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# 512 Class Note for STAT 401 at PSU

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This 17 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 25 views.

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Date Created: 02/06/15
Chapter 1 Page 1 STAT 401 INTRODUCTION Often we want to find out certain properties of a populations of objects or subjects Examples 1 Population US citizens of age 18 and oven Property proportion or fraction supporting nuclear energy 2 Population a certain component Property average length of life time Census ie examination of all members of a population is not conducted typically This is because of the cost and time required but also because the population is often hypothetical or conceptual Example 2 above is an example of a hypothetical population An alternative to census which is typically Chapter 1 Page 2 adopted is to examine a random sample Le a representative subset of the population Statistics is the science of i Collecting data sampling ii Summarizing data descriptive statistics iii Drawing conclusion from the data inferential statistics Chapter 1 Page 3 Ch1 Descriptive StatisticsI StemandLeaf Displays Example Stemandleaf of yardage N 2 40 Stem Thousands and hundreds digits Leaf Tens digits Leaf Unit 10 4 84 3 3 8 7 8 85 0 2 2 8 11 88 0 1 9 18 87 0 1 4 7 7 9 9 4 88 5 7 7 9 18 89 0 0 2 3 14 70 0 1 2 4 4 5 8 71 0 1 3 8 8 8 2 72 0 8 Note Suffices to specify leaf unit Chapter 1 Page 4 The main idea of steamandleaf and other displays is to convey a reasonable impression regarding the distribution of the data Had we chosen only thousands for stems so only two stems the display would not be as informative too crude or clumpy Occasionally we need to use repeated stems to get a better display 5H 5 5L 0 2 2 4H 6 6 7 9 4L 111 3H 566 4 4 4 4 4 03wa CONOOOO COPCDh Relative Frequency Distributions If instead of writing out the numerical values of the leafs in each stem we provide only a count of them we obtain a frequency histogram Instead of stems we now Chapter 1 Page 5 arbitrarily eg they need not have the same width Ex Class fl fl n Freq Relative Freq 1 350lt550 1 010 2 550lt750 3 030 3 750lt950 8 079 4 950lt1150 17 168 5 1150lt1350 19 188 6 1350lt1550 19 188 7 1550lt1750 11 109 8 1750lt2550 23 228 n101 1000 Histograms Histograms are pictorial representations of frequency distributions They are constructed by drawing a box above each class interval whose height is talk about class intervals which can be chosen more Page 6 2 W m 49 n w Relative Frequency Class Width EM 7 247 3 Frequency of Life Time height 7 Chapter 1 Chapter 1 Page 7 symmetric M K bimodal positively skewed negatively skewed Chapter 1 Page 8 gar Graphs for Qualitative Data Frequency distributions and histograms bar graphs can be done with qualitative or categorical data Frequency Distribution Histogram Relative Manufaturer Frequency frequency lHonda 41 34 34 2 Yamaha 27 23 3 Kawasaki 20 17 4 Suzuki 18 15 5 HarleyDavidson 3 03 6 Other 11 09 120 101 Chapter 1 Page 9 measures of Location Let X1 X2 Xn denote the data The sample mean is X X X 1 n X 1 2 nZXi n n The sample median is the value which separates the sample in two parts that with low values and that with high values To define the sample median let X1X2 XW denote the ordered values Then the sample median is f Xn1 ifn is odd 2 Xlt gtXltg1gt K 2 7 JIS if n is even Example Let n 5X1 23 X2 32 X3 18 X4 2 X5 2 ThUS Chapter 1 Page 10 f Y23321s2527 25 5 To find the median we first order the values U I I I I 11 p323 n I l Also 2 3 Thus the median is szhnyamp 25 2 Note Had the largest observation been X6 42 instead of 32 we would have X2 X225 Thus the value of is affected by extreme observations outliers Analogous to the sample mean and sample median there is the population mean u and the population median 22 Chapter 1 Page 11 a positively skewed b symmetric c negatively skewed it su MZt M gtZt A compromise between the mean and the median is the trimmed mean A 10 trimmed mean is computed by eliminating the largest 10 and the smallest 10 of the sample and then averaging the remaining values Example Let n 5 and X1X2 X5 as in previous example A 20 trimmed mean is Q 3 X4 23 25 27 25 3 3 EEI39QO Z Note Typically the trimming proportion or should be such that nor is integer and certainly nor 2 1 Thus if Q 5 it makes no sense to consider a 10 or a 5 Chapter 1 Page 12 trimmed mean Occasionally we may want to considex or such that nor gt 1 but not an integer For example we may wantitruo when n 22 Thus or 01 and nor 2 22 One way of doing it is by interpolating between 27trim 2 and 27trim 3 where 27trim 2 is obtained by eliminating the largest 2 observations and the smallest 2 observations and then averaging the remaining values 27trim 3 is defined similarly For example suppose 27trim 2 230 atrim 3 245 Then 2 I 3 22 39 24 230 233 5 Measures of Variability Location mean median trimmed mean is an important characteristic of a data set and often an important quality measure For example car manufactures want to increase the mean has mileage Another important characteristic of a data set and alsy Chapter 1 Page 14 another measure of variability The LQ is obtained Q LQ Median of smallest half of the values ie smallest 142 or n 12 as n is even or odd The UQ is obtained similarly Example Consider the n 8 values 939 704 717 1328 746 2106 1519 750 Since n is even we consider the Smallest n24 704 717 746 750 andthe Largestn224 939 1328 1519 2106 717 746 Thus LQ 731 1328 1519 UQ 1423 If there was an additional observation of 820 so 2 9 then check Chapter 1 Page 15 LQ 746 UQ 1328 The five numbers X1LQ UQ and X0 are conveniently summarized in a box plot Example Consider the above n 9 observations Thus N X 704LQ 746X 820 UQ 1328Xn 2106 397 i1 Each observation that falls between 15 IQR and 3 IQR from the edge of the box plot to which it is closest is called a mild outlier Observations that fall more than 3 IQR from the closest edge are called extreme outliers A box plot can be embellished to show such outliers see Figure 121 p43 Afinal measure of variability can be constructed using Chapter 1 Page 16 Ge deviations of each observation from the mean X1 X2 7 Xn 7 Note that n sum of deviations 7 0 1 21 Using these deviations we construct the following measure of variability Sample Variance S Xi Jot XX 14 1 1 52 1 n 14 11 The positive square root of the variance is called Sample Standard Deviation IS Computational formula Chapter 1 Page 17 Properties of 225218 Let X1 X2 Xn be a sample and C denote any constant 0 1 IfY1X1CY2 ZXzlC Yn ZXnlC then Y7C 52 53 51st 21fY12CX1Y2 CX2 Yn ICXn then 707 S2 2amp5 51 CSX Example X1 8130031 X2 813015 X3 813006X4 813011X5 812997 X6 813005 X7 813021 Code the data by subtracting 812997 and multiply by 10000 Thus the coded data are 4 18 9 14 0 8 24 It is given that the mean and variance of the coded data are respectively 110 6833 Find the mean and variance of the original data Chapter 1 Page 18 To solve this problem reason as follows To obtain the original data from the coded data we must first divide the coded data by 10000 and then add 812997 Thus we proceed by first dividing the coded data by 10000 By Property 2 the mean amp variance become 11 00011 6833 00000006833 10000 39 100002 39 Next add 812997 By Property 1 the mean and variance become 00011 l 812997 2 813008 00000006833 These are the mean and variance respectively of the original data

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