PROB & STAT THEORY I
PROB & STAT THEORY I MGS 9920
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This 43 page Class Notes was uploaded by Ms. Aaliyah Crist on Monday September 21, 2015. The Class Notes belongs to MGS 9920 at Georgia State University taught by Subhashish Samaddar in Fall. Since its upload, it has received 14 views. For similar materials see /class/209857/mgs-9920-georgia-state-university in Managerial studies at Georgia State University.
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Date Created: 09/21/15
Chapter 2 Descriptive Statistics I Tabular and Graphical Presentations Slide 1 Learning objectives 1 Single variable 11 How to use Tables and Graphs to summarize data 111 Qualitative data 112 Quantitative data 12 How to use StemandLeaf display to explore data 2 Two variables 21 How to identify and understand potential relationship between variables 211 Using Crosstabulation 7 Power of Simpson s paradox When present 212 Using Scatter Diagram and Trendline Slide 2 Outline Tabular and Graphical Procedures Qualitative Data Quantitative Data Tabular Graphical Tabular Graphical Methods Methods Methods Methods 1 l 1 0 Frequency 0 Bar Graph 0 Frequency Dot Plot Distribution 0 Pie Chart Distribution Histogram Rel Freq Dist Rel Freq Dist Ogive 0 Percent Freq 0 Cum Freq Dist OScatter Distribution 0 Cum Rel Freq Diagram 0 Crosstabulation Distribution StemandLeaf Displa Crosstabulation Slide 3 m u rIEQ LO 111 Table and Graph for Qualitative Data quot355153 I Tabular methods 39 Frequency Distribution 39 Relative Frequency Distribution 39 Percent Frequency Distribution l Graphical methods 39 Bar Graph 39 Pie Chart Slide 4 Frequency Distribution amp Graph quot3353 A freguency distribution is a tabular summary of data showing the frequency or number of items in each of several nonoverlapping classes V The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data V Slide 5 Qualitative Data Example Marada Inn Guests staying at Marada Inn were asked to rate the quality of their accommodations as being excellent above average average below average or poor The ratings provided by a sample of 20 guests are Below Above Above Below Poor Slide 6 Rating Freguencx Freguencx Poor 2 10 0 Below Average 3 3915 15 Average 5 25 25 Above Average 9 45 45 Excellent 1 05 5 Total E 100 N 100 120 05 Slide 7 Bar Graph Marada Inn Quality Ratings 10 9 8 gt 7 E q 6 5 G D E 4 7 3 7 2 7 7 7 1 7 7 n n l u a Rating Poor Below Average Above Excellent vera e Average Slide 8 Pie Chart Marada Inn Quality Ratings Average 25 Slide 9 LO 112 Table and Graph for Quantitative Data l Tabular methods 39 Frequency Distribution 39 Relative Frequency and Percent Frequency Distributions l Graphical methods 39 Dot Plot 39 Histogram 39 Cumulative Distributions called Ogive when shown as a line graph Slide 10 Quantitative Data Example Hudson Auto Repair D The manager of Hudson Auto would like to have a better understanding of the cost of parts used in the engine tune ups performed in the shop She examines 50 customer invoices for tune ups The costs of parts rounded to the nearest dollar are listed on the next slide Slide 11 Example Hudson Auto Repair l Sample of Parts Cost for 50 Tune ups Slide 12 Dot Plot 1224 53233 Tune up Parts Cost D HH HH H l HH HH 50 60 70 80 90 100 110 Cost 55 Slide 13 LCD 1112 11 1 Frequency Dlstrlbutlon c iifu i fi stmbuuun For Hudson Auto Repair if we Choose six Classes D Approximate Class Width 109 52 6 95 a Relative Percent Parts Cost FrequenCV FrequenCV FrequenCV 5059 2 04 4 6069 13 3926 26 32 32 7079 16 80 89 7 14 14 9099 7 14 14 100109 5 Q Q Total 50 100 100 H 10100 H Slide 14 anuulcy shde 15 Hjstogram Check the skewness shde 15 Discussion item lewy I Use the handout given and focus on the portion instructed by me I Identify and understand 39 what is being represented by the Histogram 39 is descriptive statistics helping these authors to make their case How Slide 17 LCD 1112 11 m Cumulative Distributions g 5353211 Cumulative frequency distribution shows the D number of items with values less than or equal to the upper limit of each class Cumulative relative frequency distribution shows D the proportion of items with values less than or equal to the upper limit of each class Cumulative percent frequency distribution shows D the percentage of items with values less than or equal to the upper limit of each class Slide 18 Cumulative Distributions Distributmn l Hudson Auto Repair V V V Cumulative Cumulative Cumulative Relative Percent Cost 155 Freguengx Freguencx Freguencx 5 59 2 04 4 D E 69 15 30 30 5 79 31 E 62 62 30100 5 89 38 76 76 5 99 45 90 9O 5 109 50 100 100 Slide 19 0 give with Cumulative Percent Frequencies 100 05 O V Cumulative Percent Frequency Tune up Parts Cost 895 76 Parts Cost Distributmn Slide 20 10 1 0 1 2 Stemrandrleaf LO 12 Exploratory Data Analysis D I The techniques of exploratory data analysis consist of silnple arithmetic and easy to draw pictures that can be used to summarize data quickly D I One such technique is the stem and leaf display 0 A stemiandileaf display shows both the rank order and shape of the distribution of the data Slide 21 Stem and Leaf Display srgrgii algi39 Hudson Auto Example 27 2222567888999 1122344555678999 0023589 a stem a leaf OkDOOVlmtn H g 01 2 Slide 22 L o 1 2 Stemrandrleaf Stretched Stem and Leaf Display D I If we believe the original stern and leaf display has condensed the data too much we can stretch the display by using two stems for each leading digits D I Whenever a stern value is stated twice the first value corresponds to leaf values of 0 4 and the second value corresponds to leaf values of 5 9 Slide23 Stretched Stem and Leaf Display Smm and39lgaf 5 2 5 7 6 2 2 2 2 6 5 6 7 8 8 8 9 9 9 7 1 1 2 2 3 4 4 D 7 5 5 5 6 7 8 9 9 9 8 0 0 2 3 8 5 8 9 9 1 3 9 7 7 7 8 9 10 1 4 10 5 5 9 Slide 24 12 Stem and Leaf Display mmrandrlgaf l Leaf Units D 39 A single digit is used to define each leaf D 39 In the preceding example the leaf unit was 1 D 39 Leaf units may be 100 10 1 01 and so on D 39 Where the leaf unit is not shown it is assumed to equal 1 Slide 25 L0 2 Qnsstabulatmn Scatter diagam LO 2 Crosstabulations and Scatter Diagrams DI Thus far we have focused on methods that are used to summarize the data for one variable at a time DI Often a manager is interested in tabular and graphical methods that will help understand the relationship between two variables DI Crosstabulation and a scatter diagram are two methods for summarizing the data for two or more variables simultaneously Slide 26 13 L0 2 Qusstabulatmn Scatter diagam LO 211 Crosstabulation DI A crosstabulation is a tabular summary of data for two variables DI Crosstabulation can be used when 39 one variable is qualitative and the other is quantitative 39 both variables are qualitative or 39 both variables are quantitative DI The left and top margin labels define the classes for the two variables Slide 27 L0 2 Qusstabulatmn Scatter diagam Crosstabulation l Example Finger Lakes Homes The number of Finger Lakes homes sold for each style and price for the past two years is shown below qualitative variable Home Style 39 Range Colonial Log Split A Frame Total 5 99000 18 6 19 12 7 gt 99000 12 14 16 3 j Total 30 20 35 15 quantitative variable Frequency distribution for the home style variable Frequency distribution for the rice variable 28 14 L0 2 ctssstalmlatmn Scatter diagam Discussion item Use the handout given and focus on the portion instructed by me Identify and understand 39 what is being represented by Table 2 39 What is being represented by Tables 3 and 4 Are these examples of crosstabulations Slide 29 Aggregate table shows Judge Kendall is better JudgeLuckett JudgeKendall Common Municipal Common Municipal Verdict lea Court Total Verdict Pleas Court Total Uheld 129 Upheld 90 90 20 30 110 Reversed Reversed Tota Lo 2 ctusstabulau Crosstabulation Simpson s Paradox 39mid wim Verdict Luckett endall Total Upheld 129 86 110 33 239 Reversed 21 14 15 12 as Total 150 100 125 100 275 1o 10 5 20 15 Total 100 100 25 100 125 I A However according to the column percentage tables above Judge Luckett seems to be better than Judge Kendall Slide 30 15 L0 2 msstabulauun LO 212 Scatter Diagram and Trendline 39 am wam D I A scatter diagram is a graphical presentation of the relationship between two Quantitative variables D I One variable is shown on the horizontal axis and the other variable is shown on the vertical axis D I The general pattern of the plotted points suggests the overall relationship between the variables D I A trendline is an approxilnation of the relationship Slide 31 L0 2 Qusstabulatmn Scatter diagam Scatter Diagram l A Positive Relationship y Slide 32 16 L0 2 Qnsstabulatmn Scatter diagam Scatter Diagram l A Negative Relationship Slide 33 L0 2 Qnsstabulatmn Scatter diagam Scatter Diagram I No Apparent Relationship y Slide 34 17 Example Panthers Football Team l Scatter Diagram The Panthers football team is interested 39 in investigating the relationship if any between interceptions made and points scored x Number of y Number of Interceptions Points Scored 1 14 D 3 24 2 18 1 17 3 30 Slide 35 L0 2 Qusstabulatmn Scatter diagam Scatter Diagram w V Number of Points Scored 0 1 2 3 4 Number of Interceptions Slide 36 18 Example Panthers Football Team L0 2 Ctusstabulatmn Scatter diagam l Insights Gained from the Preceding Scatter Diagram D 39 The scatter diagram indicates a positive relationship between the number of interceptions and the number of points scored D 39 Higher points scored are associated with a higher number of interceptions D 39 The relationship is not perfect all plotted points in the scatter diagram are not on a straight line Slide 37 Tabular and Graphical Procedures Qualitative Data Quantitative Data D Tabular Graphical Tabular Graphical Methods D Methods D Methods D Methods 0 Frequency Bar Graph Frequency Dot Plot Distribution Pie Chart Distribution Histogram Rel Freq Dist Rel Freq Dist Ogive Percent Freq Cum Freq Dist OScatter Distribution Cum Rel Freq Diagram 0 Crosstabulation Distribution StemandLeaf Displa Crosstabulation Slide 38 End of Chapter 2 Slide 39 20 Chapter 8 Interval Estimation Slide 1 Learning objectives Compute confidence interval for Population Mean when 7 is known Compute confidence interval for Population Mean when 7 is unknown Compute appropriate sample size for given confidence level Compute confidence interval for population proportion Slide 2 y o c x futurknawnv cx nrurunkmvma Samplenze 61 m Margin of Error and the Interval Estimate D A point estimator cannot be expected to provide the exact value of the population parameter An interval estimate can be computed by adding and subtracting a margin of error to the point estilnate V Point Estimate Margin of Error The purpose of an interval estimate is to provide information about how close the point estimate is to the value of the parameter V Slide 3 y o c x faruiknawnd cx nrurunkmvma Samplenze 61 m Margin of Error and the Interval Estimate The general form of an interval estimate of a population mean is D it Margin of Error V Slide 4 D rknawnv unkrmwnv information Interval Estimation of a Population Mean 6 Known In order to develop an interval estimate of a population mean the margin of error must be computed using either 39 the population standard deviation 7 or 39 the sample standard deviation 5 7 is rarely known exactly but often a good estimate can be obtained based on historical data or other We refer to such cases as the 7 known case Slide 5 LO cxm knawnd mwna Interval Estimation of a Population Mean 6 Known There is a 1 a probability that the value of a sample mean will provide a margin of error of 2M or less Sampling distribution of Slide 6 Interval Estimate of a Population Mean 6 Known E Sampling E distribution E of 1 a of all D 06 2 7 values 06 2 interval 1 E 1 does not I X include u a L interval ZaZ z 2CL includes l E l I 3 D I 7 1 Slide 7 LO learurknawna cx Esmgmma Interval Estimate of a Population Mean 6 Known 61an Interval Estimate of lLt where 7 is the sample mean 1 06 is the confidence coefficient 2W2 is the Z value providing an area of 06 2 in the upper tail of the standard normal probability distribution 7 is the population standard deviation is the sample size 3 Slide 8 mme Interval Estimate of a Population Mean 6 Known Adequate Sample Size D In most applications a sample size of n 30 is adequate If the population distribution is highly skewed or contains outliers a sample size of 50 or more is recommended V Slide 9 knawnd Interval Estimate of a Population Mean 6 Known Adequate Sample Size continued If the population is not normally distributed but is roughly symmetric a sample size as small as 15 will suffice D If the population is believed to be at least approxilnately normal a sample size of less than 15 can be used V Slide 10 Interval Estimate of Population Mean 6 Known Example Discount Sounds D Discount Sounds has 260 retail outlets throughout the United States The firm is evaluating a potential location for a r new outlet based in part on the mean D annual income of the individuals in the marketing area of the new location D A sample of size n 36 was taken k 1 the sample mean income is 31100 The population is not believed to be highly skewed The population standard deviation is estilnated to be 4500 and the confidence coefficient to be used in the interval estiInate is 95 Slide 11 Interval Estimate of Population Mean 6 Known D 95 of the sample means that can be observed are within 196 01Y of the population mean u D The margin of error is 039 4 00 Z 7196 7 0 2 K36 Thus at 95 confidence the margin of error D is 1470 Slide 12 6 Known Interval estimate of lLt is 31100 1470 D or 29630 to 32570 Interval Estimate of Population Mean D We are 95 confident that the interval contains the population mean Slide 13 In class exercise 2 b p304 3 p304 Slide 14 wnv rummawnl Interval Estimation of a Population Mean 6 Unknown D If an estiInate of the population standard deviation 7 cannot be developed prior to sampling we use the sample standard deviation 5 to estiInate 7 D This is the 7 unknown case D In this case the interval estilnate for u is based on the 15 distribution D We ll assume for now that the population is normally distributed Slide 15 t Distribution The 15 distribution is a family of silnilar probability distributions D A specific 15 distribution depends on a parameter known as the degrees of freedom Degrees of freedom refer to the number of independent pieces of information that go into the computation of s V Slide 16 V V t Distribution A 15 distribution with more degrees of freedom has less dispersion As the number of degrees of freedom increases the difference between the 15 distribution and the standard normal probability distribution becomes smaller and smaller Slide 17 t Distribution VVV 15 distribution 20 degrees of freedom Standard normal distribution 15 distribution 10 degrees of freedom Slide 18 WSW t Distribution For more than 100 degrees of freedom the standard normal Z value provides a good approxilnation to the if value V The standard normal Z values can be found in the infinite degrees 00 row of the 15 distribution table V Slide 19 gnaw t Distribution Area in Upper Tail 10 05 2009 2000 1990 1981 1960 00 he 0 O 12 12 1292 12 12 IQ IQ Ix Ix Ix CA Ix Standard normal Z values Slide 20 10 LO clfnrurkrmwnu guggggmw Interval Estimation of a Population Mean F 6 Unknown Interval Estilnate D where 1 06 the confidence coefficient togZ the 15 value providing an area of 06 2 in the upper tail of a 15 distribution with n 1 degrees of freedom 5 the sample standard deviation Slide 21 LO clfnrurkrmwnu guggggmmv Interval Estimation of a Population Mean F 6 Unknown cn Example Apartment Rents D A reporter for a student newspaper is writing an article on the cost of off campus housing A sample of 16 efficiency apartments within a half mile of campus resulted in a sample mean of 650 per month and a sample standard deviation of 55 Slide 22 11 LO lenrurkrmwnu guggggmw Interval Estimation of a Population Mean F 6 Unknown Example Apartment Rents D Let us provide a 95 confidence interval estimate of the mean rent per month for the population of efficiency apartments within a half mile of campus We will assume this population to be normally distributed Slide 23 LO lenrurkrmwnu guggggmmv Interval Estimation of a Population Mean F 6 Unknown D At 95 confidence 06 05 and 062 025 D t025 is based on n 1 16 1 15 degrees of freedom D In the 15 distribution table we see that 15025 2131 Degrees Area in Upper Tail of FreedonlWOES 010 1753 2173 1316 2120 1710 2110 1731 2101 1729 2093 Slide 24 12 D 5 2133 0 930 We are 95 confident that the mean rent per month for the population of efficiency apartments within a half mile of campus is between 62070 and 67930 V Slide 25 In class exercise 13 d p312 14 b p312 Slide 26 W Summary of Interval Estimation Procedures F for a Population Mean population standard D deviation 7 be assumed own 7 Use the sample standard deviation 5 to estimate 7 Use Use 0 039 Unknown S x Zaz Case a2 Slide 27 C iquot quot quot Sample Size for an Interval Estimate cx mu unkmwnu mm of a Population Mean D Let E the desired margin of error E is the amount added to and subtracted from the D point estilnate to obtain an interval estilnate Slide 28 14 Sample Size for an Interval Estimate of a Population Mean D Margin of Error E22 2 D Necessary Sample Size 2052 0 E2 Slide 29 Sample Size for an Interval Estimate of a Population Mean Recall that Discount Sounds is evaluating a potential location for a new retail outlet based in part on the mean annual income of the individuals in the marketing area of the new location Suppose that Discount Sounds management t eam wants an estilnate of the population mean such that there is a 95 probability that the sampling error is 500 or less How large a sample size is needed to meet the required precision Slide 30 15 Sample Size for an Interval Estimate of a Population Mean D 205 00 D At 95 confidence st 196 Recall that 0 4500 1 6 2 00 2 n 46 31117 A sample of size 312 is needed to reach a desired precision of 500 at 95 confidence Slide 31 In class exercise 23 p316 24 b p316 Slide 32 16 V Interval Estimation of a Population Proportion The general form of an interval estilnate of a population proportion is fat Margin of Error Slide 33 V V Interval Estimation of a Population Proportion The sampling distribution of f plays a key role in computing the margin of error for this interval estilnate The sampling distribution of f can be approxilnated by a normal distribution whenever my 3 5 and n1 P z 5 Slide 34 17 of a Population Proportion Normal Approxilnation of Sampling Distribution of Interval Estimation Sampling distribution 7E L of 7 D 052 1 ocofall 052 values Slide 35 Interval Estilnate where Interval Estimation of a Population Proportion 30 D 2772 1 06 is the confidence coefficient 2W2 is the Z value providing an area of 06 2 in the upper tail of the standard normal probability distribution 7 is the sample proportion Slide 36 18 Interval Estimation of a Population Proportion Example Political Science Inc Political Science Inc PSI specializes in voter polls and surveys designed to keep 39 L V political office seekers informed l 1 of their position in a race 7 r7 I Using telephone surveys PSI interviewers ask registered voters who they would vote for if the election were held that day Slide 37 Interval Estimation of a Population Proportion Example Political Science Inc In a current election campaign V 7 7 PSI has just found that 220 I 39 registered voters out of 500 1quot contacted favor a particular candidate 7 PSI wants to develop a 95 confidence interval estilnate for the proportion of the population of registered voters that favor the candidate Slide 38 19 Interval Estimation of a Population Proportion 71 f D pizw239 n D where n 500 p 220500 44 2WZ 196 44 al 440435 500 PSI is 95 confident that the proportion of all voters that favor the candidate is between 3965 and 4835 Slide 39 Sample Size for an Interval Estimate of a Population Proportion D Margin of Error D Ezzmi 7 D Solving for the necessary sample size we get 1 2 1720 17 However will not be known until after we have selected the sample We will use the planning value if for 17 Slide 40 20 Sample Size for an Interval Estimate of a Population Proportion D Necessary Sample Size Z p1p D n E2 The planning value t can be chosen by 1 Using the sample proportion from a previous sample of the same or silnilar units or 2 Selecting a prelilninary sample and using the sample proportion from this sample Slide 41 Sample Size for an Interval Estimate of a Population Proportion Q Suppose that PSI would like a 99 probabil39 that the sample proportion is within 03 of the population proportion How large a sample size is needed to meet the required precision A previous sample of silnilar units yielded 44 for the sample proportion Slide 42 21 P1 P 03 D 20124 D At 99 confidence zoos 2576 Recall that 17 44 2m2 p1 p 2576 4456 E 03 CD A sample of size 1817 is needed to reach a desired precision of i 03 at 99 confidence Slide 43 Sample Size for an Interval Estimate of a Population Proportion Note We used 44 as the best estimate of p in he at k preceding expression If no information is available about p then 5 is often assumed because it provides the highest possible sample size If we had used p 5 the recommended 71 would have been 1843 Slide 44 22 In Class exercise 32 b amp 33 p321 38 p321 Slide 45 End of Chapter 8 Slide 46 23
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