What is the 75th percentile of the adhesion distribution?
A locomotive's 'adhesion' is the locomotive's pulling force as a multiple of its weight. A diesel locomotive model has adhesion which varies in actual use according to a Normal distribution with a mean .037 and a variance of 0.0016.
Can someone tell me how to do this problem thanks
Use a z-score formula:
z = (x - mean)/sd
z = value from a z-table corresponding to the 75th percentile
x = the value you are trying to find
mean = .037
sd = square root of the variance 0.0016
Once you have the values plugged into the formula, solve for x.
To find the 75th percentile of the adhesion distribution, you need to follow a few steps:
1. Calculate the standard deviation (σ) of the adhesion distribution. The standard deviation is the square root of the variance, so in this case, it would be √0.0016 = 0.04.
2. Convert the adhesion distribution to a standard normal distribution, also called a Z-distribution. To do this, subtract the mean (µ) from the adhesion value you want to find the percentile for (in this case, the 75th percentile), and then divide that difference by the standard deviation you calculated in step 1. Mathematically, it can be represented as Z = (X - µ) / σ.
3. Once you have the Z-score (Z), you can refer to a standard normal distribution table (also known as a Z-table) or use statistical software (such as Excel or calculator functions) to find the corresponding cumulative probability of the Z-score. In this case, you want to find the cumulative probability for Z = 75th percentile, which corresponds to a probability of 0.75.
4. Now, look up the cumulative probability of 0.75 in the Z-table or use statistical software to find the Z-score that corresponds to this probability. Let's say that value is Z_75th.
5. Finally, convert the Z-score back to the original adhesion distribution using the formula: X = Z_75th * σ + µ. Substitute the value of Z_75th you found in step 4 and the values of µ and σ from the problem to calculate the 75th percentile of the adhesion distribution.
Remember, when using the Z-table or software, you may need to interpolate the values to get the exact percentile since most Z-tables provide values only for specific Z-scores.