AP Statistics Notes Unit Two: The Normal Distributions
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1 AP Statistics Ntes Unit Tw: The Nrmal Distributins Syllabus Objectives: 1.5 The student will summarize distributins f data measuring the psitin using quartiles, percentiles, and standardized scres (z-scres). This unit intrduces yu t the cncept f describing an bservatin s lcatin within a distributin. Yu will learn hw t use the Nrmal distributin t find standardized values, percentile ranks, and prprtins f bservatins n intervals. Yu will als be intrduced t methds fr assessing the Nrmality f a distributin. Measuring relative standing: Cnsider the fllwing test scres fr a small class: Jenny s scre is nted in red. Hw did she perfrm n this test relative t her peers? The mean, shwn abve n the Minitab printut, is 80. Her scre is abve average, but hw far abve average is it? One way t describe relative psitin in a data set is t tell hw many standard deviatins abve r belw the mean an bservatin is. Standardized Value: z-scre If the mean and standard deviatin f a distributin are knwn, the z-scre can be fund. x mean z = standard deviatin A z-scre tells us hw many standard deviatins away frm the mean the bservatin falls. Ex: Frm abve, the mean test scre is 80 and the standard deviatin is 6.07 pints. x T find Jenny s standardized z-scre, z = = = Jenny s scre is almst ne full standard deviatin abve the mean. Fr a persn scring 80, they wuld have a z-scre f 0, and fr a persn scring belw the mean, their z-scre wuld be negative. 1
2 Standardized values can be used t cmpare scres frm tw different distributins. Statistics Test: mean = 80, std dev = 6.07 Chemistry Test: mean = 76, std dev = 4 Jenny gt an 86 in Statistics and an 82 in Chemistry. On which test did she perfrm better? Statistics z = = Chemistry z = = Althugh Jenny had a lwer Chemistry scre, she perfrmed relatively better in Chemistry! 2
3 Syllabus Objectives: 1.11 The student will cmpare distributins f data with respect t their shapes. In Unit 1, we learned hw t plt a data set t describe its shape, center, spread and unusual features. Smetimes, the verall pattern f a large number f bservatins is s regular that we can describe it using a smth curve. Density Curve: An idealized descriptin f the verall pattern f a distributin. Area underneath the curve = 1, representing 100% f the bservatins. Density curves cme in many different shapes; symmetric, skewed, unifrm, etc. The area f a regin f a density curve represents the % f bservatins that fall in that regin. The median f a density curve cuts the area in half. The mean f a density curve is its balance pint. The median and mean are the same fr a symmetric density curve. Bth lie at the center. The mean f a skewed curve is pulled away frm the median in the directin f the lng tail. 3
4 Syllabus Objectives: 3.12 The student will describe the prperties f the Nrmal distributin. A very cmmn type f density curve, knwn as the Nrmal distributin, is widely used. These curves prvide a reasnable apprximatin t the distributin f many different variables. They play a central rle in many f the inferential prcedures discussed later. Nrmal distributins are cntinuus distributins with the fllwing prperties: The curve is single-peaked (unimdal). The shape is symmetric. Mre specifically, the distributin is bell-shaped. The curve is described by tw things: its mean, µ, and its standard deviatin, σ. Parameters are ppulatin variables. They are represented by Greek letters, like µ andσ. Statistics are sample variables. They are represented by Rman letters, like x and s. A parameter describes a ppulatin characteristic and a statistic describes a sample characteristic. Examples f Nrmal distributins: This is mre spread ut. The distr. has a larger σ. This is less spread ut. The distr. has a smaller σ. Inflectin pints the pints at which a change f curvature takes place. They are lcated at a distance σ n either side f µ. 4
5 The Rule, als knwn as the Empirical Rule In a Nrmal distributin, 68% f the bservatins fall within neσ f µ. 95% f the bservatins fall within twσ f µ. 99.7% f the bservatins fall within threeσ f µ. Ntatin: N ( µσ, ) If N(64.5,2.5), then we have a Nrmal distributin with a mean f 64.5 and a standard deviatin f 2.5. Here is the Nrmal distributin f heights with a mean f 64.5 inches and a S.D. f 2.5 inches. The Rule shws that 68% f the heights will be fund between 62 and 67 inches, 95% f the heights will be fund between 59.5 and 69.5 inches and 99.7% f the heights will be between 57 and 72 inches. Ex: Suppse that we knw that SAT I Math scres fllw an apprximately Nrmal distributin with mean 500 and standard deviatin 100. N(500,100) Using the Empirical Rule, apprximately 68% f students scred between 400 and 600 (within 1 S.D.), 95% scred between 300 and 700 (2 S.D.) and 99.7% scred between 200 and 800 (3 S.D.). Ex: Use the Empirical Rule t find percentiles. Since 500 is ur mean and median (symmetric), then 500 is the 50 th percentile. 600 is ne standard deviatin abve the mean. Because the curve is symmetric, then, 34% f the data lies between 500 and 600, s 600 is at the (50+34) 84 th percentile. 400 is ne standard deviatin belw the mean and wuld be at the (50-34) 16 th percentile. Similarly, 700 is tw standard deviatins abve the mean and using symmetry again, 47.5% f the data lies between 500 and 700, s 700 is the ( ) 97 th percentile and 300 is the ( ) 2.5 th percentile. 5
6 Syllabus Objectives: The student will slve prblems using tables f the Nrmal distributin The student will slve prblems using the Nrmal distributin as a mdel fr measurements. The Standard Nrmal Distributin A Nrmal distributin with a mean f 0 and a standard deviatin f 1. Ntatin: N( µ, σ ) N(0,1) All Nrmal distributins can be standardized by using the z-scre frmula. This gives us a cmmn scale t cmpute prbabilities, which are areas under a Nrmal curve and abve given intervals. Nrmal Distributin Calculatins Use the Empirical Rule if bservatins are 1, 2 r 3 standard deviatins frm the mean. If we are unable t use the rule, we must standardize the distributin and use the Nrmal distributin table. Step 1: Area under a density curve = prprtin f the bservatins in the distributin. Step 2: Standardize the distributin using the z-scre frmula. Step 3: Lk up the z-scre in the Standard Nrmal Prbabilities Table. It is a table f areas under the standard Nrmal curve. It reprts the area under the curve frm that value and BELOW. It gives the area t the LEFT f the scre. Using the z table Find the z scre fr the prblem using the frmula and rund t the nearest hundredth. Lk it up n the table. If yu want the prprtin t the left f that number, yur answer is the 4-digit number yu find. If yu want the area t the right f that z-scre, subtract the 4-digit number frm ne. Remember, the curve is symmetric, s if yu want the area t the right f z = 1.23, that is the same as the area t the left f z = The Nrmal distributin is describing cntinuus data. This means that the prprtin f bservatins with x > a is the same as the prprtin with x a. There is n area abve a single pint. T find the area between tw scres, find the z scre fr each value, lk bth f them up n the table and subtract the tw areas t find the area BETWEEN the tw scres. 6
7 Nrmal Tables: z* z*
8 Table Example: Find P ( z < 0.46). Clumn labeled 0.06 Rw labeled 0.4 P(z < 0.46) = Table Example 2: (a) P(z < 1.83) = (b) P(z > 1.83) = 1 P(z < 1.83) = = Symmetry Prperty: Pz ( z > ) = Pz ( < z) P(z > -2.18) = P(z < 2.18) =
9 Table Example 3: Find P(-1.37 < z < 2.34). P(Z<2.34)= P(Z<-1.37)= P(-1.37 < z < 2.34) = = Finding Nrmal Prbabilities Example 1: A cmpany prduces 20 unce jars f picante sauce. The true amunts f sauce in the jars f this brand sauce fllw a Nrmal distributin. The cntents f the jars are Nrmally distributed with a true mean f µ = 20.2 unces and a standard deviatin f σ = 0.125unces. What prprtin f the jars are under-filled? (have less than 20 unces f sauce). Step 1: Write the prbability statement and standardize the scre P ( x < 20) = = = We have standardized the distributin. Therefre, we need t find the P ( z < 1.60). Step 2: Draw the Nrmal curve, shade the apprpriate area. z = Step 3: Lk up the value f 1.60 n the z table and find the value Step 4: Summarize: That means, the prprtin f the sauce jars that are under-filled is Example 2: What prprtin f the sauce jars cntain between 20 and 20.3 unces f sauce. Step 1: Find BOTH standardized scres. We have the ne fr 20, standardize 20.3: P (20 < x < 20.3) = =
10 Step 2: Find P ( 1.60 < z < 0.80). Step 3: Lk up bth z values n the table and subtract the tw prbabilities: = Step 4: Summarize: The prprtin f sauce jars between 20 and 20.3 unces is Nrmal prbabilities n the TI-84 Graphing Calculatr The nrmalcdf cmmand can be used t find areas under a Nrmal curve. Press (DISTR) and chse 2 : nrmalcdf (. Cmplete the cmmand. It is waiting fr fur values, [lwer limit, upper limit, mean, standard deviatin]. Use 1E99 fr values t the right end f the curve, and -1E99 fr the left end f the curve. Example 2 Finding Px ( < 20) and Px ( > 20). {Nte: prbabilities add up t 1!) Nrmal Calculatins Finding a value, given a prprtin Nw find the z-scre if given a prprtin area. We use the table backwards. Step 1: Lk in the MIDDLE f the table fr the CLOSEST decimal related t the percent r prtin given in the prblem. Step 2: Read OUT alng the table s rw and clumn t find the z-scre. Example: Pz< ( 0.44) =.67, s 67% f all z values are less than 0.44, and 0.44 is the 67 th percentile f the standard nrmal distributin. Example 2 (frm abve) cntinued: 99% f the jars f this brand f picante sauce will cntain mre than what amunt f sauce? Ntice a prprtin is given in the prblem and we will find the x value that crrespnds t Step 1: Draw the curve, label and shade the apprpriate area. Step 2: Recall that the z table shws areas t the LEFT f the z value, s subtract ( ) and lk up n the z table. z
11 When we lk up in the BODY f the table, we d nt find this value. The entry clsest t is crrespnding t the z value Step 3: Use the standardized frmula t slve fr x. Our frmula: x µ z = becmes x = z σ + µ. σ Slving: x = ( 2.33)(0.125) = Step 4: Summarize 99% f the jars f picante sauce f this brand will cntain mre than unces. Finding values with invnrm n the TI-84 Graphing Calculatr The invnrm functin calculates the raw r standardized Nrmal value crrespnding t a knwn area under a Nrmal curve. Press (DISTR) and chse 3 : invnrm(. Cmplete the cmmand. It is waiting fr three values. Input the prprtin as a decimal, the mean and the standard deviatin. Example 2 cntinued Yu try: Final Nrmal prbability example: The time t first failure f a unit f a brand f ink jet printer is apprximately Nrmally distributed with a mean f 1,500 hurs and a standard deviatin f 225 hurs. (a) What prprtin f these printers will lnger than 2,000 hurs? Step 1: Px ( > 2000) = P z> = Step 2: Step 3: Find Pz> ( 2.22). Lk up 2.22 and calculate. Pz> ( 2.22) = = Step 4: The prprtin f printers that will last 2,000 hurs r mre is (b) What prprtin f these printers will last between 1,300 and 1,800 hurs? 11
12 Step 1: P(1300 < x < 1800) z = = 0.89 and z = = P(1300 < x < 1800) = P( 0.89 < z < 1.33). 225 Step 3: Lk up 0.89 and 1.33 and find the difference = Step 4: The prprtin f printers lasting between 1,300 and 1,800 hurs is (c) What shuld be the guarantee time fr these printers if the manufacturer wants nly 5% t fail within the guarantee perid? Step 1: Interested in the lwest 5%. Step 2: Lk up in the bdy f the nrmal table. It is nt there, but there are tw values that are clse: crrespnding t a z-scre f 1.64 and crrespnding t a z-scre f Since is right in between these tw values, the z value we want is Step 3: x = z σ + µ x = ( 1.645)(225) = Step 4: The guarantee perid shuld be 1130 hurs. 12
13 Is the distributin Nrmal? In the previus examples, we ve assumed that the underlying data distributin was rughly unimdal and symmetric. One must CHECK t see whether the Nrmal mdel is reasnable. Plt a histgram, stemplt and/r bxplt t determine if a distributin is bellshaped. Determine the prprtin f bservatins within ne, tw, and three standard deviatins f the mean, and cmpare with the rule fr Nrmal distributins. Cnstruct and interpret a Nrmal prbability plt. The Nrmal prbability plt plts each data pint x against its crrespnding z scre. The x-values are nrmally graphed n the hrizntal axis and the z-scres n the vertical axis. If the distributin f the data is rughly Nrmal, the plt is rughly a diagnal straight line. Deviatins frm the straight line indicate it is nt Nrmal. Graphing Calculatr Example data belw is 19 breaking strengths f cnnectins f fine wires t semicnductr wafers (in punds): Althugh the data is nt perfectly Nrmal, the histgram shws a unimdal, fairly symmetric distributin. Cllectin 1 Nrmal Quantile Plt Wire_strength Nrmal Quantile = Wire_strength Nrmal Quartile (Prbability) Plt in Fathm. Nte: All z-scres are between -2 and 2. The Empirical Rule states, 95% f the data shuld be within 2 s.d. f the mean. Skewed distributins and their Nrmal prbability plts. Skewed t the right Nrmal prbability plt falls ff n the right. Skewed t the left Nrmal prbability plt falls ff n the left. 13
, which yields. where z1. and z2
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