Statistical Inference: Sampling, Confidence Intervals, Hypothesis Testing, Exams of Statistics

Download Statistical Inference: Sampling, Confidence Intervals, Hypothesis Testing and more Exams Statistics in PDF only on Docsity! Stat 224 Fall 2004 Nov. 2-3, 2004 Discussion 7 1 The Sampling Distribution of the Mean (σ unknown) 1.1 Theorem 6.3 If X̄ is the mean of a random sample of size n taken from a normal population having the mean µ and the variance σ2, and S2 = n∑ i=1 (Xi − X̄)2 n− 1 , then t = X̄ − µ S/ √ n is a random variable having the t distribution with the parameter ν = n− 1. 2 Confidence Interval (1− α)× 100% Confidence Interval for µ (when σ is known or the same size is large):( x̄− zα/2 · σ√ n , x̄ + zα/2 · σ√ n ) (1− α)× 100% Confidence Interval for µ (when σ is unknown):( x̄− tn−1,α/2 · s√ n , x̄ + tn−1,α/2 · s√ n ) where s is the sample standard deviation. 2.1 Example (Exercise 7.80, page 276, Text) While performing a certain task under simulated weightlessness, the pulse rate of 32 astronaut trainees increased on the average by 26.4 beats per minute with a standard deviation of 4.28 beats per minute. What can one assert with 95% confidence about the maximum error if x̄ = 26.4 is used as a point estimate of the true average increase in the pulse rate of astronaut trainees performing the given task? 2.2 Example (Exercise 7.81, page 276, Text) With reference to the preceding exercise, construct a 95% confidence interval for the true average increase in the pulse rate of astronaut trainees performing the given task. 1231 MSC ting-li@stat.wisc.edu Ting-Li Lin