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Statistical Inference: Methods and Applications

Learn about the field of statistical inference, including parameter estimation and hypothesis testing, and understand how to make decisions and draw conclusions about populations based on sample data.

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Statistical Inference: Methods and Applications

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  1. 7-1 Introduction • The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population. • These methods utilize the information contained in a samplefrom the population in drawing conclusions. • Statistical inference may be divided into two major areas: • Parameter estimation • Hypothesis testing

  2. 7-1 Introduction Definition

  3. 7-1 Introduction

  4. 7-1 Introduction

  5. 7.2 Sampling Distributions and the Central Limit Theorem Statistical inference is concerned with making decisions about a population based on the information contained in a random sample from that population. Definitions:

  6. 7.2 Sampling Distributions and the Central Limit Theorem

  7. 7.2 Sampling Distributions and the Central Limit Theorem Figure 7-1 Distributions of average scores from throwing dice. [Adapted with permission from Box, Hunter, and Hunter (1978).]

  8. 7.2 Sampling Distributions and the Central Limit Theorem Example 7-1

  9. 7.2 Sampling Distributions and the Central Limit Theorem Figure 7-2Probability for Example 7-1

  10. 7.2 Sampling Distributions and the Central Limit Theorem Approximate Sampling Distribution of a Difference in Sample Means

  11. 7-3 General Concepts of Point Estimation 7-3.1 Unbiased Estimators Definition

  12. 7-3 General Concepts of Point Estimation Example 7-1

  13. 7-3 General Concepts of Point Estimation Example 7-1 (continued)

  14. 7-3 General Concepts of Point Estimation 7-3.2 Variance of a Point Estimator Definition Figure 7-5The sampling distributions of two unbiased estimators

  15. 7-3 General Concepts of Point Estimation 7-3.2 Variance of a Point Estimator

  16. 7-3 General Concepts of Point Estimation 7-3.3 Standard Error: Reporting a Point Estimate Definition

  17. 7-3 General Concepts of Point Estimation 7-3.3 Standard Error: Reporting a Point Estimate

  18. 7-3 General Concepts of Point Estimation Example 7-5

  19. 7-3 General Concepts of Point Estimation Example 7-5 (continued)

  20. 7-3 General Concepts of Point Estimation 7-3.4 Mean Square Error of an Estimator Definition

  21. 7-3 General Concepts of Point Estimation 7-3.4 Mean Square Error of an Estimator

  22. 7-3 General Concepts of Point Estimation 7-3.4 Mean Square Error of an Estimator Figure 7-6A biased estimator that has smaller variance than the unbiased estimator

  23. 7-4 Methods of Point Estimation Definition Definition

  24. 7-4 Methods of Point Estimation Example 7-7

  25. 7-4 Methods of Point Estimation 7-4.2 Method of Maximum Likelihood Definition

  26. 7-4 Methods of Point Estimation Example 7-9

  27. 7-4 Methods of Point Estimation Example 7-9 (continued)

  28. 7-4 Methods of Point Estimation Figure 7-7Log likelihood for the exponential distribution, using the failure time data. (a) Log likelihood with n = 8 (original data). (b) Log likelihood if n = 8, 20, and 40.

  29. 7-4 Methods of Point Estimation Example 7-12

  30. 7-4 Methods of Point Estimation Example 7-12 (continued)

  31. 7-4 Methods of Point Estimation Properties of the Maximum Likelihood Estimator

  32. 7-4 Methods of Point Estimation The Invariance Property

  33. 7-4 Methods of Point Estimation Example 7-13

  34. 7-4 Methods of Point Estimation • Complications in Using Maximum Likelihood Estimation • It is not always easy to maximize the likelihood function because the equation(s) obtained from dL()/d = 0 may be difficult to solve. • It may not always be possible to use calculus methods directly to determine the maximum of L().

  35. 7-4 Methods of Point Estimation Example 7-14

  36. 7-4 Methods of Point Estimation Figure 7-8The likelihood function for the uniform distribution in Example 7-13.

  37. 7-4 Methods of Point Estimation 7-4.3 Bayesian Estimation of Parameters

  38. 7-4 Methods of Point Estimation 7-4.3 Bayesian Estimation of Parameters

  39. 7-4 Methods of Point Estimation Example 7-16

  40. 7-4 Methods of Point Estimation Example 7-16 (Continued)

  41. 7-4 Methods of Point Estimation Example 7-16 (Continued)

  42. 7-4 Methods of Point Estimation Example 7-16 (Continued)

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