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Statistics for Business and Economics

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

2 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 ...

3 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?

4 Introduction to Estimation

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

6 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

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

8 Estimation Methods Estimation Interval Estimation Point Estimation 14

9 Point Estimation

10 Estimation Methods Estimation Interval Estimation Point Estimation 14

11 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

12 Interval Estimation

13 Estimation Methods Estimation Interval Estimation Point Estimation 14

14 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

15 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

16 Confidence Limits for Population Mean
Parameter = Statistic ± Error © T/Maker Co. 27

17 Many Samples Have Same Interval
X=  ± Zx x _ X 90% Samples +1.65x -1.65x 99% Samples -2.58x +2.58x 95% Samples +1.96x -1.96x 29

18 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%

19 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

20 Factors Affecting Interval Width
Data dispersion Measured by  Intervals extend from X – ZX toX + ZX 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.

21 Confidence Interval Estimates
Confidence Intervals Mean Proportion σ Known σ Unknown 38

22 Confidence Interval Estimate Mean ( Known)

23 Confidence Interval Estimates
Confidence Intervals Mean Proportion σ Known σ Unknown 43

24 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

25 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. 49

26 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.

27 Confidence Interval Solution*
52

28 Confidence Interval Estimate Mean ( Unknown)

29 Confidence Interval Estimates
Confidence Intervals Mean Proportion σ Known σ Unknown 43

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

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

32 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

33 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

34 Confidence Interval Mean ( Unknown)

35 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

36 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.

37 Confidence Interval Solution*
x = 3.7 s = n = 6, df = n - 1 = = 5 t.05 = 2.015 72

38 Confidence Interval Estimate of Proportion

39 Confidence Interval Estimates
Confidence Intervals Mean Proportion σ Known σ Unknown 43

40 Confidence Interval Proportion
Assumptions Random sample selected Normal approximation can be used if Confidence interval estimate

41 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

42 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?

43 Confidence Interval Solution*
84

44 Finding Sample Sizes

45 Finding Sample Sizes for Estimating 
SE = Sampling Error I don’t want to sample too much or too little! 89

46 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

47 Finding Sample Sizes for Estimating p
SE = Sampling Error If no estimate of p is available, use p = q = .5 89

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

49 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?

50 Sample Size Solution* 93

51 Finite Population Correction Factor

52 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:

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

54 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

55 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.

56 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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