Statistics and probability

Question

Every year "Nathan's hotdog stand of Coney Island NY holds a hot dog eating contest. Assume the that number of hotdogs eaten by contestants last year was normally distributed with a mean of 40 and a standard deviation of 5.

What percentage of people ate between 30 and 45 hot dogs?

Answer 1

x = number of hot dogs

want to determine P(30 < x < 45) when mean of x is 40 and standard deviation = 5.

utilize z-score = (x - mean)/standard deviation to translate x to standard normal distribution (mean 0 standard deviation 1)

P(30 < x < 45) = P( (30-40)/5 < z < (45-30)/5) = P(-2 < z < 3) = P(z < 3) - P(z < -2) = 0.9987 - 0.0228 = 0.9759

Answer 2

From the emperical rule:

1. 34% are within the mean and + 1 standard deviation, so the percentage between 40 and 45 is 34%
2. 34% are within the mean and -1 standard deviation, so the percentage between 35 and 40 is 34%
3. 13.5% are within -2 and - 1 standard deviations, so the percentage between 30 and 35 is 13.5%

Add all these up and you get 68+13.5 = 81.5%