The Journal of Behavioral Decision Making (Jan. 2007) published a study of how g

Question

The Journal of Behavioral Decision Making (Jan. 2007) published a study of how guilty feelings impact on-the-job decisions. In one experiment, 57 participants were assigned to a guilty state through a reading/writing task. Immediately after the task, the participants were presented with a decision problem where the stated option had predominantly negative features (e.g., spending money on repairing a very old car). Of these 57 participants, 45 chose the stated option. Suppose 10 of the 57 guiltystate participants are selected at random. Define x as the number in the sample of 10 who chose the stated option.

Find P(x=5)

Find P(x=8)

What is the expected value (mean) of x?

Explanation / Answer

Here, if we approximate this as a binomial distribution,

p = 45/57 = 0.789473684 = probability of 1 success

a)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    10      
p = the probability of a success =    0.789473684      
x = the number of successes =    5      
          
Thus, the probability is          
          
P (    5   ) =    0.031961031 [answer]

***********

b)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    10      
p = the probability of a success =    0.789473684      
x = the number of successes =    8      
          
Thus, the probability is          
          
P (    8   ) =    0.300972322 [answer]

****************

c)

E(x) = n p = 10 * 0.789473684 = 7.89473684 [answer]

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