Thisassumptionisreasonablesince if x 1 x 2 x n

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Unformatted text preview: dom variables θ low and θ high such that: ˆ ˆ P(θ low < θ < θ high ) = 1 − α ↑ Greek letter alpha 1 − α is called the confidence level. ˆ ˆ [θ low , θ high ] gives a 100 (1 − α) % confidence interval estimator for θ . 9 Chapter 8 Interval Estimation for the Population Mean Results established in previous lecture notes are first reviewed. A point estimator for the population mean μ is: 1n X = ∑ Xi n i=1 X has a sampling distribution with the properties: E( X ) = μ ( X is an unbiased estimator of μ ) and Var ( X ) = σ n 2 The standard error of X is: se( X ) = σ n To proceed further, assume that X follows a normal distribution. This assumption is reasonable since: • If X 1 , X 2 , . . . , X n follow a normal distribution then a result is that X also has a normal distribution. • Even if X 1 , X 2 , . . . , X n are not normally distributed, by the Central Limit Theorem, X will tend to the normal distribution. A standard normal random variable is: Z= 10 X−μ X−μ ~ N(0 , 1) = se( X ) σ n Chapter 8 A procedure for interval estimation is now developed. A critical value zc can be found so that: P(− zc < Z < zc ) = 1 − α where 1 − α is set to a desired level. Rearrange the probability statement to obtain: ⎛ ⎞ ⎜− z < X − μ < z ⎟ P( − zc < Z < zc ) = P⎜ c c⎟ σ ⎜ ⎟ n ⎝ ⎠ ( = P X − zc σ n < μ < X + zc σ n ) This gives the 100 (1 − α) % confidence interval estimator for the population mean μ as: [ X − zc σ n , X + zc σ n This can be written as: 11 X ± zc σ n Chapter 8 To finish off, the critical value zc must be set. For a 90% confidence interval estimator, with α = 0.10 , find a number zc such that: P(− zc < Z < zc ) = 0.90 An illustration is below. PDF of Z f(z) Area = 0.9 Lower Tail Area = 0.05 -zc Note that the area in each tail is: 12 Upper Tail Area = 0.05 0 z zc α = 0.10 = 0.05 2 2 Chapter 8 This says find zc such that: P(Z < zc ) = F(zc ) = 1 − α 2 = 0.95 Appendix Table 1 can be used to get an answer. Another option is to use Microsoft Excel. Select Insert Function: NORMSINV(0.95) This returns the answer: zc = 1.645 The confidence level 1 − α can be set to any desired probability. A table of some popular choices is below. Confidence level α 2 Microsoft Excel NORMSINV probability zc 0.90 0.05 0.95 1.645 0.95 0.025 0.975 1.96 0.99 0.005 0.995 2.576 13 Chapter 8 An application can now proceed. Collect a data set with numeric observations: x 1 , x 2 , . . . , x n . The point estimate for the population mean μ is the calculated sample mean: 1n x = ∑ xi n i =1 Assume that the population standard deviation σ is known from previous research....
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