Question

The oil production at a refinery is normally distributed with a mean of 230,000 barrels per day with a standard deviation of 7,500 barrels.  

1. Justify the following sentence without calculation using the characteristics of the normal distribution: “On any given day, there is an equal probability of producing more than 240,000 barrels as the probability of producing less than 220,000 barrels”.  
2. What is the probability of producing less than 245,000 barrels? [Provide an answer accurate to 6 decimal places]
3. What is the probability of producing at least 220,000 barrels? [Provide an answer accurate to 6 decimal places]
4. Finish the sentence: “There’s an 25% chance of producing more than ____ barrels on any given day” [Provide an answer accurate to the nearest barrel]
5. According to the Central Limit Theorem, describe the distribution of a 20 day average production at this refinery.   Make special reference to the shape, mean and spread of the distribution.   Assume that every day’s production is independent of every other day’s production.

Answer 1

1. As the normal distribution is symmetrical about its mean, the P(X>Mean+a) = P(X<mean-a). So here the probability of producing more than 240,000 is same as probability of production less than 230,000.

2. P(X>245,000) = P((X-230,000)/7500> 2) = 1- 0.9772499 = 0.022750

3. P(X<220,000) = P((X-230,000)/7500 < -1.333) = 0.09121127 = 0.091211

4. There is 25% chance of producing more than 235,058.7 (235,059) barrels per day.

Which is basically 0.6744898*7500 + 230,000

Thanks