2379-chapter10 - MAT 2379 Introduction to Biostatistics...

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MAT 2379, Introduction to Biostatistics, Lecture Notes for Chapter 10 1 MAT 2379, Introduction to Biostatistics Chapter 10. Confidence Intervals The goal of the statistical inference is to draw conclusions about an unknown parameter which describes a feature of the population (given by a measurement X ). Examples of parameters include: (a) the population mean μ ; (b) the population variance σ 2 ; (c) the proportion p of individuals with a certain characteristic. In this chapter, we introduce the method of estimation by confidence intervals , for estimating the unknown parameter. This method produces an interval of possible values that includes the unknown parameter with a high probability. Our conclusions will be based on the observed values x 1 , . . . , x n of the measurement X for a particular sample drawn from the population. Note that x 1 , . . . , x n can be regarded as the observed values which correspond to some random measurements X 1 , . . . , X n . The measurements X 1 , . . . , X n are independent and have the same distribution as X . A point estimator for the population mean μ is the theoretical quantity: ¯ X = X 1 + . . . + X n n . The sample mean ¯ x is the observed value of this estimator for our particular sample, and is called an estimate of μ . A point estimator for the population variance σ 2 is the theoretical quantity: S 2 = 1 n - 1 n X i =1 ( X i - ¯ X ) 2 . The sample variance s 2 is the observed value of this estimator for our particular sample, and is called an estimate of σ 2 . In summary, we have the following table: Parameter Estimator Estimate μ ¯ X ¯ x σ 2 S 2 s 2 Example 1. (Example 10.1 in the book) A research project supported in part by the Canadian Wildlife Federation, shows that in the recent years, an increased number of polar bears in the Beauford Sea are eating less, possibly due to a decrease in the number of ringed seals (the bear’s main food source), during a critical spring feeding period. Further indication that the bears are fasting are smaller weights of their cubs at birth. The following data gives the weights at birth (in grams), for a sample of n = 5 cubs: x 1 = 785 , x 2 = 825 x 3 = 671 x 4 = 981 x 5 = 732 .
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