class: center, middle, inverse, title-slide .title[ # Comparing Sampling Plans through SBA Loan Data ] .subtitle[ ## Presented for STA 490 ] .author[ ### Alice Xiang ] .author[ ### Joshua Zhong ] .date[ ### 2024-04-01 ] --- ## Table of contents - Introduction to the Dataset - Identifying a Stratification Variable - Research Question - Sampling Plans - Performance Analysis - Interpretation - Conclusion --- class: inverse center middle ## Introduction to the Dataset --- ## Introduction - How data was collected - Variable Inspection - Response Variable --- ## Identifying A Stratification Variable - Stratify by US Census Bureau regions - Similarities within each region <center><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Census_Regions_and_Division_of_the_United_States.svg/390px-Census_Regions_and_Division_of_the_United_States.svg.png" alt="State Regions" height="400px" /></center> --- ## Data Preprocessing - Removal of certain observations <ul> <li> MIS_Status </li> <li> State </li> <li> Zip Code </li> </ul> - Approval Date <div style="border: 1px solid #ddd; padding: 0px; overflow-y: scroll; height:300px; "><table class="table" style="margin-left: auto; margin-right: auto;"> <caption>Observations by Region</caption> <thead> <tr> <th style="text-align:left;position: sticky; top:0; background-color: #FFFFFF;"> Region </th> <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Count </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> East North Central </td> <td style="text-align:right;"> 117014 </td> </tr> <tr> <td style="text-align:left;"> East South Central </td> <td style="text-align:right;"> 32941 </td> </tr> <tr> <td style="text-align:left;"> Middle Atlantic </td> <td style="text-align:right;"> 115542 </td> </tr> <tr> <td style="text-align:left;"> Mountain </td> <td style="text-align:right;"> 91745 </td> </tr> <tr> <td style="text-align:left;"> Northeast </td> <td style="text-align:right;"> 68575 </td> </tr> <tr> <td style="text-align:left;"> Pacific </td> <td style="text-align:right;"> 168138 </td> </tr> <tr> <td style="text-align:left;"> South Atlantic </td> <td style="text-align:right;"> 114659 </td> </tr> <tr> <td style="text-align:left;"> West North Central </td> <td style="text-align:right;"> 84015 </td> </tr> <tr> <td style="text-align:left;"> West South Central </td> <td style="text-align:right;"> 95713 </td> </tr> </tbody> </table></div> --- class: inverse center middle ## Research Question --- class: center middle ### <span style= "font-size:36px; color: black;"> What is the most effective sampling plan for accurately calculating SBA loan default rates when the data is stratified by US Census Bureau regions? </span> --- class: inverse center middle ## Sampling Plans --- ## Sampling plans - All samples of around 3000 observations, without replacement - Study Population - Four sampling plans - Simple Random Sample - Systematic Sample - Stratified Sample - Cluster Sample --- ## Visualizing our Sampling Plans <center><img src="https://www.qualtrics.com/m/assets/wp-content/uploads/2022/02/1381633_SystemicRandomSamplingRefresh_01-560x315_060722.png" alt="Sampling Plans Visual" height="500px" /></center> --- ## Simple Random Sample - Random 3000 observations from the dataset <table> <thead> <tr> <th style="text-align:right;"> Sample Size </th> <th style="text-align:right;"> Variable Count </th> </tr> </thead> <tbody> <tr> <td style="text-align:right;"> 3000 </td> <td style="text-align:right;"> 30 </td> </tr> </tbody> </table> --- ## Systematic Random Sample - Ordered by Approval Date - First number chosen randomly - Jump size <table> <thead> <tr> <th style="text-align:right;"> Sample Size </th> <th style="text-align:right;"> Variable count </th> </tr> </thead> <tbody> <tr> <td style="text-align:right;"> 3001 </td> <td style="text-align:right;"> 30 </td> </tr> </tbody> </table> --- ## Cluster Sampling - Unique Zip Code as cluster - Two-stage cluster sampling <table> <caption>Actual Sample Size and Proportion Needed</caption> <thead> <tr> <th style="text-align:right;"> Actual Sample Size </th> <th style="text-align:right;"> Proportion Needed </th> <th style="text-align:right;"> Final Sample Size </th> </tr> </thead> <tbody> <tr> <td style="text-align:right;"> 3837 </td> <td style="text-align:right;"> 0.7818608 </td> <td style="text-align:right;"> 3003 </td> </tr> </tbody> </table> <div style="border: 1px solid #ddd; padding: 0px; overflow-y: scroll; height:300px; "><table class="table" style="margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;position: sticky; top:0; background-color: #FFFFFF;"> ZipCode </th> <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> InitialSize </th> <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> FinalSize </th> <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Proportion </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> 70394 </td> <td style="text-align:right;"> 20 </td> <td style="text-align:right;"> 16 </td> <td style="text-align:right;"> 0.8000000 </td> </tr> <tr> <td style="text-align:left;"> 06857 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 97430 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 63049 </td> <td style="text-align:right;"> 50 </td> <td style="text-align:right;"> 39 </td> <td style="text-align:right;"> 0.7800000 </td> </tr> <tr> <td style="text-align:left;"> 64854 </td> <td style="text-align:right;"> 7 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 0.7142857 </td> </tr> <tr> <td style="text-align:left;"> 89445 </td> <td style="text-align:right;"> 53 </td> <td style="text-align:right;"> 41 </td> <td style="text-align:right;"> 0.7735849 </td> </tr> <tr> <td style="text-align:left;"> 12835 </td> <td style="text-align:right;"> 13 </td> <td style="text-align:right;"> 10 </td> <td style="text-align:right;"> 0.7692308 </td> </tr> <tr> <td style="text-align:left;"> 30293 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 35747 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 75652 </td> <td style="text-align:right;"> 22 </td> <td style="text-align:right;"> 17 </td> <td style="text-align:right;"> 0.7727273 </td> </tr> <tr> <td style="text-align:left;"> 29697 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 33910 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 0.6666667 </td> </tr> <tr> <td style="text-align:left;"> 37042 </td> <td style="text-align:right;"> 75 </td> <td style="text-align:right;"> 59 </td> <td style="text-align:right;"> 0.7866667 </td> </tr> <tr> <td style="text-align:left;"> 06058 </td> <td style="text-align:right;"> 8 </td> <td style="text-align:right;"> 6 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 55386 </td> <td style="text-align:right;"> 23 </td> <td style="text-align:right;"> 18 </td> <td style="text-align:right;"> 0.7826087 </td> </tr> <tr> <td style="text-align:left;"> 29052 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 60029 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 61546 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 0.8000000 </td> </tr> <tr> <td style="text-align:left;"> 80622 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 89442 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 03041 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 33296 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 87504 </td> <td style="text-align:right;"> 13 </td> <td style="text-align:right;"> 10 </td> <td style="text-align:right;"> 0.7692308 </td> </tr> <tr> <td style="text-align:left;"> 13068 </td> <td style="text-align:right;"> 12 </td> <td style="text-align:right;"> 9 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 45377 </td> <td style="text-align:right;"> 42 </td> <td style="text-align:right;"> 33 </td> <td style="text-align:right;"> 0.7857143 </td> </tr> <tr> <td style="text-align:left;"> 56662 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 59719 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 70354 </td> <td style="text-align:right;"> 7 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 0.7142857 </td> </tr> <tr> <td style="text-align:left;"> 47715 </td> <td style="text-align:right;"> 101 </td> <td style="text-align:right;"> 79 </td> <td style="text-align:right;"> 0.7821782 </td> </tr> <tr> <td style="text-align:left;"> 61350 </td> <td style="text-align:right;"> 72 </td> <td style="text-align:right;"> 56 </td> <td style="text-align:right;"> 0.7777778 </td> </tr> <tr> <td style="text-align:left;"> 02858 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 14125 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 54104 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 60204 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 33027 </td> <td style="text-align:right;"> 114 </td> <td style="text-align:right;"> 89 </td> <td style="text-align:right;"> 0.7807018 </td> </tr> <tr> <td style="text-align:left;"> 22949 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 21244 </td> <td style="text-align:right;"> 83 </td> <td style="text-align:right;"> 65 </td> <td style="text-align:right;"> 0.7831325 </td> </tr> <tr> <td style="text-align:left;"> 56741 </td> <td style="text-align:right;"> 6 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 0.8333333 </td> </tr> <tr> <td style="text-align:left;"> 76630 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 24504 </td> <td style="text-align:right;"> 13 </td> <td style="text-align:right;"> 10 </td> <td style="text-align:right;"> 0.7692308 </td> </tr> <tr> <td style="text-align:left;"> 99661 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 27542 </td> <td style="text-align:right;"> 7 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 0.7142857 </td> </tr> <tr> <td style="text-align:left;"> 64444 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 43105 </td> <td style="text-align:right;"> 23 </td> <td style="text-align:right;"> 18 </td> <td style="text-align:right;"> 0.7826087 </td> </tr> <tr> <td style="text-align:left;"> 38620 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 84773 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 56550 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 0.8000000 </td> </tr> <tr> <td style="text-align:left;"> 90640 </td> <td style="text-align:right;"> 208 </td> <td style="text-align:right;"> 163 </td> <td style="text-align:right;"> 0.7836538 </td> </tr> <tr> <td style="text-align:left;"> 23116 </td> <td style="text-align:right;"> 36 </td> <td style="text-align:right;"> 28 </td> <td style="text-align:right;"> 0.7777778 </td> </tr> <tr> <td style="text-align:left;"> 95934 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 71742 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 48217 </td> <td style="text-align:right;"> 16 </td> <td style="text-align:right;"> 13 </td> <td style="text-align:right;"> 0.8125000 </td> </tr> <tr> <td style="text-align:left;"> 31047 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 0.8000000 </td> </tr> <tr> <td style="text-align:left;"> 68433 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 20188 </td> <td style="text-align:right;"> 6 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 0.8333333 </td> </tr> <tr> <td style="text-align:left;"> 21531 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 93464 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 0.6666667 </td> </tr> <tr> <td style="text-align:left;"> 14075 </td> <td style="text-align:right;"> 214 </td> <td style="text-align:right;"> 167 </td> <td style="text-align:right;"> 0.7803738 </td> </tr> <tr> <td style="text-align:left;"> 14102 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 0.6666667 </td> </tr> <tr> <td style="text-align:left;"> 66040 </td> <td style="text-align:right;"> 9 </td> <td style="text-align:right;"> 7 </td> <td style="text-align:right;"> 0.7777778 </td> </tr> <tr> <td style="text-align:left;"> 30073 </td> <td style="text-align:right;"> 17 </td> <td style="text-align:right;"> 13 </td> <td style="text-align:right;"> 0.7647059 </td> </tr> <tr> <td style="text-align:left;"> 53925 </td> <td style="text-align:right;"> 27 </td> <td style="text-align:right;"> 21 </td> <td style="text-align:right;"> 0.7777778 </td> </tr> <tr> <td style="text-align:left;"> 17504 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 81123 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 29458 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 19903 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 0.8000000 </td> </tr> <tr> <td style="text-align:left;"> 08751 </td> <td style="text-align:right;"> 17 </td> <td style="text-align:right;"> 13 </td> <td style="text-align:right;"> 0.7647059 </td> </tr> <tr> <td style="text-align:left;"> 65256 </td> <td style="text-align:right;"> 6 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 0.8333333 </td> </tr> <tr> <td style="text-align:left;"> 40356 </td> <td style="text-align:right;"> 99 </td> <td style="text-align:right;"> 77 </td> <td style="text-align:right;"> 0.7777778 </td> </tr> <tr> <td style="text-align:left;"> 50029 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 84790 </td> <td style="text-align:right;"> 235 </td> <td style="text-align:right;"> 184 </td> <td style="text-align:right;"> 0.7829787 </td> </tr> <tr> <td style="text-align:left;"> 21076 </td> <td style="text-align:right;"> 55 </td> <td style="text-align:right;"> 43 </td> <td style="text-align:right;"> 0.7818182 </td> </tr> <tr> <td style="text-align:left;"> 49601 </td> <td style="text-align:right;"> 36 </td> <td style="text-align:right;"> 28 </td> <td style="text-align:right;"> 0.7777778 </td> </tr> <tr> <td style="text-align:left;"> 69357 </td> <td style="text-align:right;"> 17 </td> <td style="text-align:right;"> 13 </td> <td style="text-align:right;"> 0.7647059 </td> </tr> <tr> <td style="text-align:left;"> 96733 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 0.6666667 </td> </tr> <tr> <td style="text-align:left;"> 99203 </td> <td style="text-align:right;"> 51 </td> <td style="text-align:right;"> 40 </td> <td style="text-align:right;"> 0.7843137 </td> </tr> <tr> <td style="text-align:left;"> 51031 </td> <td style="text-align:right;"> 90 </td> <td style="text-align:right;"> 70 </td> <td style="text-align:right;"> 0.7777778 </td> </tr> <tr> <td style="text-align:left;"> 28752 </td> <td style="text-align:right;"> 31 </td> <td style="text-align:right;"> 24 </td> <td style="text-align:right;"> 0.7741935 </td> </tr> <tr> <td style="text-align:left;"> 94086 </td> <td style="text-align:right;"> 191 </td> <td style="text-align:right;"> 149 </td> <td style="text-align:right;"> 0.7801047 </td> </tr> <tr> <td style="text-align:left;"> 06807 </td> <td style="text-align:right;"> 7 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 0.7142857 </td> </tr> <tr> <td style="text-align:left;"> 21163 </td> <td style="text-align:right;"> 13 </td> <td style="text-align:right;"> 10 </td> <td style="text-align:right;"> 0.7692308 </td> </tr> <tr> <td style="text-align:left;"> 29067 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 0.6666667 </td> </tr> <tr> <td style="text-align:left;"> 63863 </td> <td style="text-align:right;"> 15 </td> <td style="text-align:right;"> 12 </td> <td style="text-align:right;"> 0.8000000 </td> </tr> <tr> <td style="text-align:left;"> 28348 </td> <td style="text-align:right;"> 26 </td> <td style="text-align:right;"> 20 </td> <td style="text-align:right;"> 0.7692308 </td> </tr> <tr> <td style="text-align:left;"> 66614 </td> <td style="text-align:right;"> 80 </td> <td style="text-align:right;"> 63 </td> <td style="text-align:right;"> 0.7875000 </td> </tr> <tr> <td style="text-align:left;"> 96803 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 29907 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 0.8000000 </td> </tr> <tr> <td style="text-align:left;"> 52330 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 99803 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 0.8000000 </td> </tr> <tr> <td style="text-align:left;"> 58645 </td> <td style="text-align:right;"> 7 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 0.7142857 </td> </tr> <tr> <td style="text-align:left;"> 58106 </td> <td style="text-align:right;"> 23 </td> <td style="text-align:right;"> 18 </td> <td style="text-align:right;"> 0.7826087 </td> </tr> <tr> <td style="text-align:left;"> 53517 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 0.6666667 </td> </tr> <tr> <td style="text-align:left;"> 43205 </td> <td style="text-align:right;"> 45 </td> <td style="text-align:right;"> 35 </td> <td style="text-align:right;"> 0.7777778 </td> </tr> <tr> <td style="text-align:left;"> 01937 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 02743 </td> <td style="text-align:right;"> 31 </td> <td style="text-align:right;"> 24 </td> <td style="text-align:right;"> 0.7741935 </td> </tr> <tr> <td style="text-align:left;"> 52742 </td> <td style="text-align:right;"> 70 </td> <td style="text-align:right;"> 55 </td> <td style="text-align:right;"> 0.7857143 </td> </tr> <tr> <td style="text-align:left;"> 67053 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 0.6666667 </td> </tr> <tr> <td style="text-align:left;"> 20374 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 49715 </td> <td style="text-align:right;"> 8 </td> <td style="text-align:right;"> 6 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 62421 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 32145 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 99161 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 85041 </td> <td style="text-align:right;"> 38 </td> <td style="text-align:right;"> 30 </td> <td style="text-align:right;"> 0.7894737 </td> </tr> <tr> <td style="text-align:left;"> 92225 </td> <td style="text-align:right;"> 23 </td> <td style="text-align:right;"> 18 </td> <td style="text-align:right;"> 0.7826087 </td> </tr> <tr> <td style="text-align:left;"> 50645 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 0.6666667 </td> </tr> <tr> <td style="text-align:left;"> 45358 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 95830 </td> <td style="text-align:right;"> 7 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 0.7142857 </td> </tr> <tr> <td style="text-align:left;"> 45061 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 38646 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 0.8000000 </td> </tr> <tr> <td style="text-align:left;"> 44106 </td> <td style="text-align:right;"> 40 </td> <td style="text-align:right;"> 31 </td> <td style="text-align:right;"> 0.7750000 </td> </tr> <tr> <td style="text-align:left;"> 71403 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 76071 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 0.7500000 </td> </tr> <tr> <td style="text-align:left;"> 18251 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 0.6666667 </td> </tr> <tr> <td style="text-align:left;"> 99918 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 24637 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 38957 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 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style="text-align:right;"> 0.6666667 </td> </tr> <tr> <td style="text-align:left;"> 25356 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 46626 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 95204 </td> <td style="text-align:right;"> 56 </td> <td style="text-align:right;"> 44 </td> <td style="text-align:right;"> 0.7857143 </td> </tr> <tr> <td style="text-align:left;"> 66783 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 10098 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1.0000000 </td> </tr> <tr> <td style="text-align:left;"> 72801 </td> <td 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style="text-align:left;"> East South Central </td> <td style="text-align:right;"> 111 </td> </tr> <tr> <td style="text-align:left;"> Middle Atlantic </td> <td style="text-align:right;"> 390 </td> </tr> <tr> <td style="text-align:left;"> Mountain </td> <td style="text-align:right;"> 310 </td> </tr> <tr> <td style="text-align:left;"> Northeast </td> <td style="text-align:right;"> 232 </td> </tr> <tr> <td style="text-align:left;"> Pacific </td> <td style="text-align:right;"> 568 </td> </tr> <tr> <td style="text-align:left;"> South Atlantic </td> <td style="text-align:right;"> 387 </td> </tr> <tr> <td style="text-align:left;"> West North Central </td> <td style="text-align:right;"> 284 </td> </tr> <tr> <td style="text-align:left;"> West South Central </td> <td style="text-align:right;"> 323 </td> </tr> </tbody> </table> --- class: inverse center middle ## Performance Analysis --- ## Population Default rates - Performance Metric - Calculations <table> <caption>Population size, default counts, and population default rates</caption> <thead> <tr> <th style="text-align:left;"> </th> <th style="text-align:right;"> default </th> <th style="text-align:right;"> no.default </th> <th style="text-align:right;"> default.rate </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> East North Central </td> <td style="text-align:right;"> 21599 </td> <td style="text-align:right;"> 95415 </td> <td style="text-align:right;"> 18.5 </td> </tr> <tr> <td style="text-align:left;"> East South Central </td> <td style="text-align:right;"> 6071 </td> <td style="text-align:right;"> 26870 </td> <td style="text-align:right;"> 18.4 </td> </tr> <tr> <td style="text-align:left;"> Middle Atlantic </td> <td style="text-align:right;"> 21291 </td> <td style="text-align:right;"> 94251 </td> <td style="text-align:right;"> 18.4 </td> </tr> <tr> <td style="text-align:left;"> Mountain </td> <td style="text-align:right;"> 15239 </td> <td style="text-align:right;"> 76506 </td> <td style="text-align:right;"> 16.6 </td> </tr> <tr> <td style="text-align:left;"> Northeast </td> <td style="text-align:right;"> 8203 </td> <td style="text-align:right;"> 60372 </td> <td style="text-align:right;"> 12.0 </td> </tr> <tr> <td style="text-align:left;"> Pacific </td> <td style="text-align:right;"> 29743 </td> <td style="text-align:right;"> 138395 </td> <td style="text-align:right;"> 17.7 </td> </tr> <tr> <td style="text-align:left;"> South Atlantic </td> <td style="text-align:right;"> 26714 </td> <td style="text-align:right;"> 87945 </td> <td style="text-align:right;"> 23.3 </td> </tr> <tr> <td style="text-align:left;"> West North Central </td> <td style="text-align:right;"> 10298 </td> <td style="text-align:right;"> 73717 </td> <td style="text-align:right;"> 12.3 </td> </tr> <tr> <td style="text-align:left;"> West South Central </td> <td style="text-align:right;"> 17491 </td> <td style="text-align:right;"> 78222 </td> <td style="text-align:right;"> 18.3 </td> </tr> </tbody> </table> --- ## Sample Default Rates Then, we calculate the default rates for each sampling plan. <table> <caption>Comparing Default Rates between Population and Our Sampling Plans</caption> <thead> <tr> <th style="text-align:left;"> </th> <th style="text-align:right;"> Population </th> <th style="text-align:right;"> SRS </th> <th style="text-align:right;"> Systematic </th> <th style="text-align:right;"> Cluster </th> <th style="text-align:right;"> Stratified </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> East North Central </td> <td style="text-align:right;"> 18.5 </td> <td style="text-align:right;"> 15.9 </td> <td style="text-align:right;"> 23.4 </td> <td style="text-align:right;"> 18.1 </td> <td style="text-align:right;"> 16.5 </td> </tr> <tr> <td style="text-align:left;"> East South Central </td> <td style="text-align:right;"> 18.4 </td> <td style="text-align:right;"> 23.5 </td> <td style="text-align:right;"> 21.8 </td> <td style="text-align:right;"> 27.4 </td> <td style="text-align:right;"> 17.1 </td> </tr> <tr> <td style="text-align:left;"> Middle Atlantic </td> <td style="text-align:right;"> 18.4 </td> <td style="text-align:right;"> 17.0 </td> <td style="text-align:right;"> 18.7 </td> <td style="text-align:right;"> 14.4 </td> <td style="text-align:right;"> 22.1 </td> </tr> <tr> <td style="text-align:left;"> Mountain </td> <td style="text-align:right;"> 16.6 </td> <td style="text-align:right;"> 14.1 </td> <td style="text-align:right;"> 15.7 </td> <td style="text-align:right;"> 23.2 </td> <td style="text-align:right;"> 19.0 </td> </tr> <tr> <td style="text-align:left;"> Northeast </td> <td style="text-align:right;"> 12.0 </td> <td style="text-align:right;"> 9.6 </td> <td style="text-align:right;"> 11.4 </td> <td style="text-align:right;"> 8.8 </td> <td style="text-align:right;"> 12.1 </td> </tr> <tr> <td style="text-align:left;"> Pacific </td> <td style="text-align:right;"> 17.7 </td> <td style="text-align:right;"> 15.7 </td> <td style="text-align:right;"> 19.5 </td> <td style="text-align:right;"> 17.5 </td> <td style="text-align:right;"> 18.1 </td> </tr> <tr> <td style="text-align:left;"> South Atlantic </td> <td style="text-align:right;"> 23.3 </td> <td style="text-align:right;"> 22.1 </td> <td style="text-align:right;"> 21.9 </td> <td style="text-align:right;"> 27.0 </td> <td style="text-align:right;"> 24.8 </td> </tr> <tr> <td style="text-align:left;"> West North Central </td> <td style="text-align:right;"> 12.3 </td> <td style="text-align:right;"> 12.7 </td> <td style="text-align:right;"> 13.7 </td> <td style="text-align:right;"> 12.4 </td> <td style="text-align:right;"> 12.0 </td> </tr> <tr> <td style="text-align:left;"> West South Central </td> <td style="text-align:right;"> 18.3 </td> <td style="text-align:right;"> 19.0 </td> <td style="text-align:right;"> 17.7 </td> <td style="text-align:right;"> 21.7 </td> <td style="text-align:right;"> 20.1 </td> </tr> </tbody> </table> --- ## Visualizing our Default 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