Answer:
Step-by-step explanation:
Hello!
Using the information of the triathlon times of men and women you have to find:
a. The cutoff time for the fastest 5% men, if the times of the male triathletes have a normal distribution, this value will separate the top 5% from the rest of the distribution, symbolically:
P(X≥x₀)= 0.05
Where x₀ represents the cutoff time that you need to find.
Now if this value has 5% of the distribution above it, then 95% of the distribution will be below it, symbolically:
P(X<x₀)= 0.95
Next is to use the standard normal distribution, you have to find the Z value that accumulates 0.95 of probability: P(Z<z₀)= 0.95
z₀= 1.648
Using the formula: Z= (X-μ)/δ what is left to do is to reverse the standardization and reach the value of the variable:
z₀= (x₀-μ)/δ
x₀= (z₀*δ)+μ
x₀= (1.648*583)+4313
x₀= 5273.784 seconds
The cutoff time for the fastest 5% is 5273.784 seconds.
b. The cutoff time for the slowest 10% women, if you are taking the "slowest" into account, then this is the lowes 10% of the distribution. Meaning that the value of time separates the "lowest" 10% of the distribution from the rest, symbolically:
P(X≤x₀)=0.10
Same as before, you have to use the standard normal distribution:
P(Z≤z₀)= 0.10
z₀= - 1.283
Now using this value you have to reverse the standardization:
z₀= (x₀-μ)/δ
x₀= (z₀*δ)+μ
x₀= (-1.283*807)+5261
x₀= 4225.619 seconds
The cutoff time for the slowest 10% is 4225.619 seconds.
I hope it helps!
