Question

6. Answer each question, given is a sample of 1000 observations which distribution follow a Gaussian distribution.

a. The mean is 100 and the standard deviation is 3. Based on this information what is the expected maximum amount of observations that is bigger than 109?

b. The mean is 100 and the variance is 9. Based on this information what is the expected maximum amount of observations that is bigger than 109?

c. Given is a dataset of 1000 observations which is normally distributed (Gaussian). The mean is 100 and the standard deviation is 20. How many observations are expected to be between 60 and 140?

a. 950, b. 667, c. 550, d. not enough information to answer

Answer 1

$(a)~P(X>109)=1-\Phi((109-100)/3)=0.0013\\ Expected~maximum~amount~of~observations~that~is~bigger~than~109\\ =0.0013*1000=1.3 \approx 1\\ (b)~a.~and~b.~are~same~question.\\ (c)~P(60\leq X\leq 140)=\Phi((140-100)/20)-\Phi((60-100)/20)=0.9545\\ no.~of~observations~are~expected~to~be~between~60~and~140\\ =0.9544*1000=954.4\approx 950\\ Option~a.$