Activity 6A Answer Sheet Name: Rev 1.5 Before you begin: Q1: What are the two Branches of Statistics? Q2: Theorize what the distributions would approximately look like for both the right-hand and left-hand digits of the populations of towns and cities from your selected state. Show a sketch in the rectangles below. 0 1 2 3 4 5 6 7 Right-hand Digit Distribution 8 (1) 9 (2) 1 2 3 4 5 6 7 8 Left-hand Digit Distribution You have to choose one of three options to complete this activity. Your total score depends on the option chosen. Option 1: Regular score (7.0 points) Option 2: 80% of the Regular score (5.6 points) Option 3: 50% of the Regular score (3.5 points) Which Option did you choose? If Option 1, which State? 9 Analysis: Now with both side-by-side bar graphs completed, answer the questions below. 1. What type of distribution shape would you say fits (models) the following counts? Right Actual Count ______________ Right Theoretical Count _____________ Left Actual Count ________________ Left Theoretical Count _______________ 2. Which of the two types of actual digits fits its theoretical count distribution the best? LEFT-HAND RIGHT-HAND (circle one) 3. Referring to the one you did not circle above, describe in a brief sentence what characteristic(s) of the shape made it not be the best fit. Answer: Discussion: Before addressing the items below, modify the formula for the Left-hand Theoretical Count by using the logarithmic formula from Benford's Law. A brief Internet search on "Benford's Law" and/or "first digit phenomenon" would be very helpful. 1. Why do you suppose the actual count distributions are not as smooth as the theoretical? Answer: 2. By changing the right-hand theoretical count to reflect the logarithmic formula from Benford's Law, discuss how the shape improved the fit of the actual count Answer: . 3. Why would the right-hand digit's distribution be approximately uniform (flat)? Answer: 4. Why would the left-hand digit's distribution be roughly right-skewed? (see #5 below) Answer: 5. To better understand why there is a built-in bias for the lower digits in the left-hand distribution, scan the sorted populations of your state from low to high. Discuss why a city, as it grows in population, would remain with a left-hand digit of a 1 longer than a 2, or why longer with a 2 than a 3, etc. You may come to a better appreciation for the first digit phenomenon that occurs in certain kinds of data by noting how there is a 100% increase from 1 to 2 but then a dramatically tapering percentage thereafter. Fill in the rest of the table and discuss how this might apply to population changes in a town or city. Digit %Increas e 1 n/a 2 100% 3 4 5 6 7 8 Answer: 6. Based on your Internet research, discuss a practical application of Benford's Law that interested you and why. Also, include what the Benford ratios are for digits 1 through 9. Answer: 9 1