Question 1034997: a convenience store makes over $200 approximately 75% of the days it is open. use the normal approximation to determine the probability that the store makes over $200 at least 265 out of 365 days. round your answer to the nearest thousandth.
I think the probability of success is (265/365) or (53/73) in simplist form and I also this the probability of failure is (100/365) or (20/73) in simplist form. I know we have to use this formula:
P(at least r)= p ( r ) + p(r-1) + p( r-2) ....+p(0)
And we use nCr P(success)^r * P(failure)^n-r
But I don't understand what n or r is I was thinking r equaled .75 but I am not sure.
I know the answer is .868 but I do not know how to get there.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! p = .75
q = .25
n = 365
x = 265
mean = n*p = .75 * 365 = 273.75
s = standard error of the mean = sqrt(n*p*q) = sqrt(365*.75*.25) = 8.272696054
adjust the value of x to account for discrete versus continuous by subtracting .5 from it.
x becomes 264.5
z = (x-m)/s = (24.5 - 273.75) / 8.272696054 = -1.118136088
p(z > -1.118136088) = .8682455167
round that to 3 decimal places and you get p(x > -1.118136088) = .868.
since the binomial distribution is discrete and the normal approximation is continuous, there is an adjustment that needs to be made.
that adjustment is to subtract .5 from the score and to add .5 to the score.
to find the probability of that score happening, you find the z-score for that score - .5 and the z-score for that score + .5 and then find the probability of getting between those z-score.
to find the probability of getting less than that score, you find the z-score for that score + .5 and then look for the probability that you can get below that z-score.
to find the probability of getting more than that score, you find the z-score for that score - .5 and then look for the probability that you can get above that z-score.
the last one is what we did.
the score was 265.
we subtracted .5 from that to get 264.5.
we calculated the z-score for (264.5 - 273.75) / 8.272696054.
that gave us the z-score of -1.118136088.
we then looked for the probability that we would get a z-score greater than that.
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