PDB_Stat_100_Lecture_17_Printable

PDB_Stat_100_Lecture_17_Printable - STA 100 Lecture 17 Paul...

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STA 100 Lecture 17 Paul Baines Department of Statistics University of California, Davis February 11th, 2011
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Admin for the Day I Homework 4 due today by 5pm! I !!! PLEASE TURN IN A PRINTED COPY OF YOUR R OUTPUT !!! I Hwk 4, Q1(f) – check the forum post ‘Question 1.f’! I Hwk 4, Q3(e/f) – what are the assumptions? independent? normal? same mean μ for each month? same SD σ for each month? I Hwk 4, Q3(e/f) – is it normal, are consecutive weeks independent? does it matter? Either answer is fine – just justify it! I Stop the prejudice – let means be free (and negative) , I Project details coming soon. . . References for Today: Rosner, Ch 6.5-6.8, Ch 6.5 (7th Ed.) References for Friday: Rosner, Ch 7.1-7.4 (7th Ed.)
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More General CI’s In general, if σ is known, X i iid N ( μ, σ 2 ) for i = 1 , . . . , n , then a 100(1 - α )% CI for μ is given by: ± ¯ x - z 1 - α 2 · σ n , ¯ x + z 1 - α 2 · σ n ² . where z 1 - α 2 is the (1 - α 2 )th percentile of a normal distribution. Depending on what level confidence interval you want. . . 1. 90% CI: α = 0 . 10, (1 - α 2 ) = 0 . 950, z 0 . 950 = 1 . 64; 2. 95% CI: α = 0 . 05, (1 - α 2 ) = 0 . 975, z 0 . 975 = 1 . 96; 3. 99% CI: α = 0 . 01, (1 - α 2 ) = 0 . 995, z 0 . 995 = 2 . 58.
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CI for unknown σ As we discussed last time, generally you don’t know σ . If X i iid N ( μ, σ 2 ) for i = 1 , . . . , n , and you don’t know σ then a 100(1 - α )% CI for μ is given by: ± ¯ x - t n - 1 , 1 - α 2 · s n , ¯ x + t n - 1 , 1 - α 2 · s n ² . where t n - 1 , 1 - α 2 is the (1 - α 2 )th percentile of a t n - 1 distribution (i.e., the (1 - α 2 )th percentile of a t - distribution with n - 1 degrees of freedom. To find what t n - 1 , 1 - α 2 is in R , type: qt(1-alpha/2,df=n-1) where you replace n and alpha with the appropriate numbers! 1. 90% CI: α = 0 . 10, (1 - α 2 ) = 0 . 950, t n - 1 , 0 . 950 = qt(0.950,df=n-1) ; 2. 95% CI: α = 0 . 05, (1 - α 2 ) = 0 . 975, t n - 1 , 0 . 975 = qt(0.975,df=n-1) ; 3. 99% CI: α = 0 . 01, (1 - α 2 ) = 0 . 995, t n - 1 , 0 . 995 = qt(0.995,df=n-1) ;
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Understanding Confidence Intervals Now that we have the mechanics we can see what a confidence really means for an actual example. . . Lets stick with those trusty parsnips. You obviously don’t know the true distribution of parsnip weights. . . . . . but suppose you have magic powers. . . , It turns out that parsnips weights are normally distributed with mean μ and variance σ 2 . (This is the true distribution . Again, you would need magic powers to know this!)
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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 True Distribution of Parsnip Weights Weight (lb) Probability Density True value of mu = 0.52
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Back to Reality Back in the real world, we don’t have magic powers. . . / . . . but we do have data! , Suppose we weigh a crate of 200 parsnips. . .
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Parnsip Weights Actual weights of 200 parsnips (in lbs): ---------------------------------------- [1] 0.51 0.69 0.57 0.17 0.45 0.58 0.39 0.52 0.54 0.26 0.57 0.62 0.61 0.51 0.49 [16] 0.63 0.66 0.51 0.42 0.53 0.57 0.60 0.51 0.31 0.42 0.60 0.38 0.54 0.52 0.32 [31] 0.71 0.59 0.62 0.66 0.51 0.31 0.58 0.59 0.49 0.51 0.70 0.56 0.69 0.48 0.61 [46] 0.47 0.46 0.41 0.52 0.54 0.60 0.67 0.56 0.20 0.73 0.47 0.74 0.56 0.47 0.41 [61] 0.54 0.46 0.43 0.51 0.55 0.36 0.61 0.47 0.53 0.44 0.49 0.50 0.64 0.59 0.49 [76] 0.65 0.55 0.50 0.66 0.37 0.45 0.64 0.68 0.53 0.77 0.27 0.60 0.45 0.44 0.55 [91] 0.43 0.49 0.77 0.47 0.54 0.66 0.55 0.51 0.80 0.57 0.59 0.73 0.49 0.69 0.40 [106] 0.43 0.38 0.50 0.51 0.48 0.59 0.59 0.53 0.45 0.41 0.56 0.60 0.24 0.47 0.55 [121] 0.63 0.48 0.30 0.50 0.48 0.49 0.54 0.28 0.77 0.40 0.61 0.30 0.47 0.54 0.66 [136] 0.71 0.56 0.61 0.58 0.45 0.58 0.55 0.40 0.44 0.48 0.72 0.28 0.58 0.54 0.39 [151] 0.68 0.75 0.55 0.38 0.67 0.40 0.54 0.35 0.24 0.58 0.40 0.79 0.49 0.49 0.43 [166] 0.32 0.66 0.61 0.50 0.54 0.39 0.51 0.31 0.59 0.53 0.52 0.52 0.45 0.47 0.43
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PDB_Stat_100_Lecture_17_Printable - STA 100 Lecture 17 Paul...

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