C1000__23[1]

# C1000__23[1] - Confidence intervals for the mean continued...

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1 Confidence intervals for the mean - continued Sample mean: X Population Mean - μ

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2 Reminder Point estimator for μ: Limitations of point estimators Interval estimation for μ X
3 A ( 1- α )% confidence interval for μ A ( 1-α)% confidence interval for μ is: Z ( 1-α )% X α/2% α/2% n X 2 1 σ α - ± Z

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4 Example A test for the level of potassium in the blood is not perfectly precise. Moreover, the actual level of potassium in a person’s blood varies slightly from day to day. Suppose that repeated measurements for the same person on different days vary normally with σ=0.2.
5 (a) Julie’s potassium level is measured three times and the mean result is . Give a 90% confidence interval for Julie’s mean blood potassium level. 4 . 3 X = 3 2 . 0 , ~ - level pottasium Mean μ N X 90% confidence interval for : X n X 2 1 σ α - ± Z 3 2 . 0 3.4 3 2 . 0 4 . 3 95 . 2 1 . 0 1 = ± = ± = - Z Z 19 . 0 3.4 ) 1155 . 0 ( 645 . 1 3.4 ± = ± = =[3.21,3.59]

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6 (b) A confidence interval of 95% level would be: (i) wider than a confidence interval of 90% level (ii) narrower than a confidence interval of 90% level (c) Give a 95% confidence interval for Julie’s mean blood potassium level. 95% confidence interval for : X n X 2 1 σ α - ± Z 3 2 . 0 3.4 3 2 . 0 4 . 3 975 . 2 05 . 0 1 = ± = ± = - Z Z 226 . 0 3.4 ) 1155 . 0 ( 96 . 1 3.4 ± = ± = =[3.174,3.626]
7 (d) Julie wants a 99% confidence interval of [3.3,3.5]. What sample size should she take to achieve this (=how many times should she measure her potassium blood level?) [3.3,3.5]=3.4±Z 0.995 (0.2/√n) 3.4 - Z 0.995 (0.2/√n)=3.3 3.4 + Z 0.995 (0.2/√n)=3.5 solve for n: subtract the first equation from the second 2Z 0.995 (0.2/√n)=0. 2 Z 0.995 (0.2/√n)=0.1 2.575(0.2/√n)=0.1 (0.2/√n)=0.0388 √n=5.15 n=26.5225 she needs 27 blood tests to achieve a 99% CI [3.3,3.5].

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8 2 2 1 = - m z n σ α Suppose we want a specific interval with a confidence level 1-α.
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