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# Professional Issues and Ethics 101 120

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This 14 page Class Notes was uploaded by Haleigh Nitzsche on Friday October 23, 2015. The Class Notes belongs to 101 120 at University of Iowa taught by Staff in Fall. Since its upload, it has received 26 views. For similar materials see /class/228106/101-120-university-of-iowa in Physical Science at University of Iowa.

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Date Created: 10/23/15

Chapter 6 RANDOM SAMPLING AND DATA DESCRIPTION Part 1 Random Sampling Numerical Summaries StemnLeaf plots Histograms and Box plots Sections 671 to 674 Random Sampling ln statistics7 were usually interested in a value or parameter that describes a particular population Such as o the mean cholesterol level of all 50 year old men 7 value of interest is the mean 7 population is all 50 year old men 0 the mean height of all NBA basketball play GTS value of interest is the mean 7 population is all NBA basketball players 0 The mean number of worker related failures occurring on any given Friday 7 value of interest is the mean 7 population is all Fridays o The mean hole diameter of manufactured wash ers value of interest is the mean 7 population is all manufactured washers Gathering data on all individuals in a popula tion is usually not realistic though the census attempts this every 10 years But we can get info on a population by looking at subset of the population To get at the population parameters such as the population mean M7 we collect data on a subset of the full population Often7 this is done with a simple random sample of the population7 which means the observations were taken totally at random What do we do with the data once we collect it We can summarize it in a useful manner One option is to report a statistic from the data o Statistic A statistic is a summary value calculated from a sample of observations Usually7 a statistic is an estimator of some population parame ter Suppose we collect n observations in a sample 21 22 13 7 from a particular population7 Estimates the Statistic population parameter Sample mean u 7 E 3 i 1711 1 Sample variance 02 2 Elna7932 S Computation of 52 Original formula and alternatives 2 i LNM igt2 71 1 71 2 Egl ilz 11 1T 71 1 7 ELM n52 i n 1 Note that the divisor for sample variance is n 1 We subtract 1 from the sample size be cause we had to estimate u with f in order to compute the sample variance were interested in how the observations are dis persed around u7 but we only have information on how the observations are dispersed around 5 lf we didnt make this adjustment7 our estimate for 02 ie our 52 value7 would consistenty be too small in estimating the true population vari ance We also say7 s2 is based on n l degrees of freedom Welll discuss this more later Another measure of sample spread is the sample range 0 Sample Range lf the n observations in a sample are denoted by 21 227 13 7 the sample range is r mommy msz This is as a single value7 not 2 individual val ues Stemn leaf diagrams Consider the following set of n 80 data points Section 672 which are compressive strengths in pounds per square inch of 80 specimens of a new aluminum The mean and variance are quantities that give lithium alloy undeTgOing GVahlatiOH us information on the center and spread of the 105 97 245 163 207 134 218 199 160 196 data respectively These are important Silrn 221 154 228 131 180 178 157 151 175 201 Inark Ofadk r mluont 183 153 174 154 190 76 101 142 149 200 186 174 199 115 193 167 171 163 87 176 B t d t b t h th 121 120 181 160 194 184 165 145 160 150 u many 1S 1 u Ions can ave esame mean 181 168 158 208 133 135 172 171 237 170 and variance and yet be different distributions 180 167 176 158 156 229 158 148 150 118 143 141 110 133 123 146 169 158 135 149 We can use graphical displays to consider the whole W For this data 2 16266 and 52 114063 These give a measure of center and spread We can look at a stem n leaf diagram to get a feel for the full distribution of the data 7 I 6 8 I 7 9 I 7 10 I 15 11 I 058 12 I 013 13 I 133455 14 I 12356899 15 I 001344678888 16 I 0003357789 17 I 0112445668 18 I 0011346 19 I 034699 20 I 0178 21 I 8 22 I 189 23 I 7 24 I 5 The decimal point is 1 digits to the right of the The minimum value is 76 77 is the stem7 and 67 is the leaf The maximum value is 245 247 is the stem7 and 57 is the leaf The legend7 tells us where the decimal is at This stem n leaf suggests this distribution can be described as bell shaped and unimodal ie has one peak Steps for making a Stemn Leaf Dia gram 1 Separate each observation into a stern con sisting of all but the nal rightmost digit and a leaf the nal digit Stems may have as many digits as needed7 but each leaf con tains only a single digit 2 Write the stems in a vertical column with the smallest at the top and draw a vertical line at the right of this column 3 Write each leaf in the row to the right of its stem in increasing order out from the stem 15 If there are too many values for each stem you can also do a split stem n leaf diagram by split ting the values for each stem Stem Leaf Stem Leaf Stem Leaf v 34556 6L 134 02 1 739 011357889 5U 5561 it 3 8 1344788 7L 1113 of 455 l 235 TU 57889 65 e a 8L 3 4 4 6e EiU T 8 8 7392 0 l 0L 2 3 Ft 3 L U 5 quotif 5 I 75 739 Te 8 9 82 8t 3 8f 4 4 8s 7 8t 3 8 92 0t 2 139 9f 5 5 9 C 16 Mode Quartiles and Percentiles lf n is odd an actual data point is the me dian Once welve ordered the data as in the stem n leaf diagrami we can eaSlly Pun 011t some Other lf n is even7 the median falls between the 2 113er1 data features data points at the middle use the average of these two data points Consider the followmg stem n leaf diagram The decimal point is 1 digits to the The median is a measure of central tendency7 rlght of the l and is denoted by 13 6 I 134 6 I 5568 7 I 0113 7 I 57 We see that n 137 the min is 617 the max is 39 MOde 7739 This is the most frequently occurring data point 0 Median This is the value at which 50 fall below and 7 There are two mOdeS m thls data set 65 and 71 We would call this distribution 50 fall above bimodal ie has 2 peaks 7 The median is 68 for this data set o Quartiles The positions that break the data into 4 quad rants7 each containing 25 of the data are the quartiles The rst quartildql7 the sec ond quartil Q2 also called the median7 and the third quartildqg This data set has ql 645 C12 58 613 72 There are a number of ways to nd positions the break the data into the 25 proportions since the data is discrete But here7s one op tion ql is the interpolated value between the data points at ordered positions of and lnTTll qg is the interpolated value between the L301le data points at ordered positions of and 1 These are symbols for rounded down L j and rounded up 17 respectively The interquartile range QR is equal to qg ql and is a measure of variability The lQR is less sensitive to extremes than the ordinary sample range 0 Percentiles The lOOkth percentile is a data value such that approximately 100k of the observa tions are at or below this value and approxi mately 1001 k of them are above it for 0 lt k lt 1 0 Example Mean and Median A manufacturer of electronic components is interested in determining the lifetime of a certain type of battery A sample7 in hours of life7 is as follows 123116122110175126125111118117 a Find the sample mean and median b What feature in this data set is responsi ble for the substantial difference between the mean and median Frequency Distributions and Histograms Section 673 A frequency distribution is a table that divides a set of data into a suitable number of classes categories7 showing also the number of items belonging to each class Consider the following stem n leaf diagram for humidity readings rounded to the nearest per cent Stem Leaf We might group these data into the following frequency distribution Cumulative The histogram is a visual display of a fre Class Class Frequency Relative Relative quency distribution hNErval nndponm f equency equency 10719 3 320 015 20729 2455 8 820 0540 0555 0 Example Recall the n 80 compressive 30739 345 5 520 0325 0 80 Strengths from earlier 40749 4455 3 320 0 15 095 5059 545 l 120005 100 105 97 245 163 207 134 218 199 160 196 221 154 228 131 180 178 157 151 175 201 There were 5 my or 7 or intervals for this 183 153 174 154 190 76 101 142 149 200 frequency table 186 174 199 115 193 167 171 163 87 176 121 120 181 160 194 184 165 145 160 150 181 168 158 208 133 135 172 171 237 170 180 167 176 158 156 229 158 148 150 118 143 141 110 133 123 146 169 158 135 149 Using 10 loins7 we can create the frequency distribution Cumulative Class Class Frequency Relative Relative lnterval midpoint f frequency frequency 6180 1 180 00125 817100 905 2 280 00250 00375 1017120 1105 6 680 00750 01125 1217140 1305 8 880 01000 02125 1417160 1505 23 2380 02875 05000 1617180 1705 19 1980 02375 07375 1817200 1905 12 1280 01500 08875 2017220 2105 4 480 00500 09375 2217240 2305 4 480 00500 09875 2417260 2505 1 180 00125 10000 The histogram for this frequency table Histogram o dma Frequency TL l 100 150 data 1 l 200 250 We can see this is a unimodal distribution with a bell shape NOTE The loin widths can alter the shape of a histogram For instance7 if 1 only chose 3 bins Hiaogram of data Fvequency mi l l l l u an mm mm 2am 2am sun data This is not as informative ln general7 you dont want too many or too few observations in each bin relative to n7 and you can play around with bin size for the best scenario We summarize data in a histogram by lump ing a lot of individual observations together in a cell7 so we lose some information But this loss is usually small compared to the information gained in the visual7 and the ease of interpreta tion gained in the graph 0 Some possible descriptions of histograms Symmetric Skewed asymmetric7 long tail to one side Right tail stretched out positive skew Left tail stretched out negative skew Unimodal one peak Bimodal two peaks Bell shaped uniformly distributed flat Symmetric If the distribution is symmetric7 the mean median Rightskewed If the distribution is right skewed7 mean gt me dian Leftskewed If the distribution is left skewed7 mean lt me dian Left skewed Symmetric Right skewed 29 The histogram of the sample data at the bot tom of the slide gives us a feel for the population from which the sample was drawn The top plot is of the conceptual population from which the sample was drawn Population I a Sample 31 32 af3 spill l sample average quot39 a sample standard deviation lll i siogra m HI 5 30 Box Plots Section 6 4 Boxplots are another graphical tool for visual izing data Boxplots utilize the quartiles to give us a feel for the data distribution Values in the boxplot C11 q2 these values form the boa 93 15 X IQR forms the whiskers distance from Q1 or Q3 outliers values past the Whiskers past ql 15 X IQR or past Q3 15 X IQR seen at either tail 31 250 245 23725 237 15 IQR l 200 181 T 113 181 ED MR 112 1615 0 1435 150 1121435 15 10R 1 100 97 8725 1 87 76 Whisker extends to Whisker extends to smallest data point within largest data point within 15 interquartile ranges from 15 interquartile ranges first quartile from third quartile First quartile Second quartile Third quartile o o o o o Outliers Outliers Extreme outlier llt 151QR gtllt 15lQR L l lQR 32 151QR gtllt151QR rl

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