Double checking my math?

I suck at probability so I am double checking my answers if you could show work that would help me:





Q. a study of past participants indicates that the mean length of time spent on the program is 500 hours and that in normally distributed random variable that has a standard deviation of 100 hours.





A-what is the probability that a candidate selected at random will take more than 700 hours to complete the program?





B-suppose the training-program director wants to know the probability that a participant chosen at random would require between 550 and 650 hours to complete required work?





C- what is the probability that a candidate selected at random will require less than 580 hours to complete the program?

Double checking my math?
You have a normal distribution with a mean of 500 and a standard deviation of 100.





First thing you do is standardize the distribution.





In (A) you want Pr(X%26gt;700). 700=500+2*100 so it is the mean plus twice the standard deviation. So you are looking for the probability that the event is 2 standard deviation higher than the mean.





With a normal distribution you can standardize this. In general you refer to a Z-score as Z=(X-mean)/standard deviation to standardize the distribution to a 0 mean and a variance of 1. The probabilities for both are the same.





Z=(700-500)/100=2. The cumulative distribution from the table is 0.97725 so the probability is 1-0.97725 = 2.275%.





B - Here you are looking for Pr(550%26lt;X%26lt;650) = Pr(500+100/2%26lt;X%26lt;500+3/2*100) so between 1/2 and 3/2 standard deviations above the mean.





The cumulative probability for 3/2 is 0.9332


The cumulative probability for 1/2 is 0.6915





You want the difference 0.9332-0.6915 = 0.2417.





C - Now just note that 580=500+0.8*100





The cumulative probability for 0.8 is 0.7881 so that is your answer.