Question

Consider 3400 marathon runners where each is 32% likely to drop out of the race. Apply the normal approximation to calculate the probability that no more than 2380 runners but more than 2295 runners eventually cross the finish line. First calculate without the histogram correction. Then recalculate with the histogram correction.

Answer 1

Let X is a random variable shows the number of marathon runners drop out from the race. Here X has binomial distribution with following parameters: n = 3400, p=0.32 Checking: Since np = 1088 and n(1-p) = 2312 both are greater than 5 so normal approximation is appropriate. Using normal approximation distributon of X will be approximately normal with mean and SD as follow: Il = np = 1088 o= Inp(1-p) = sqrt((3400 * (1 -0.32) * 0.32) = 27.2

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Without the histogram correction:

We need to find the probability

P(2295 <X <= 2380)

The 2-score for X=2295 is: X- - = (2295 - 1088)/27.2 0 = 44.38 The Z-score for X=2380 is: X- 2 = - = (2380 - 1088)/27.2 o = 47.5 Using z table, the probability that X is between 2295 and 2380 is: P( 2295 < X <= 2380 ) = P( 44.38<z<= 47.5) = P(Z <= 47.5) - P(Z <= 44.38 ) = 1 - 1 = 0 Answer: 0 Excel can be used to find the probabilites is =ROUND(NORMSDIST(47.5) - NORMSDIST(44.38), 4).

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With the histogram correction:

We need to find the probability

P(2295 <X <= 2380) = P(2295.5 < X < 2380.5)

The Z-score for X=2295.5 is: Z=- - = (2295.5 - 1088)/27.2 o - 44.39 The Z-score for X=2380.5 is: Z =- - = (2380.5 - 1088)/27.2 = 47.52 0 Using z table, the probability that X is between 2295.5 and 2380.5 is: P( 2295.5 <x< 2380.5 ) = P( 44.39<z<= 47.52) = P(Z <= 47.52 ) - P(Z <= 44.39 ) = 1 - 1 = 0 Answer: 0 Excel can be used to find the probabilites is =ROUND(NORMSDIST(47.52) - NORMSDIST(44.39), 4).

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