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[]
[null,null,"The Binomial Distribution"]
[null,null,"( OR … are mice cleverer than we think?)"]

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["BINOMIAL","Results for the 5 mice finding their way out of the maze",null,null,null,null,null,null,null,null,null,"#Successes","Frequency",null,null,null,"BinRange","Emp. Freq.","Theor. Freq.","Emp. Prob.","Theor. Prob."]
["©GDIB - UCT - 2009",null,null,null,null,null,null,null,null,null,null,0,166,null,"Emp. Mean",{"$f":"SUM(heads*freq)/Nexp","v":2.5082},0,{"$f":"M2","v":166},{"$f":"U2*Nexp","v":156.25},{"$f":"M2/Nexp","v":3.32E-2},{"$f":"(FACT(nMice)/(FACT(nMice-Q2)*FACT(Q2)))*(pp^Q2)*(1-pp)^(nMice-Q2)","v":3.125E-2},null,null,null,null,null,6.2910672400000003]
["xl2007 ver 14","If Mouse solves maze  =  1",null,null,null,null,null,null,null,null,null,1,782,null,"Theoretical mean",{"$f":"nMice*pp","v":2.5},1,782,781.25,0.15640000000000001,0.15625,null,null,null,null,null,2.2746672399999999]
[null,"If Mouse fails to solve maze = 0",null,null,null,null,null,null,null,null,null,2,1505,null,"Emp.Var",{"$f":"SUM(freq*heads2)/Nexp","v":1.2699327600000001},2,1505,1562.5,0.30099999999999999,0.3125,null,null,null,null,null,0.25826724000000001]
["# Experiments (< 10 000)","Experiment #","Mouse #1","Mouse #2","Mouse #3","Mouse #4","Mouse #5","Total successful Mice",null,null,null,3,1601,null,"Theoretical Var",{"$f":"nMice*pp*(1-pp)","v":1.25},3,1601,1562.5,0.32019999999999998,0.3125,null,null,null,null,null,0.24186724000000001]
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["#Mice (Fixed)",2,0,1,1,0,1,3,null,null,null,5,162,null,null,null,5,162,156.25,3.2399999999999998E-2,3.125E-2,null,null,null,null,null,6.2090672400000004]
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["P(Mouse solves maze)",4,1,0,1,0,0,2,null,null,null,"To Print Graph"]
[0.5,5,1,0,0,1,1,3,null,null,null,"1. Click on graph to select"]
[null,6,0,0,0,1,0,1,null,null,null,"2. Click on Office button (top-left)"]
[null,7,0,1,1,1,0,3,null,null,null,"3. Print (preview first)"]
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[null,4997,1,1,1,0,0,3]
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{"sheet":"Notes","index":2,"n_rows":31,"n_cols":2,"rows_kept":31,"cut":false,"hidden":false,"merged":[]}
[null,"Aims Sta1"]
[]
[null,"Use a fixed n to simplify the problem"]
[null,"Demonstrate that as we increase the #repetitions, the empirical distribution converges upon the theoretical"]
[null,"compare the theoretical means and variances with the empirical means and variances "]
[]
[]
[]
[null,"We want to test whether mice have \"intelligence\" wrt to a particular simple maze"]
[null,"if the mice make random decisions they should have about a 50% of navigating the maze"]
[null,"if they have intelligence they should have more than a 50% chance of navigating the maze"]
[]
[null,"Say we take 5 mice and one makes (4 don't) it through the maze, what do we conclude?"]
[]
[]
[null,"first lets assume p=0.5 and consider the DISTRIBUTION of the number of mice succeeding"]
[null,"Under the assumption that p=0.5 we see the prob. of only 1 mouse succeeding is 13.5% "]
[null,"Is this consistent with p=0.5?"]
[]
[null,"So, we would probably want to repeat the experiment with many batches of 5 mice to get a more precise estimate?"]
[]
[null,"We could then test our number of successful mice against the theoretical number of successful mice (if p=0.5)"]
[]
[null,"If the numbers were about the same then the mice probably aren’t intelligent"]
[null,"if the average number who are succesful is greater than the theoretical number (p=0.5) the mice probably are not intelligent"]
[]
[null,"We could also establish that for p=0.5 (any p in fact between 0 and 1) the number of successful mice follows a BINOMIAL distribution"]
[null,"We can use this information to formulate an hypothesis test"]
[null,"Firstly we could calculate the difference between the observed (or empirical) number of successes and the expected number (assuming p=0.5)"]
[null,"We could then also use the variance of the BINOMIAL distribution to formulate a statistcal hypothesis test"]
[null,"the variance gives us an idea of how much the number of successes will vary even when p=0.5"]

{"sheet":"Exercises","index":3,"n_rows":5,"n_cols":3,"rows_kept":5,"cut":false,"hidden":false,"merged":[]}
[]
[null,null,"For the #experiments(replications) = 500"]
[null,1,"consider different p's. In particular if the mice have intelligence (say p=0.8) consider the expected maze results"]
[null,2,"Print the graphs for p=0.5 and p=0.8"]
[null,3,"Repeat the above with #experiments=100 #experiments = 5 000, comparing the results"]