PDB_Stat_100_Lecture_16_Printable

PDB_Stat_100_Lecture_16_Printable - STA 100 Lecture 16 Paul...

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STA 100 Lecture 16 Paul Baines Department of Statistics University of California, Davis February 9th, 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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Computing Normal Probabilities Let X be normally distributed with mean μ and standard deviation σ , (i.e., X N ( μ, σ 2 )). We are often asked to compute things like P ( X > k ), or P ( X < k ), or P ( a < X < b ). We can do all of this with R , but what did people do before computers? They used normal probability tables. Did they have a separate table for every different mean μ and standard deviation σ ? No! That’s a lot of tables. . . ,
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z -scores Instead, they used rules for linear transformations so they could just have one table. How? I Let X N (100 , 10 2 ) and let Z N (0 , 1 2 ). What is P ( X < 120)? = 0 . 977 What is P ( Z < 2)? = 0 . 977 I Let X N (20 , 5 2 ) and let Z N (0 , 1 2 ). What is P ( X < 12)? = 0 . 055 What is P ( Z < - 1 . 6)? = 0 . 055 If we can just convert everything to mean = 0, sd = 1, then we only need one table!
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z - scores The way we do this, is to form a z - score by subtracting the mean and dividing by the standard deviation. This z - score tells us ‘how many standard deviations away from the mean’ we are. If X N ( μ, σ 2 ), then: Z = X - μ σ , Z N (0 , 1) You can check this yourself –find a and b and plug into the formulae for linear transformations!
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Using z - scores Example: Tax bills (or refunds, if negative) for UC Davis students are normally distributed with mean - $400 and standard deviation $250 i.e., X N ( - 400 , 250 2 ). You only have access to a table of N (0 , 1) probabilities. Q: What proportion of UC Davis students will receive a refund? A: We want P ( X < 0). So. . . subtract the mean from both sides, and divide both sides by the standard deviation. . . P ( X < 0) = P X - ( - 400) 250 < 0 - ( - 400) 250 P ( Z < 1 . 60) = 0 . 945 I Note that X - ( - 400) 250 N (0 , 1)! I A normal distribution with mean = 0 and variance = 1 is known as a standard normal distribution I We use letter Z for standard normal distributions, i.e., Z N (0 , 1).
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About z - scores Much (but not all) of what we do with z -scores, and standardization of normal variables (i.e., transforming X from N ( μ, σ 2 ) to N (0 , 1)) is a legacy from pre-computer days, but. . . Understanding the idea of ‘how many standard deviations from the mean’ is still very useful.
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