(Coins & Distributions & Bias s)

Transcription

1 (Coins & Distributions & Bias s) If you were to take 10 coins and flip them and count the heads, how many coins would you expect to be heads up? Try a brief experiment. Flip 10 coins, count the number of heads that appear, write the number that appeared on a post it note. Finally, place the post notes on the board to create a frequency dot plot This is a great way to make a demonstration of an approximate normal distribution. However to create a large sample takes a long time. Using Fathom, you can simulate this event hundreds even thousands of time in a very short amount of time. Open a new Fathom Workspace. Drag a collection,, from the shelf into the workspace. Open the collection by positioning the cursor at the lower right hand corner of the collection icon (as shown below). Finally, click and drag when the cursor changes to a resizing icon. Make the collection larger. This will effectively open the collection so that you can see what is inside of it. With the open collection selected choose New Cases under the Collection menu (as shown at the right). Choose 10 cases and press OK. M. Winking p.19

2 10 new gold icons should be created in the collection after the last step. Double click on one of the icons to open up the inspector window. In the Inspector window under the Tab Cases and the Column Attribute click on the word <new> and type in the word Flip To the right of the row you just labeled Flip double click the space under Formula In the calculator window Type space to open the Calculator/Formula window. and press OK. This should cause either an H or T to appear in the value cell. Next, at the top of the Inspect Collection 1 window click on the Display Tab. Then, double click on the cell to the right of caption and under Formula as shown below. When the Formula window opens up click the next to the word Attributes. Double click on the word Flip. Finally, press OK. space space and to then open double the click Calculator/Formula on the Attribute window. Flip The last step should have caused the title underneath each gold ball representing a case to change to either H or T depending on which the case is representing. Fathom can even take it a step further and have the icon be a picture of a coin Heads up or Tails up depending on the value of the Flip for each case. M. Winking p.20

3 This time in the Inspect Collection 1 window and under the formula cell for image double click to open the formula editor. In the formula editor type:. You can select the different Icons by clicking on the next to Icon Names. space and to then open enter the Calculator/Formula this conditional statement. window. First click on the Measures tab in the Inspect Collection window. Click on the text <new> and type in NumberofHeads Click here and change the title to NumberofHeads Double click on the cell to the right of NumberofHeads and under Formula. Then in the formula window type in and click on OK. space space and to then open enter the the count Calculator/Formula statement shown. window. M. Winking p.21

4 With the collection highlighted, select Collect Measures from the Collection menu. The last step will cause Fathom to rerandomize the 10 coin flips and then record the information from that group in a new collection. Fathom at its default setting will collect 5 sets of coin flips. Open the newly created measures by resizing the measures box created. Double click on one of the green cases inside. Double click on the green case to open this inspector window. In the Inspect Measures from Collection window select the Display tab and double click on the Formula cell to the right of caption. This will open the formula window again. Click the next to Attributes and double click the attribute NumberofHeads. Finally, click OK. space space and to then open double the Calculator/Formula click on NumberofHeads window. The last step will change the title underneath each Measure to the number of Heads flipped in the sample. Currently, we only have 5 sets of 10 coins being flipped. Lets quickly create a 1000 more sets of 10 coin flips. Click on the Collect Measures tab in the inspector window. M. Winking p.22

5 First, unselect the Animations on and change the number of measures from the default of 5 to as many trials as you would like. Finally, click on. Any time you are creating large samples of measures it is CRITICAL make sure Animations on is NOT checked. Click on the Cases tab in the inspector window and then drag a Graph, the workspace., into Drag the word NumberofHeads from the Inspector window onto the x-axis of the graph. M. Winking p.23

6 Click on the arrow to the right of word and change the graph into a histogram as shown at the right. The histogram looks fairly normally distributed. In order to overlay an actual normal curve on top of the histogram to see how normal the distribution looks, it will be easier if we look at the histogram using a relative frequency rather than just the actual frequency. With the graph selected, under the Graph menu change the Scale to Relative Frequency as shown at the right. While the graph is still highlighted, this time select Plot Function from the Graph menu. M. Winking p.24

7 In the formula window type in: and push OK. You can experiment with taking new collections in the measure menu and very consistently demonstrate that this is consistently a very normal distribution. Several studies have shown that actual real world coin flips don t always have a true 50% chance of landing on one side or the other. Some studies suggest that a coin that is flipped is just slightly more likely to land on the side that it starts with when flipped. Other studies show a huge bias when spinning a penny on a table. Let s consider trying a similar experiment to the one above. Spin the same coin on a table 10 times (making sure to alternate sides that the spinning begins from), letting it fall on its own, and then record the number of tails that showed up. Flip the same coin 10 times (making sure to alternate sides that the coin is initially flipped from) and then record the number of tails. The ideas on the internet suggest tails are significantly more likely to appear when spinning a penny. Lets make a hypothesis the mean of the difference of spinning tails and flipping tails is greater than 0. This experiment takes a fair amount of time. However, maybe we could collect at least samples. With this small of a sample, we will need to use something like a Student T test to determine whether our results reasonably suggest the hypothesis to be true. Open a new fathom window. Drag a Table from the shelf into the window. Click on the word <new> and type in SpinTails Click on the next <new> that appears and type in FlipTails M. Winking p.25

8 Begin entering the results from your trials in each of the columns. Then on the next column to the right, click on the word <new> and change the title to Difference With the Table selected, select Show Formulas from the Table menu as shown at the right. Double click on the cell just below Difference which should open the Formula Editor. Click the next to Attributes and double click on the appropriate attributes to type in the editor window:. Double click in the area right below Difference and using the Attribute menu to type the formula shown in the formula editor. Next, let s determine the mean of this difference. Drag a Summary Table into the workspace. Then, drag the word Difference from the Table into the Summary Table (as shown at the right). First, drag a summary table into the workspace Second, drag the word difference into the summary table. M. Winking p.26

9 To see if having a mean of greater than 0 with such a small sample was just a fluke because of high variability with small samples, we will need to perform a student T test. To do this drag the Test Icon,, into the workspace. Click the arrow next to the word and change to Test Mean Drag the word Difference into the top of the Test window where it says Attribute (as shown at the right). Any of the words in blue can be altered. In our original hypothesis we were trying to determine if the mean would be greater than zero. Click on the blue words that say is not equal to 0 and from the drop down menu select is greater than With the test still highlighted select Show Test Statistic Distribution under the Test menu as shown below. The number at the bottom represents the probability that mean being greater than 0 was just a fluke. M. Winking p.27