Normal Distributions

Transcription

1 Normal Distributios Stacey Hacock Look at these three differet data sets Each histogram is overlaid with a curve : A B C A) Weights (g) of ewly bor lab rat pups B) Mea aual temperatures ( F ) i A Arbor, Michiga C) Heights (cm) of 14 year old boys i Oxford, Eglad What differs betwee these three distributios? What characteristics are similar? May distributios we look at have a shape similar to those above: most of the data lies close to the mea, ad the left ad right sides are symmetric We call it bell-shaped ad the best example is called the Normal distributio Normal distributios all have the same shape; they differ oly i mea µ ad stadard deviatio σ Importat fact: Statistics vary from sample to sample, ad the patter is predictable For may statistics, the patter of the samplig distributio resembles a ormal distributio with a bell-shaped curve Studyig the ormal distributio will allow us to fid probabilities for statistical iferece which do ot require ruig simulatios 1 Properties of Normal Distributios The Stadard Normal Distributio has mea µ = 0 ad stadard deviatio σ = 1 We ca stadardize ay ormal distributio to make it have mea 0 ad stadard deviatio 1 by subtractig µ ad dividig by σ If a radom variable, X, has a ormal distributio with mea µ ad stadard deviatio σ, the Z = X µ σ has a Stadard Normal Distributio We use the stadardized versios to say how may stadard deviatios (σ s) a observatio is from the mea (µ) For example, suppose birth weights of full term babies have a ormal distributio with mea µ = 3000 grams ad stadard deviatio σ = 700 grams 1

2 The a baby who weighs 3500 grams is ( )/700 = 0714 stadard deviatios above the mea birth weight The Empirical Rule states that for variable X that has a ormal distributio with mea µ ad stadard deviatio σ, approximately 68% of the values of X fall withi 1 stadard deviatio of the mea i either directio (µ ± σ) 95% of the values of X fall withi 2 stadard deviatios of the mea i either directio (µ ± 2σ) 997% (almost all) of the values of X fall withi 3 stadard deviatios of the mea i either directio (µ ± 3σ) For example, approximately 95% of babies have birth weights betwee (700) = 1600 grams ad (700) = 4400 grams We calculate probabilities of a ormally distributed radom variable by fidig areas uder the ormal curve The applet will calculate these areas for us Use the applet to aswer the followig questios about the distributio of birth weights specified previously (mea 3000 grams ad stadard deviatio 700 grams) For all questios below, eter Birth weight for the Variable ame, 3000 for the Mea, ad 700 for the SD The click Scale to Fit (You ca igore the secod row of Mea ad SD this would add aother ormal distributio to the plot 1 What proportio of babies have birth weights below 2000 grams? Aswer: Check the box ext to the first row of probability calculatios ad select < Iput 2000 for X 2

3 2 What is the probability that a radomly selected baby will have a birth weight betwee 3100 ad 4500 grams? Aswer: Check both boxes of probability calculatios Iput 3100 for X i the first row ad 4500 for X i the secod row 3 What is the 25th percetile of birth weights? That is, at what birth weight do 25% of babies fall below ad 75% of babies fall above? Aswer: grams Check the box ext to the first row of probability calculatios ad select < Iput 025 for Probability 2 Samplig Distributios of Statistics A samplig distributio is a probability distributio of a sample statistic For large sample sizes, the samplig distributio of a sample proportio has a approximate ormal distributio with a 3

4 mea equal to the populatio proportio π ad stadard deviatio SD() = The sample size is large eough for this approximatio to be valid if both π ad (1 π) are at least 10 (or if the umber of successes ad the umber of failures i your sample are both at least 10) Example: Suppose i the populatio of all messages, 80% are spam, ad 20% are ot spam We pla o takig a simple radom sample of s ad measurig the sample proportio of s that are spam The for large, the samplig distributio of sample proportios is approximately ormal 08(1 08) with mea E() = 080 ad stadard deviatio SD() = For istace, if = 50, the π = 50(08) = 40 ad (1 π) = 50(02) = 10 are both at least 10, so our sample size is large eough ad the samplig distributio of sample proportios will be 08(1 08) approximately ormal with mea 080 ad stadard deviatio 50 = The figure below visualizes this process: We start with the populatio of all s, for which 80% are spam ad 20% are ot spam (Note that each observatioal uit is oe The variable is whether a is spam or ot spam, which is categorical) Imagie selectig a sample of s ad calculate the sample proportio of s that are spam, We repeat this process may times, each time givig us a ew value for these may samples are represeted by pik circles Lastly, we plot all these values of o a dotplot, which will have the shape of the plot o the right as log as we have a large eough sample size Probability Populatio Distributio π = 08 Not Spam Spam POPULATION (eg, all messages) SAMPLING DISTRIBUTION of a Sample Proportio May may radom samples of size, with varyig sample proportios Samplig Distributio of the Sample Proportio (if π ad (1-π) are both greater tha or equal to 10) Std Dev = Populatio Proportio = π (eg, proportio of all messages that are spam) Sample Proportio = (eg, proportio of spam messages i our sample) Mea = π 4

5 We use this property of sample proportios whe we assess statistical sigificace usig a stadardized statistic I Exploratio 12, you leared how to calculate a stadardized sample proportio as: observed statistic mea of ull distributio SD of ull distributio The stadard deviatio of the ull distributio ca be calculated through simulatio, or through the formula above If the ceter of the ull distributio is π, this stadardizatio matches 3 Refereces π Robiso-Cox, J (2016) Stat 216 Course Pack Fall 2016: Activities ad Notes Licese: Creative Commos BY-SA 30 Utts, J M, & Heckard, R F (2015) Mid o Statistics, 5th ed, Chapters 7 ad 8 Stamford, CT: Cegage Learig 5