mini 4-2

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University Of Connecticut *

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1000Q

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Statistics

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Apr 3, 2024

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Assignment 4 Please follow the instructions in each exercise to run your simulations. Each graph must be labeled with a title, axis labels, and a footnote with your name and section number. If the assignment requires you to print the Session Window, click in the Session Window and type your name and section number so you can identify your printout. 1. The Birthday Paradox: Suppose you are in a room with 24 other people. You would like to answer the following question: What are the chances that at least two people in the room have the same birthday? a. Now that you know something about probability, take an “educated guess” at the chances that at least two people in the room have the same birthday. If you think it is “very unlikely” that at least two people have the same birthday, then assign a small probability to this random event. If you think it is “very likely” that at least two people have the same birthday, then assign a large probability to this random event. Write your answer here: I think that it would be very unlikely that 2 people would share a birthday, I estimate to be around 10%. The probability of this random event is difficult to calculate mathematically; however, it is an ideal simulation problem. Let’s simulate the problem and see how closely your “guess” agrees with the simulated results. To run the simulation, begin with the following information: 1. There are 365 days in a year. (We will ignore leap years). 2. There are 25 people in the room. (You and 24 others). 3. Each person could have any day of the year as his or her birthday, so each birthday is equally likely. Start your simulation with 1 room of 25 people: Click on: CALC ® RANDOM DATA ® INTEGER In the screen that appears, Generate 25 rows of data (representing the 25 people), Store in column(s) C1 (representing 1 room), Minimum value is 1 (representing January 1), Maximum value is 365 (representing December 31).
The question is: Are there at least two people with the same birthday? In other words, are there at least two numbers in C1 that are the same? The easiest way to answer this question is to summarize the data. Click on: STAT ® TABLES ® TALLY INDIVIDUAL VARIABLES On the screen that appears, select C1 and click on COUNTS. Next click on OK, and in the Session Window, you will see a summary table of C1. You will need to maximize the Session Window to see the entire summary table. This table lists each of the different birthdays that occurred in this simulation, as well as a count of the number of people who had that birthday. Notice that most counts are 1’s. If you see a count of “2” or more anywhere in the COUNTS column, then you have at least two people in the room with the same birthday. b. Did you find at least one match in C1? No, I did not In order to calculate the probability of finding at least two people with the same birthday you must repeat this experiment several times. You can generate several rooms with 25 people at the same time. To do this, click on: CALC ® RANDOM DATA ® INTEGER In the screen that appears, Generate 25 rows of data, Store in columns C2-C50, Minimum value is 1, Maximum value is 365. You now have 49 more rooms with 25 people in each room. Now summarize each of the 50 columns so that you can look for matches (people with the same birthday). When you use STAT ® TABLES ® TALLY Individual Variable command, you can select C1-C50, and the summary tables for all 50 columns will appear in the Session Window. c. Maximize the Session Window. You will see the column tallies for all 50 columns. You will look through the counts for each column to see whether you have a match in that column. A match will be a ‘2’ or ‘3’ in the count column indicating that 2 or 3 people have that same birthday. Start with the first five columns (C1 – C5), how many of these columns have at least one match? Record this number in the table below next to C1-C5. Scroll through the Session Window and repeat this process for the remaining columns. Here are the results of one set of simulations for Col1 – Col5. Tally C1 C o C2 C o C3 C o C4 Count C5 Count
u nt u n t u nt 31 1 29 1 6 1 4 1 20 1 54 1 38 1 7 1 12 1 52 1 57 1 43 1 14 1 19 1 67 1 79 1 51 1 52 1 28 1 77 1 85 1 53 1 53 1 38 1 112 1 95 1 56 1 84 1 39 1 119 1 105 1 69 1 86 1 48 1 121 1 121 1 112 1 104 1 56 1 131 1 123 1 123 1 105 1 62 1 149 1 125 1 150 1 109 1 67 1 161 1 139 1 166 1 120 1 102 1 173 1 140 2 173 1 137 1 127 1 193 1 173 1 179 1 145 1 129 1 195 1 217 1 186 1 151 1 156 1 216 1 219 1 188 1 160 1 181 1 223 1 248 1 193 1 215 1 190 1 241 1 250 1 227 1 220 1 221 1 272 1 251 1 236 1 268 1 236 1 281 1 254 1 251 1 272 1 267 1 291 1 266 1 258 1 288 1 269 1 294 1 299 1 269 1 292 1 270 1 303 1 302 1 325 1 316 1 273 1 309 1 317 1 327 1 334 1 280 1 313 1 340 1 335 1 350 1 319 1 314 1 N= 2 5 361 1 357 1 346 1 362 1 N= 2 5 N= 2 5 N= 25 N= 25 In this simulation, C1 had a match (2 people born on Day #140). Columns 2 through 5 did not have any matches. So, for this simulation, you would enter ‘1’ in the space next to Col1-Col5 below. Col1-Col5: __1__ Col 26-Col30: _3___ Col6-Col10: ___2_ Col31-Col35: __2__ Col11-Col15: _1___ Col36-Col40: __5__
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Col16-Col20: _5___ Col41-Col45: __5__ Col21-Col25: __5__ Col46-Col50: __7__ d. Of the 50 columns that you generated, how many of them had at least one matching pair? Out of the 50 columns, 33/50 have at least one pair e. Based on your simulation, what is the probability that, in a room with 25 people, at least 2 people have the same birthday? Note: the probability is equal to (# of columns with a matching pair / 50) The probability is 0.66, or 66% f. How close is your experimental probability to the theoretical probability, which is .57? The experimental probability is very close, 0.09 (9%) more than the theoretical probability. This is significantly more than my estimated probability, which was 10%.