Question

The yield, in tons of ore per day, at a given coal mine is approximately normally distributed with mean 500 tons and standard deviation 50. a) Find the probability that at least 480 tons of ore will be mined on a given day. b) Find the proportion of working days in which anywhere from 510 to 580 tons is mined. c) Find the probability that on a given day, the yield will be below 605 tons Question 4: Most graduate schools of business require applicants for admission to take the Graduate Management Admission Council's GMAT examination. Scores on the GMAT are roughly normally distributed with a mean of 527 and a standard deviation of 112. What is the probability of an individual scoring above 500 on the GMAT? How high must an individual score on the GMAT in order to score in the highest 5967 Question 5: The length of human pregnancies from conception to birth approximates a normal distribution with a mean of 266 days and a standard deviation of 16 days. What proportion of all pregnancies will last between 240 and 270 days (roughly between 8 and 9 months)? What length of time marks the shortest 70% of all pregnancies? Question 6: The average number of acres burned by forest and range fires in a large New Mexico county is 4,300 acres per year, with a standard deviation of 750 acres. The distribution of the number of aros bumed normal. What is the probability that between 2,500 and 4,200 acres will be burned in any given year? What number of burnt acres corresponds to the 38th percentile? Question 7: Scientists in Weather Forecasting Department have been working really hard and finally came to the conclusion that the annually rainfall of Xuân Lộc - Đông Nai is normally distributed with mean 2.956 mm, and standard deviation of 0.615 mm. Note that the annually rainfall are independent to each other a) What is the probability this year's rainfall will over 3.22 mm? b) What is the probability that the sum of the next 2 years' rainfall will less than 5.68 mm? c) What is the probability that the sum of the next 3 years' rainfall will exceed 7.89 mm? Hint: E(X,+X)=E(X1)+E(X) Var (Xi+X)=6'(x))+ 6'(X)

Answer 1

Q.1) X = Yield in tons of ore per day

X ~ N(500,2500)

let Z ~ N(0,1)

we know that, by property of normal distribution,

X-500/50 ~N(0,1)

the values below are obtained using standard normal table

a) P(X is atleast 480) = P(X > 480) = P(Z > 480-500/50) = P(Z > -0.4) = 0.65542

Thus, probability that atleast 480 tons are mined in a day is 0.65542

b) P(X is between 510 to 580) = P(510 < X < 580) = P(X > 510) - P(X > 580) = 0.42074 - 0.05480 = 0.36594

Proportion of working days in which ore is mined between 510 tons to 580 tons is 0.36594

c) P(X is below 605) = P(X < 605) = 0.98214

probability that yield is below 605 tons is 0.98214