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 = 0.39 and standard deviation σ = 0.036
What proportion of adhesions (± 0.001) measured in use are higher than 0.32?
What proportion of adhesions (± 0.001) are between 0.32 and 0.45?
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To answer these questions, we need to find the proportions based on a given normal distribution with a mean μ of 0.39 and a standard deviation σ of 0.036.
1. Proportion of adhesions higher than 0.32:
To find the proportion of adhesions higher than 0.32, we need to calculate the area under the curve to the right of 0.32 in the normal distribution.
Step 1: Standardize the value 0.32 by subtracting the mean and dividing by the standard deviation:
z = (0.32 - μ) / σ
Substituting the given values:
z = (0.32 - 0.39) / 0.036
Step 2: Look up the corresponding z-value in the standard normal distribution table or use a calculator to find the area to the left of that z-value.
The area to the left of the z-value -2.111 is approximately 0.0179.
Step 3: Subtract this value from 1 to find the proportion to the right of 0.32:
Proportion = 1 - 0.0179 = 0.9821
Therefore, approximately 98.21% of adhesions are higher than 0.32.
2. Proportion of adhesions between 0.32 and 0.45:
To find the proportion of adhesions between 0.32 and 0.45, we need to calculate the area under the curve between these two values in the normal distribution.
Step 1: Standardize the values 0.32 and 0.45 by subtracting the mean and dividing by the standard deviation:
z1 = (0.32 - μ) / σ
z2 = (0.45 - μ) / σ
Substituting the given values:
z1 = (0.32 - 0.39) / 0.036
z2 = (0.45 - 0.39) / 0.036
Step 2: Look up the corresponding z-values in the standard normal distribution table or use a calculator to find the area to the left of those z-values.
The area to the left of z1 ≈ -2.111 is approximately 0.0179.
The area to the left of z2 ≈ 1.667 is approximately 0.9522.
Step 3: Subtract the smaller area from the larger area to find the proportion between 0.32 and 0.45:
Proportion = 0.9522 - 0.0179 = 0.9343
Therefore, approximately 93.43% of adhesions are between 0.32 and 0.45.