is greater than.05 use the finite correction factor Since

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Finite Population Correction Factor Example You want to estimate a population mean, μ, where x =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for μ.

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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 6.Discussed Finite Population Correction Factor

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