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# Download Introduction to Chance Models (Section 1.1) Introduction A key step

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```4. Is it possible that this difference could happen even if dolphin therapy was not effective,
simply due simply to the random nature of putting subjects into groups (i.e., the luck of the
draw)?
Sure, this is possible. Consider the following scenario: Assume that the 13 people in the study
whose depression symptoms improved would have improved whether they swam with dolphins
or not, letâs call them the âimproversâ. Now, what if, by chance, of the 13 improvers (people who
improve no matter what), 10 randomly ended up in the âswim with dolphinsâ group. Recall the
important fact that subjects were randomly assigned to swim with dolphins or not. So,
randomly, it is possible that 10 improvers end up in the swim with dolphins group and only 3 in
the not swim with dolphins group. Thus, it is possible we would see 67% (10/15) of dolphin
swimmers improving and only 20% (3/15) non-dolphin swimmers improving even if swimming
with dolphins doesnât actually make a difference.
So, it is possibleâ¦.but how unlikely is it? In order to answer this question we will once again
turn to simulation. Remember that simulation is a method we have used to estimate
probabilities. So far you used coin flipping and an applet to simulate the p-value (the probability
we would obtain the observed statistic or something more extreme if the null hypothesis was
true) for different tests of significance. Because the dolphin experiment is more complex than
the previous studies we looked at, we canât simply flip a coin any longer. Instead, to do our
simulation, weâre going to use playing cards and a computer applet.
To estimate the p-value for this study you will need 30 index cards, 13 of which are blue to
represent the âimproversâ and 17 of which are green to represent the ânon-improvers.â The null
hypothesis assumes that 13 people (the blue cards) will get better no matter whether they swim
with dolphins or not. So those that improved were going to do so (perhaps just from getting to fly
to Honduras and swim in the water) regardless of which treatment group they were assigned to.
5. Now shuffle your 30 cards and deal them into two stacks of fifteen. One of the two stacks
represents the people who got to swim with dolphins and the other stack represents people
who didnât. Decide which stack is which and then fill in the table below.
Showed substantial improvement
Did not show substantial improvement
Total
Dolphin therapy Control group Total
13
17
15
15
30
Difference in proportions of improvement (dolphin group minus control group):
Repeat this shuffling and dealing process a second time:
Dolphin therapy Control group Total
Showed substantial improvement
13
Did not show substantial improvement
17
Total
15
15
30
Difference in proportions of improvement (dolphin group minus control group):
June 27, 2014
MAA PREP workshop
66
```
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