Statistics and the Common Core
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Simulation LabUnderstanding randomness using simulation NAME___________________________ A penny for your thoughts: What is the probability of getting “heads” after flipping a penny? How do you know? After 50 flips, how many “heads” to you expect? How many “heads” in a row would make you suspect that the coin was unfair? Are some coins more fair than others? Can a coin be weighted to come up more than 50% “heads?” What is the likelihood of a coin having a streak of 10 “heads” in a row? Will the likelihood of “heads” be different if you spin the penny on a table? 1. Flip a penny 50 times, and record each spin in consecutive order on the Stirling Recording Sheet as shown below. Suppose your first 5 flips were H, H, T, H, T. It would look like this: Continue to flip your penny until you have 50 flips. You can read your percent “heads” on the bottom row of the chart. How many heads did you get? ________ What is your percent “heads?” _________
_______________________________________________________ 2. What was your longest streak of heads in a row? ________ Do you think yours was unusual? What do you think the distribution of the class’s longest streaks will show? Sketch and label a dot plot of the class distribution of longest “heads” streaks below: _____________________________________________________ 3. What if you and a friend recorded both of your 50 flips on the same sheet. What is the probability that your two paths will cross? Simulate this by taking your finished Stirling sheet and another student’s sheet. Put them on top of each other, hold them up to a light, and see if they cross. Do this several times with other students to simulate the results. What percent of the time did your two paths cross? __________% GOING A LITTLE DEEPER: 4. What everyone in the class flipped a coin only twice? What would the graph for percent heads look like? What would be a reasonable range for this distribution? ________% to ________% Take a look at the class graph for percent heads in problem #1. What is the range for the percent heads for 50 flips? ________% to ________% Assuming we would expect to get 25 heads out of 50 flips, find out the most extreme number of heads in the class (high and low). How far were they from 25? Furthest away above 25: _______ Furthest away below 25: ________ So a reasonable range for the number of heads is ________ heads to ________ heads. 5. Now imagine a MUCH bigger sheet that can record 10,000 flips. Would it be unusual to be 50 heads away from what you would expect? Explain. 6. What happens to the cumulative percent heads as the number of flips increases? 7. On a new Stirling sheet, record the results of spinning a penny 50 times. Percent heads: ________ Dot plot of class results: _______________________________________________ 8. Compare and contrast the two class distributions: flipped vs. spun 9. What do you think caused the difference in results? 50 Heads 25 Heads 0 Heads Heads Tails Start Stirling Recording Sheet 10. Another way to produce heads and tails from a penny is to stand the penny upright on a flat table. Once the penny is balanced, lightly tap the table with your fist. The penny should fall. Are these results similar to flipping or spinning, or do they have their own unique distribution? Jelly BlubbersBasic principles of sampling NAME__________________________ Final Question: A student decides to generate a random sample by closing her eyes and pointing at the sheet of blubbers randomly. She chooses the blubber to which her finger is closest. Comment on this method of generating a SRS. Alf Landon: Statistical InfamyThe 1936 Presidential Election NAME__________________________ Investigative Task—Using Google, Wikipedia, etc. find out what happened in the predictions and surveys before the 1936 Presidential election. In particular, search for: the Literary Digest survey and George Gallup’s survey. You might also want to research a bit about candidate Alf Landon. 1. Who was Alf Landon? 2. Who did the Literary Digest predict would win the 1936 election? _____________________ 3. How successful had the Literary Digest been in the past for predicting Presidential elections? 4. Why was the Literary Digest’s prediction so far off? Explain fully. 5. Why was George Gallup’s prediction more accurate? Other Random Sampling Activities: One way of determining the “readability” of texts is to find the average word length. The larger the average is, the higher the reading level. 1. Gettysburg Address: estimate the average word length by taking a random sample of 10 words and averaging your results. Compare your mean to the rest of the means in the class by sketching a dotplot: __________________________________________________________ 2. Bush’s 2nd Inaugural Address vs. Obama’s 2nd Inaugural Address: find texts of the speeches, and decide whose speech contained a higher reading level, Bush’s or Obama’s. Devise a good random sampling method to estimate the average word length of the entire speeches using the mean of one good random sample from each speech. 3. Do English books have a larger mean word length than math books? Devise a good random sampling method to answer this question. 1 Four 55 We 109 cannot 163 for 217 they 2 score 56 are 110 dedicate, 164 us 218 gave 3 and 57 met 111 we 165 the 219 the 4 seven 58 on 112 cannot 166 living, 220 last 5 years 59 a 113 consecrate, 167 rather, 221 full 6 ago, 60 great 114 we 168 to 222 measure 7 our 61 battlefield 115 cannot 169 be 223 of 8 fathers 62 of 116 hallow 170 dedicated 224 devotion, 9 brought 63 that 117 this 171 hear 225 that 10 forth 64 war. 118 ground. 172 to 226 we 11 upon 65 We 119 The 173 the 227 here 12 this 66 have 120 brave 174 unfinished 228 highly 13 continent 67 come 121 men, 175 work 229 resolve 14 a 68 to 122 living 176 which 230 that 15 new 69 dedicate 123 and 177 they 231 these 16 nation: 70 a 124 dead, 178 who 232 dead 17 conceived 71 portion 125 who 179 fought 233 shall 18 in 72 of 126 struggled 180 here 234 not 19 liberty, 73 that 127 here 181 have 235 have 20 and 74 field , 128 have 182 thus 236 died 21 dedicated 75 as 129 consecrated 183 far 237 in 22 to 76 a 130 it, 184 so 238 vain, 23 the 77 final 131 far 185 nobly 239 that 24 proposition 78 resting 132 above 186 advanced. 240 this 25 that 79 place 133 our 187 It 241 nation, 26 all 80 for 134 poor 188 is 242 under 27 men 81 those 135 power 189 rather 243 God, 28 are 82 who 136 to 190 for 244 shall 29 created 83 here 137 add 191 us 245 have 30 equal. 84 gave 138 or 192 to 246 a 31 Now 85 their 139 detract. 193 be 247 new 32 we 86 lives 140 The 194 here 248 birth 33 are 87 that 141 world 195 dedicated 249 of 34 engaged 88 that 142 will 196 to 250 freedom, 35 in 89 nation 143 little 197 the 251 and 36 a 90 might 144 note, 198 great 252 that 37 great 91 live. 145 nor 199 task 253 government 38 civil 92 It 146 long 200 remaining 254 of 39 war, 93 is 147 remember 201 before 255 the 40 testing 94 altogether 148 what 202 us, 256 people, 41 whether 95 fitting 149 we 203 that 257 by 42 that 96 and 150 say 204 from 258 the 43 nation, 97 proper 151 here, 205 these 259 people, 44 or 98 that 152 but 206 honored 260 for 45 any 99 we 153 it 207 dead 261 the 46 nation 100 should 154 can 208 we 262 people, 47 so 101 do 155 never 209 take 263 shall 48 conceived 102 this. 156 forget 210 increased 264 not 49 and 103 But, 157 what 211 devotion 265 perish 50 so 104 in 158 they 212 to 266 from 51 dedicated, 105 a 159 did 213 that 267 the 52 can 106 larger 160 here. 214 cause 268 earth. 53 long 107 sense, 161 It 215 for 54 endure. 108 we 162 is 216 which Pick’s Theorem and Multiple Regression1 Going deeper with a fun Geometry Theorem NAME____________________________
a. The number of dots on the perimeter (“border points”) b. The number of dots in the interior (“interior points”) c. The area of the shape in square units. 1
2. Now draw nine more “dot polygons” (all vertices must be on dots) and find the same three calculations for your new shapes. Record your data in the table. 3. Graph border points vs. area. Describe and sketch the association below. 4. Graph interior points vs. area. Describe and sketch the association below. 5. What percent of the variation in area is explained by a linear model on border points?
6. What percent of the variation in area is explained by a linear model on interior points? 7. Given a known amount of border points, how much of the variation in area can be explained by a linear model on interior points? Hmm…that’s a much deeper question… What GENERALLY should be true about this type of analysis? (FYI: This statistical idea is connected to #6 on the 2014 AP Statistics Exam…) 8. In “real-life” data, it’s very common that more than one variable will have a relationship with another variable, just like in this geometry example. But analyzing them separately, the relationship between all three variables is not so clear: A cool statistics analysis can take all three variables and find the relationship. Using border points and interior points as TWO explanatory variables, a process called multiple regression will find the relationship between the two x variables and the y variable (area). It looks like this: Parameter Estimates
See if you can find the formula for Pick’s Theorem using the coefficients from the table above. Area = ____•(border points) + ____•(inside points) + _____ 9. Verify that this formula “works” for the data you collected on page 1. Download 283.31 Kb. Share with your friends:
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