2) The amount of gasoline sold every week at a gas station is Normally distributed with a mean of 30,000 gallons and a Standard Deviation of 3,000 gallons. What is the Probability that in a given week that the amount of gasoline sold is between 28,000 and 35,000 gallons?
28000 is 0.666 std below the mean
35000 is 2.666 std above the mean.
Now go to your Z table. Or, you can play around with this stuff at
http://davidmlane.com/hyperstat/z_table.html
To find the probability that the amount of gasoline sold in a given week is between 28,000 and 35,000 gallons, we need to calculate the area under the Normal distribution curve within this range.
To do this, we can use the standard Normal distribution, which has a mean of 0 and a standard deviation of 1. We'll need to standardize the values of 28,000 and 35,000 to Z-scores using the formula:
Z = (X - μ) / σ
Where:
X = the value we want to standardize (in this case, 28,000 and 35,000)
μ = the mean of the distribution (30,000 gallons)
σ = the standard deviation of the distribution (3,000 gallons)
Let's find the Z-scores for both 28,000 and 35,000:
Z1 = (28,000 - 30,000) / 3,000
Z2 = (35,000 - 30,000) / 3,000
Calculating these values:
Z1 = -2,000 / 3,000 = -0.67
Z2 = 5,000 / 3,000 = 1.67
Now that we have the Z-scores, we can look up the corresponding probabilities in the standard Normal distribution table.
Looking up the value for Z = -0.67 in the table, we find that the cumulative probability up to this point is approximately 0.2514.
Similarly, looking up the value for Z = 1.67 in the table, we find that the cumulative probability up to this point is approximately 0.9525.
To find the probability that the amount of gasoline sold in a given week is between 28,000 and 35,000 gallons, we subtract the probability of Z = -0.67 from the probability of Z = 1.67:
P(28,000 ≤ X ≤ 35,000) = P(Z1 ≤ Z ≤ Z2) = P(Z ≤ 1.67) - P(Z ≤ -0.67) = 0.9525 - 0.2514 = 0.7011
Therefore, the probability that the amount of gasoline sold in a given week is between 28,000 and 35,000 gallons is approximately 0.7011, or 70.11%.