Please help with the following statistics question There is

Please help with the following statistics question,

There is some evidence to suggest that businesses are moving out of states where unions are prevalent. In California, 18.4% of all workers belong to a union. Suppose 26 workers from California are selected at random. Demonstrate that X = the number of workers that belong to a union is a binomial random variable. Find the following probabilities: At most three California workers belong to a union. At least four but at most seven California workers belong to a union. Exactly eight California workers belong to a union. Less than twenty California workers do not belong to a union. More than nine but at most fourteen California workers do not belong to a union. What is the mean number of California workers that do belong to a union? Use the correct notation. What is the standard deviation for this probability distribution? Use the correct notation. Find P(mu - 2sigma

Solution

a)

X is a binomial random variable as it only has two possible outcomes, that is, \"belonging to a union\" and \"not belonging to a union\". It also has a constant probability of success, 0.184.

Also, it seems here that the persons choose independently whether to join a union or not. Hence, X is a binomial random variable.

b)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    26      
p = the probability of a success =    0.184      
x = the maximum number of successes =    3      
          
Then the cumulative probability is          
          
P(at most   3   ) =    0.269050207 [answer]

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c)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    4      
x2 =    7      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    26      
p = the probability of a success =    0.184      
          
Then          
          
P(at most    3   ) =    0.269050207
P(at most    7   ) =    0.910147397
          
Thus,          
          
P(between x1 and x2) =    0.64109719   [answer]

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d)

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 =    26      
p = the probability of a success =    0.184      
x = the number of successes =    8      
          
Thus, the probability is          
          
P (    8   ) =    0.052810822 [answer]
  

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