An important measure of the performance of a locomotive is its "adhesion," which is the locomotive's pulling force as a multiple of its weight. The adhesion of one 4400-horsepower diesel locomotive model varies in actual use according to a Normal distribution with mean µ = 0.35 and standard deviation s = 0.045
What proportion of adhesions (± 0.001) measured in use are higher than 0.41?
What proportion of adhesions (± 0.001) are between 0.41 and 0.6
a handy Z table calculator is at
davidmlane.com/hyperstat/z_table.html
you can explore the various values there.
To answer these questions, we will use the concept of the standard normal distribution. In order to find the proportion of adhesions higher than 0.41, we need to calculate the z-score for this value. The z-score measures how many standard deviations a given value is from the mean.
To calculate the z-score, we use the formula:
z = (x - µ) / s
Where:
x = the value we want to find the proportion for (0.41 in this case)
µ = the mean of the distribution (0.35)
s = the standard deviation of the distribution (0.045)
Plugging in these values, we get:
z = (0.41 - 0.35) / 0.045
z ≈ 1.33
Next, we can look up the proportion of adhesions higher than this z-score in a standard normal distribution table or use a calculator.
According to the standard normal distribution table, we find that the proportion of adhesions higher than 1.33 is approximately 0.0918.
So, the proportion of adhesions (± 0.001) measured in use that are higher than 0.41 is approximately 0.0918.
To find the proportion of adhesions between 0.41 and 0.6, we repeat the same process but with different values.
First, we calculate the z-scores for both values:
For 0.41:
z1 = (0.41 - 0.35) / 0.045 ≈ 1.33
For 0.6:
z2 = (0.6 - 0.35) / 0.045 ≈ 5.56
Next, we can subtract the proportion of adhesions less than the lower z-score from the proportion of adhesions less than the higher z-score to find the proportion between them.
Using the standard normal distribution table or calculator, we find that the proportion of adhesions less than 1.33 is approximately 0.9082, and the proportion of adhesions less than 5.56 is approximately 1.
So, the proportion of adhesions (± 0.001) that are between 0.41 and 0.6 is approximately 1 - 0.9082 = 0.0918.
Therefore, the proportions of adhesions (± 0.001) that are higher than 0.41 and between 0.41 and 0.6 are both approximately 0.0918.