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. (a) What is the
probability of producing 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?

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions that correspond to the Z scores.

What do you mean "Z" scores in relation to the problem above? sorry, I'm totally confused and overwhelmed

To find the probabilities in this scenario, we need to use the concept of the standard normal distribution and Z-scores.

(a) To find the probability of producing at least 232,000 barrels, we need to find the area under the normal distribution curve to the right of 232,000 barrels. To do this:

1. Calculate the Z-score: Z = (X - μ) / σ, where X is the value (232,000), μ is the mean (232,000), and σ is the standard deviation (7,000).
Z = (232,000 - 232,000) / 7,000 = 0

2. Look up the Z-score in the standard normal distribution table. The Z-score of 0 corresponds to a cumulative probability of 0.5000.

So, the probability of producing at least 232,000 barrels is 0.5000.

(b) To find the probability of producing between 232,000 and 239,000 barrels, we need to find the area under the normal distribution curve between these two values. To do this:

1. Calculate the Z-scores for both values:
Z1 = (232,000 - 232,000) / 7,000 = 0
Z2 = (239,000 - 232,000) / 7,000 ≈ 1

2. Look up the corresponding probabilities for both Z-scores in the standard normal distribution table.
The probability for Z1 (0) is 0.5000.
The probability for Z2 (approximately 1) is 0.8413.

To find the probability between these two values, subtract the probability corresponding to the smaller Z-score from the probability corresponding to the larger Z-score:
0.8413 - 0.5000 = 0.3413

So, the probability of producing between 232,000 and 239,000 barrels is approximately 0.3413.

(c) To find the probability of producing less than 239,000 barrels, we need to find the area under the normal distribution curve to the left of 239,000 barrels. To do this:

1. Calculate the Z-score: Z = (239,000 - 232,000) / 7,000 ≈ 1

2. Look up the Z-score in the standard normal distribution table. The probability for Z ≈ 1 is 0.8413.

So, the probability of producing less than 239,000 barrels is approximately 0.8413.

(d) To find the probability of producing less than 245,000 barrels, we follow the same steps:

1. Calculate the Z-score: Z = (245,000 - 232,000) / 7,000 ≈ 1.857

2. Look up the Z-score in the standard normal distribution table. The probability for Z ≈ 1.857 is approximately 0.9686.

So, the probability of producing less than 245,000 barrels is approximately 0.9686.

(e) To find the probability of producing more than 225,000 barrels, we need to find the area under the normal distribution curve to the right of 225,000 barrels. To do this:

1. Calculate the Z-score: Z = (225,000 - 232,000) / 7,000 ≈ -1

2. Look up the Z-score in the standard normal distribution table. The probability for Z ≈ -1 is approximately 0.1587.

Since we want the probability of "more than," we subtract the probability from 1:
1 - 0.1587 = 0.8413

So, the probability of producing more than 225,000 barrels is approximately 0.8413.