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The brand name of Domino's has a 75% recognition rate (each consumer is independent from each other). Assume 12 randomly selected consumers are surveyed. Use this situation to answer the questions below.
\n" ); document.write( "(a)Recognizing that this is a binomial situation, give the meaning/values of S, F, n, p, and q.
\n" ); document.write( " S is: n =
\n" ); document.write( " F is: p =
\n" ); document.write( " q =
\n" ); document.write( "(b)Construct the complete binomial probability distribution for this situation in a table at the right.
\n" ); document.write( "(c)Find the probability all twelve consumers recognize the Domino's brand.
\n" ); document.write( "(d)Find the probability less than five consumers recognize the Domino's brand.
\n" ); document.write( "(e)Find the mean and standard deviation of this binomial probability distribution.
\n" ); document.write( "(f)By writing a sentence, interpret the meaning of the mean found in (e).
\n" ); document.write( "(g)Would it be unusual for exactly six of the twelve consumers to recognize the Domino's brand? Explain your answer.
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\n" ); document.write( "Solution:
\n" ); document.write( "
\n" ); document.write( "(a) according to standard binomial distribution nomenclature,
\n" ); document.write( "n=sample size
\n" ); document.write( "p=probability of success
\n" ); document.write( "q=probability of failure, with condition that p+q=1, or q=1-p
\n" ); document.write( "S: event of success, in this case, brand is recognized.
\n" ); document.write( "F: event or failure, in this case, brand is not recognized.
\n" ); document.write( "
\n" ); document.write( "(b) The binomial distribution, with variable x and above parameters, is given by
\n" ); document.write( "P(x;n;p)=
\n" ); document.write( "where C(n,x) is the binomial coefficient of choosing x object out of n, and where
\n" ); document.write( "C(n,x)=n!/(x!(n-x)!)
\n" ); document.write( "so the distribution is a table of x versus P(x;n;p) for x=0 to 12.
\n" ); document.write( "Here I will give you a kickstart:
\n" ); document.write( "Example, for x=9, n=12, p=0.75 (q=0.25), then
\n" ); document.write( "P(9,12,0.75)=
\n" ); document.write( "Proceeding to construct the table:
\n" ); document.write( "x P(x,12,0.75)
\n" ); document.write( "0 0.0000000596
\n" ); document.write( "1 0.0000021458
\n" ); document.write( "2 0.0000354052
\n" ); document.write( "3 0.0003540516
\n" ); document.write( "4 0.0023898422
\n" ); document.write( "5 0.0114712715
\n" ); document.write( "6 0.0401494503
\n" ); document.write( "... (please fill in the missing values as practice)
\n" ); document.write( "12 0.031676352
\n" ); document.write( "You can check your answers by adding up all (13) probabilities and they should add up to exactly one.
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\n" ); document.write( "(c) If all consumers recognize the brand, that makes 12 successes, or x=12.
\n" ); document.write( "Consult table above to find probability.
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\n" ); document.write( "(d) Less than 5 include the cases x=0,1,2,3,4. So you will need to sum the probabilities for x=0,1,2,3,4 accordingly.
\n" ); document.write( "
\n" ); document.write( "(e) μ & σ
\n" ); document.write( "Mean of the binomial distribution is μ=np=12*0.75=9
\n" ); document.write( "standard deviation is given by σ=npq=12*0.75*0.25=2.25
\n" ); document.write( "
\n" ); document.write( "(f) Mean of a distribution is the summation (discrete distribution) or integral (continuous distribution) of the value
\n" ); document.write( "In this case, μ=∑
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\n" ); document.write( "(g) From the table above, the probability that six of the 12 sampled will recognize the brand is P(6)=0.0401494503, or about 4%, or approximately 1 in 25 samples. Comment on whether this is an unusual occurrence. \n" ); document.write( "