Please help me with this calculus based statistics question which involves using

Question

Please help me with this calculus based statistics question which involves using binomial distribution to find probabilities along with using Chebyshev's Theorem to find one of the probabilities. Please be DETAILED by showing all work and make sure to double check answer since I've had some wrong answers lately. If not much work is shown, I will rate the answer poorly since I really need to understand this material. I'm mainly struggling with Chebyshev's Theorem (I struggle with inputting the values in the theorem equation, which requires no solving, just substituting the right values).

Thank you

Explanation / Answer

a)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    52      
x2 =    76      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    32000000      
p = the probability of a success =    0.000002      
          
Then          
          
P(at most    51   ) =    0.055201128
P(at most    76   ) =    0.937683963
          
Thus,          
          
P(between x1 and x2) =    0.882482835   [ANSWER]

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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 =    32000000      
p = the probability of a success =    0.000002      
x = the number of successes =    63      
          
Thus, the probability is          
          
P (    63   ) =    0.049802944 [ANSWER]

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c)

Here,

u = mean = np =    64      
          
s = standard deviation = sqrt(np(1-p)) = 8

Hence, between 56 and 72 is within 1 standard deviation from the mean.

Hence, by Chebyshev's theorem, as k = 1,

1 - 1/k^2 = 1 - 1/1^2 = 0

Hence,

P(56<X<72) > 0 [ANSWER, which is trivial]

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Hi! These are exact answers, as I used binomial distirbution. If you use another method of approximation (probably Poisson), please resubmit this question indicating that method. That way we can continue helping you! Thanks!

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