Ch5_CI_2-1

# Ch5_CI_2-1 - OMS 2550(001 Lecture Notes Chapter 5...

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OMS 2550 (001) Lecture Notes Chapter 5 – Inferences Based on a Single Sample: Estimation with Confidence Intervals (2) Updated 10/10/2011 1 / 15

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Population Parameters, Estimators, Standard Errors Parameter Estimator Standard Error Estimated Standard Error of the Estimator of the Estimator μ ¯ x σ n s n p ˆ p pq n ˆ p ˆ q n 2 / 15
C.I. for μ when σ is known Suppose we take a random sample of size n ... 1 Case 1: Normal population, any sample size ¯ x N μ, σ n 2 Case 2: Large sample size (i.e. n 30 ), any population Based on CLT, ¯ x is approximately N μ, σ n . For both cases, the 100(1- α )% C.I. for μ is ¯ x ± z α / 2 σ ¯ x = ¯ x ± z α / 2 σ n 3 / 15

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C.I. for μ when σ is unknown Suppose we take a random sample of size n... If σ is unknown, we can use s to estimate σ . The estimated distribution for the standardized sample mean ( ¯ x - μ s / n ) has a t-distribution with ( n - 1 ) d.f.. 1 Case 1: Large sample size (i.e. n 30) s is a good estimate for σ , and the t-distribution is close to the standard normal. ¯ x ± t α / 2 s n ¯ x ± z α / 2 s n (1) 2 Case 2: Small sample size; normal population. ¯ x ± t α / 2 s n (2) 3 Case 3: Small sample size; non normal population. (Other methods, not covered in OMS 2550) 4 / 15
Confidence Interval for p 1 Case 1: Large Sample Conditions required for a valid large-sample confidence interval for p : A random sample is selected from the target population.

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