Jenny is up to the million dollar question on “Who Wants to be a Millionaire”. T

Question

Jenny is up to the million dollar question on “Who Wants to be a Millionaire”. The question is,

In statistical theory, which of the following distributions would be most suitable for the random variable “the number of joggers running past a certain bus stop in an hour”?

A: Binomial B: Normal C: Poisson D: Geometric

Jenny thinks for a while and says “well, it can’t be B: Normal, because that’s a continuous distribution and this is a discrete random variable. Having said that, I haven’t studied any of this in many years, so I have no idea which of the other three it might be!”        The host, Eddie, reminds Jenny that she still has her “ask the audience” lifeline remaining, and that she may as well use it. Jenny says to the audience “if you don’t know the answer, choose B”.  

If 10% of the general population definitely know the answer is C, 25% think they know the answer, and will guess either A, C or D with probability 15%, 60%, and 25% respectively, and 65% definitely don’t know the answer and will choose B, express the approximately probability that in a randomly selected audience of 578 people, more than 150 answer correctly. Use a normal approximation with continuity correction, and leave your answer in terms of the standard Normal cumulative distribution function, ?(z)=Pr(2<z).

Explanation / Answer

Expected number of people who will tell the correct answer = 578 * 10/100 + 578* 25/100 * 65/100 = 151.725

Here we can also calculate standard error of the sampling s.

Here when we calculated expected number of people we can see that 578* 10/100 = 57.8 people always told the truth. and there is variation only in the number of 25% people from that means

Here n = 578 * 0.25 = 144.5 and proportion of people who choose option c is lets say p = 0.65

so standard deviation of the answer C option is s = sqrt [ p(1-p) *n] = sqrt [ 0.65 * 0.35 * 144.5] = 5.7336

so We have to calculate now the probability of more than 150 people answered correctly.

so we have to calculate Pr(X>150) where X is the number of people gave correct answer.

by continuity correction Pr(X>150) = Pr(X >150+0.5) = Pr(X > 150.5)

so we have to calculate Pr(X >150.5) where mean = 151.725 and standard deviation = 5.7336

so Z - value = (150.5 - 151.725)/ 5.7336 = -0.2137

so Pr(X>150) = 1- (z) = 1- (-0.214)

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