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# ps03 - Statistics 403 Problem Set 3 Due in lab on Friday...

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Statistics 403 Problem Set 3 Due in lab on Friday, October 1st 1. Suppose we are interested in estimating a quantity μ , and we can obtain independent measurements X i with EX i = μ . We intend to use ¯ X to estimate μ , and we wish to have the probability of the estimation error being greater than 0 . 1 units be equal to 0.1. What sample size is required if the standard deviation σ = SD( X i ) is (i) σ = 0 . 5 or (ii) σ = 0 . 2? You can treat ¯ X as being normally distributed. Solution: For this problem, I was thinking of the error as being the absolute difference between the estimate and the true value, | ¯ X - μ | . But if you use the signed error ¯ X - μ , that’s OK this time. The following calculation gives the probability of the absolute error being greater than 0 . 1 units: P ( | ¯ X - μ | > 0 . 1) = 2 P ( ¯ X - μ > 0 . 1) = 2 P ( n ¯ X - μ σ > 0 . 1 n/σ ) = 2 P ( Z > 0 . 1 n/σ ) We want an error larger than 0.1 to occur only 0.1 fraction of the time, so we set 2 P ( Z > 0 . 1 n/σ ) = 0 . 1 , or P ( Z > 0 . 1 n/σ ) = 0 . 05 .

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