A Review and Some Connections. Lecture 3: The Normal Distribution and Statistical Inference. The Normal Distribution. Normal Distribution

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1 A Review ad Some Coectios Lecture 3: The Normal Distributio ad Statistical Iferece Sady Eckel The Normal Distributio The Cetral Limit Theorem Estimates of meas ad proportios: uses ad properties Cofidece itervals ad Hypothesis tests 24 April / 36 2 / 36 The Normal Distributio Normal Distributio A probability distributio for cotiuous data Characterized by a symmetric bell-shaped curve (Gaussia curve Normal Desity, f( µ + Symmetric about its mea µ Uder certai coditios, ca be used to approimate Biomial(,p distributio p>5 (1-p>5 3 / 36 Takes o values betwee ad + Mea = Media = Mode Area uder curve equals 1 Notatio for Normal radom variable: X N(µ, σ 2 Parameters µ = mea σ = stadard deviatio 4 / 36

2 Formula: Normal Probability Desity Fuctio (pdf Stadard Normal Normal Desity, f( µ + The ormal probability desity fuctio for X N(µ, σ 2 is: f ( = 1 2πσ e ( µ2 /2σ 2, < < + Note: π 3.14 ad e 2.72 are mathematical costats Defiitio: a Normal distributio N(µ, σ 2 with parameters µ = 0 ad σ = 1 Its desity fuctio is writte as: f ( = 1 2π e 2 /2, < < + We typically use the letter Z to deote a stadard ormal radom variable (Z N(0, 1 Importat! We use the stadard ormal all the time because if X N(µ, σ 2, the X µ σ N(0, 1 This process is called stadardizig a ormal radom variable 5 / 36 6 / Rule I Rule II 68% of the desity is withi oe stadard deviatio of the mea 95% of the desity is withi two stadard deviatios of the mea Normal Desity, f( µ 1σ µ µ + 1σ + Normal Desity, f( µ 2σ µ µ + 2σ + 7 / 36 8 / 36

3 Rule III Differet Meas 99.7% of the desity is withi three stadard deviatios of the mea Normal Desity, f( µ 3σ µ µ + 3σ + Normal Desity µ 1 µ 2 µ 3 9 / 36 Three ormal distributios with differet meas µ 1 < µ 2 < µ 3 10 / 36 Differet Stadard Deviatios Stadard Normal N(0,1 Normal Desity σ 1 σ 2 σ 3 Normal Desity σ=1 Three ormal distributios with differet stadard deviatios σ 1 < σ 2 < σ 3 11 / µ=0 12 / 36

4 Eample: Birthweights (i grams of ifats i a populatio Normal Probabilities Desity We are ofte iterested i the probability that z takes o values betwee z 0 ad z 1 P(z 0 z z 1 = z1 z 0 1 2π e z2 /2 dz Weights Cotiuous data Mea = Media = Mode = 3000 = µ Stadard deviatio = 1000 = σ The area uder the curve represets the probability (proportio of ifats with birthweights betwee certai values 13 / 36 How do we calculate this probability? Equivalet to fidig area uder the curve Cotiuous distributio, so we caot use sums to fid probabilities Performig the itegratio is ot ecessary sice tables ad computers are available 14 / 36 Z Tables But...we ll use R For stadard ormal radom variables Z N(0,1 we ll use 1 porm(? to fid P(Z? 2 porm(?, lower.tail=f to fid P(Z? <? >??? For ay ormal radom variable X N(µ, σ 2 (but takig X N(2,3 2 as a eample we ll use 1 porm(?, mea=2, sd=3 to fid P(X? 2 porm(?, mea=2, sd=3, lower.tail=f to fid P(X? 15 / / 36

5 Eample: Birthweights (i grams Questio I What is the probability of a ifat weighig more tha 5000g? Desity Weights µ = 3000 σ = 1000 X = birthweight Z = X µ σ 17 / 36 P(X > 5000 = P( X µ > σ = P(Z > 2 = Get this usig porm(2, lower.tail=f (sice we stadardized 18 / 36 Questio II Questio III What is the probability of a ifat weighig less tha 3500g? What is the probability of a ifat weighig betwee 2500 ad 4000g? P(X < 3500 = P( X µ < σ = P(Z < 0.5 = P(2500 < X < 4000 = P( < X µ 1000 σ = P( 0.5 < Z < 1 < = 1 P(Z > 1 P(Z < 0.5 = = / / 36

6 Statistical Iferece Defiitios Populatios ad samples Samplig distributios Statistical iferece is the attempt to reach a coclusio cocerig all members of a class from observatios of oly some of them. (Rues 1959 A populatio is a collectio of observatios A parameter is a umerical descriptor of a populatio A sample is a part or subset of a populatio A statistic is a umerical descriptor of the sample 21 / / 36 Populatio vs. Sample Estimatig the populatio mea, µ Populatio populatio size = N µ = mea, a measure of ceter σ 2 = variace, a measure of dispersio σ = stadard deviatio Sample from the populatio is used to calculate sample estimates (statistics that approimate populatio parameters sample size = X = sample mea s 2 = sample variace s = sample stadard deviatio Usually µ is ukow ad we would like to estimate it We use X to estimate µ We kow the samplig distributio of X Defiitio: Samplig distributio The distributio of all possible values of some statistic, computed from samples of the same size radomly draw from the same populatio, is called the samplig distributio of that statistic Populatio: parameters Sample: statistics 23 / / 36

7 Samplig Distributio of X The Cetral Limit Theorem (CLT Populatio Distributio of X Distributio of Sample Mea X X~N(µ,σ 2 X~N(µ,σ 2 Desity µ X Desity µ X =10 =30 =100 Whe samplig from a ormally distributed populatio X will be ormally distributed The mea of the distributio of X is equal to the true mea µ of the populatio from which the samples were draw The variace of the distributio is σ 2 /, where σ 2 is the variace of the populatio ad is the sample size We ca write: X N(µ, σ 2 / Whe samplig from a populatio whose distributio is ot ormal ad the sample size is large, use the Cetral Limit Theorem 25 / 36 Give a populatio of ay distributio with mea, µ, ad variace, σ 2, the samplig distributio of X, computed from samples of size from this populatio, will be approimately N(µ, σ 2 / whe the sample size is large I geeral, this applies whe 25 The approimatio of ormality becomes better as icreases 26 / 36 What if a radom variable has a Biomial distributio? Biomial CLT First, recall that a Biomial variable is just the sum of Beroulli variable: S = i=1 X i Notatio: S Biomial(,p X i Beroulli(p = Biomial(1, p for i = 1,..., I this case, we wat to estimate p by ˆp where ˆp = S = i=1 X i = X ˆp is just a sample mea! So we ca use the cetral limit theorem whe is large For a Beroulli variable µ = mea = p σ 2 = variace = p(1-p X N(µ, σ 2 / as before Equivaletly, ˆp N(p, p(1 p 27 / / 36

8 Distributio of Differeces Distributio of Differeces: Notatio Ofte we are iterested i detectig a differece betwee two populatios Differeces i average icome by eighborhood Differeces i disease cure rates by age Populatio 1: Size = N 1 Mea = µ 1 Stadard deviatio = σ 1 Populatio 2: Size = N 2 Mea = µ 2 Stadard deviatio = σ 2 Samples of size 1 from Populatio 1: Mea = µ X 1 = µ 1 Stadard deviatio = σ 1 / 1 = σ X 1 Samples of size 2 from Populatio 2: Mea = µ X2 = µ 2 Stadard deviatio = σ 2 / 2 = σ X2 29 / / 36 Distributio of Differeces: CLT result Differece i proportios? Now by CLT, for large : X 1 N(µ 1, σ 2 1 / 1 X 2 N(µ 2, σ 2 2 / 2 ad X 1 X 2 N(µ 1 µ 2, σ σ2 2 2 We re doe if the uderlyig variable is cotiuous. What if the uderlyig variable is Biomial? The X 1 X 2 N(µ 1 µ 2, σ2 1 is replaced by: 1 + σ2 2 2 ˆp 1 ˆp 2 N(p 1 p 2, p 1(1 p p 2(1 p / / 36

9 Summary of Samplig Distributios Statistical iferece Samplig Distributio Statistic Mea Variace σ X µ 2 X 1 X σ1 2 µ 1 - µ σ2 2 pq ˆp p ˆp p pq ˆp 1 ˆp 2 p 1 p 2 p 1 q p 2q Two methods Estimatio (Cofidece itervals Hypothesis testig Both make use of samplig distributios Remember to use CLT 33 / / 36 Rest of material moved to lecture 4 Lecture 3 Summary We did t get a chace to cover the rest of the material, so it has bee moved to lecture 4. The Normal Distributio The Cetral Limit Theorem Samplig distributios Net time, we ll discuss Cofidece itervals for populatio parameters The t-distributio Hypothesis testig (p-values 35 / / 36