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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 State What Is Estimated**

Distinguish Point & Interval Estimates Explain Interval Estimates Compute Confidence Interval Estimates for Population Mean & Proportion Compute Sample Size Discuss Finite Population Correction Factor As a result of this class, you will be able to ...

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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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**Descriptive Statistics Inferential Statistics**

Statistical Methods Statistical Methods Descriptive Statistics Inferential Statistics Hypothesis Testing Estimation 5

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**I am 95% confident that is between 40 & 60.**

Estimation Process Population Mean X= 50 Random Sample I am 95% confident that is between 40 & 60. Mean, , is unknown Sample 7

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**Unknown Population Parameters Are Estimated**

Estimate Population Parameter... with Sample Statistic Mean x ^ Proportion p Variance 2 s Differences 1 - 2 x -

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Estimation Methods Estimation Interval Estimation Point Estimation 14

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Point Estimation

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Estimation Methods Estimation Interval Estimation Point Estimation 14

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**Point Estimation Provides a single value**

Based on observations from one sample Gives no information about how close the value is to the unknown population parameter 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 14

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**Interval Estimation Provides a range of values**

Based on observations from one sample Gives information about closeness to unknown population parameter Stated in terms of probability Knowing exact closeness requires knowing unknown population parameter 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. 24

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**Confidence Limits for Population Mean**

Parameter = Statistic ± Error © T/Maker Co. 27

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**Many Samples Have Same Interval**

X= ± Zx x _ X 90% Samples +1.65x -1.65x 99% Samples -2.58x +2.58x 95% Samples +1.96x -1.96x 29

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

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**Intervals & Confidence Level**

Sampling Distribution of Sample Mean _ s x a /2 a /2 1 - a _ X m = m ` x (1 – α)% of intervals contain μ α% do not Intervals extend from X – ZσX to X + ZσX Large number of intervals

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**Factors Affecting Interval Width**

Data dispersion Measured by Intervals extend from X – ZX toX + ZX Sample size Have students explain why each of these occurs. Level of confidence can be seen in the sampling distribution. Level of confidence (1 – ) Affects Z © T/Maker Co.

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**Confidence Interval Estimates**

Confidence Intervals Mean Proportion σ Known σ Unknown 38

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**Confidence Interval Estimate Mean ( Known)**

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**Confidence Interval Estimates**

Confidence Intervals Mean Proportion σ Known σ Unknown 43

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**Confidence Interval Mean ( Known)**

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

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**Estimation Example Mean ( Known)**

The mean of a random sample of n = 25 isX = 50. Set up a 95% confidence interval estimate for if = 10. 49

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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 2 liter © T/Maker Co.

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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 Mean Proportion σ Known σ Unknown 43

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**Confidence Interval Mean ( Unknown)**

Assumptions Population standard deviation is unknown Population must be normally distributed Use Student’s t–distribution

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**Student’s t Distribution**

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

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**Degrees of Freedom (df)**

Number of observations that are free to vary after sample statistic has been calculated Example Sum of 3 numbers is 6 X = 1 (or any number) X = 2 (or any number) X = 3 (cannot vary) Sum = 6 degrees of freedom = n - 1 = = 2

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**Student’s t Table t / 2 2.920 t values v t 1 3.078 6.314 2 1.886**

Assume: n = 3 df = n - 1 = 2 = .10 /2 =.05 t / 2 v t .10 .05 .025 1 3.078 6.314 12.706 2 1.886 2.920 4.303 3 1.638 2.353 3.182 Confidence intervals use /2, so divide ! t values 2.920 59

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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 . 70

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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? Allow students about 20 minutes to solve.

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**Confidence Interval Solution***

x = 3.7 s = n = 6, df = n - 1 = = 5 t.05 = 2.015 72

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**Confidence Interval Estimate of Proportion**

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**Confidence Interval Estimates**

Confidence Intervals Mean Proportion σ Known σ Unknown 43

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**Confidence Interval Proportion**

Assumptions Random sample selected Normal approximation can be used if 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. 82

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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! 89

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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. 91

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**Finding Sample Sizes for Estimating p**

SE = Sampling Error If no estimate of p is available, use p = q = .5 89

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Sample Size Example What sample size is needed to estimate p with 90% confidence and a width of .03? 91

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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* 93

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**Finite Population Correction Factor**

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**Finite Population Correction Factor**

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 Stated What Is Estimated**

Distinguished Point & Interval Estimates Explained Interval Estimates Computed Confidence Interval Estimates for Population Mean & Proportion Computed Sample Size Discussed Finite Population Correction Factor

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Estimation: Confidence Intervals Based in part on Chapter 6 General Business 704.

Estimation: Confidence Intervals Based in part on Chapter 6 General Business 704.

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