The times for completing one circuit of a bicycle course are normally distributed with a mean of 81.7 minutes and a standard deviation of 8.6 minutes. An association wants to sponsor a race and provide prizes for the top (that is, fastest) 15% of riders. Where should they expect to set the cutoff time to earn a prize?
http://davidmlane.com/hyperstat/z_table.html
To determine the cutoff time to earn a prize for the top 15% of riders, we can use the standard normal distribution.
Here's how you can calculate it:
Step 1: Calculate the Z-score corresponding to the desired percentile (15%).
Z-score = InvNorm(Percentile)
To calculate the Z-score, we can use the inverse normal distribution function, typically denoted as InvNorm or ZInv. If you don't have access to that function, you can refer to a Z-score table or use an online calculator.
In this case, the desired percentile is 15%, so the Z-score is:
Z-score = InvNorm(0.15)
Step 2: Convert the Z-score into the actual cutoff time using the formula:
Cutoff time = Mean + (Z-score * Standard Deviation)
Given the mean (μ) of 81.7 minutes and the standard deviation (σ) of 8.6 minutes, we can now substitute the values into the formula:
Cutoff time = 81.7 + (Z-score * 8.6)
For example, let's say the Z-score is -1.04 (hypothetical value). The cutoff time can be calculated as follows:
Cutoff time = 81.7 + (-1.04 * 8.6)
After performing the calculation, you will have the cutoff time that the association should set to earn a prize for the top 15% of riders.