Daily output of Marathon's Garyville, Lousiana, refinery is normally distributed with a mean of 232,000 barrels of crude oil per day with a standard deviation of 7,000 barrels.
Find the probability of producing..
a.
at least 232,000 barrels
b.
between 232,000 and 239,000 barrels
c.
less than 239,000 barrels
d.
less than 245,000 barrels
e.
more than 225,000 barrels
A wonderful little applet here:
http://davidmlane.com/hyperstat/z_table.html
Just enter the mean and sd, and then click on the suitable "between", "below" etc with the given values.
(To my way of thinking, I really see no difference between using the traditional use of tables, graphs, or charts for these kind of problems and this particular little program.)
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To find the probabilities, we can use the standard normal distribution and convert the given values to z-scores.
The z-score formula is calculated as:
z = (x - μ) / σ
where:
z = z-score
x = given value
μ = mean
σ = standard deviation
a. To find the probability of producing at least 232,000 barrels, we need to find the area to the right of 232,000 in the z-table.
z = (232,000 - 232,000) / 7,000 = 0
Using the z-table, the area to the right of z = 0 is 0.5. Therefore, the probability of producing at least 232,000 barrels is 0.5.
b. To find the probability of producing between 232,000 and 239,000 barrels, we need to find the area between the z-scores of the two values.
z1 = (232,000 - 232,000) / 7,000 = 0
z2 = (239,000 - 232,000) / 7,000 = 1
Using the z-table, the area between z = 0 and z = 1 is 0.3413. Therefore, the probability of producing between 232,000 and 239,000 barrels is 0.3413.
c. To find the probability of producing less than 239,000 barrels, we need to find the area to the left of 239,000 in the z-table.
z = (239,000 - 232,000) / 7,000 = 1
Using the z-table, the area to the left of z = 1 is 0.8413. Therefore, the probability of producing less than 239,000 barrels is 0.8413.
d. To find the probability of producing less than 245,000 barrels, we need to find the area to the left of 245,000 in the z-table.
z = (245,000 - 232,000) / 7,000 = 1.8571
Using the z-table, the area to the left of z = 1.8571 is 0.9671. Therefore, the probability of producing less than 245,000 barrels is 0.9671.
e. To find the probability of producing more than 225,000 barrels, we need to find the area to the right of 225,000 in the z-table.
z = (225,000 - 232,000) / 7,000 = -1
Using the z-table, the area to the right of z = -1 is 0.8413. Since we want the probability of producing more than 225,000 barrels, we subtract this value from 1 to get 1 - 0.8413 = 0.1587. Therefore, the probability of producing more than 225,000 barrels is 0.1587.
To find the probabilities for different scenarios, we can use the cumulative distribution function (CDF) of the normal distribution. The CDF gives us the probability that a random variable is less than or equal to a specific value.
In this case, we have a normally distributed random variable with a mean of 232,000 barrels and a standard deviation of 7,000 barrels.
a. To find the probability of producing at least 232,000 barrels, we can calculate the area under the curve from 232,000 barrels to positive infinity. This can be found by subtracting the probability of producing less than 232,000 barrels from 1. We can use the standard normal distribution table or a statistical calculator to find the probability.
b. To find the probability of producing between 232,000 and 239,000 barrels, we need to calculate the area under the curve between these two values. This can be done by finding the probability of producing less than or equal to 239,000 barrels and subtracting the probability of producing less than or equal to 232,000 barrels.
c. To find the probability of producing less than 239,000 barrels, we can calculate the area under the curve from negative infinity to 239,000 barrels. This can be found by using the CDF function with the given mean and standard deviation.
d. To find the probability of producing less than 245,000 barrels, we can calculate the area under the curve from negative infinity to 245,000 barrels using the CDF function.
e. To find the probability of producing more than 225,000 barrels, we can subtract the probability of producing less than or equal to 225,000 barrels from 1. This can be calculated using the CDF function.
Using a statistical calculator or software, we can input the mean, standard deviation, and the desired values to calculate the probabilities for each scenario.