A random variable X is distributed normally with mean 450 and standard deviation 20. Find P(X ≤ 475), and find a which given that P(X > a) = 0.27
Given `mu=450,sigma=20` :
(1) Find `P(x<=475)` .
First convert 475 to a z score: `z=(x-mu)/sigma=(475-450)/20=1.25` .
Then `P(x<=475)=P(z<=1.25)` . Using a graphing calculator or consulting `z` tables we find the percentage of results less than 1.25 is .8943 or 89.43%.
Thus ` P(x<=475)~~89.4% `
(2) Find `a` such that `P(x>a)=.27` Again consulting a graphing calculator or tables, we find that for a `z` value of .6128, 73% of values lie below this `z` (or conversely, 27% of values are greater than this `z` )
So `P(z>.6128)=.27` . Convert this `z` back: `.6128=(x-450)/20 => x=462.256` So `P(x>=462.256)=.27`
Thus the `a` we require is approximately 462.26.