Sampling distribution of sample mean

[Shackman]

Homework Statement

Let X1,X2,...X10 be independent normal random variables with mean 10 and std dev 4. Let Y = X1+X2+...+X10. Let X be Y/10. What is P(X < 8)

Homework Equations

E(X) = 10 and Var(X) = 4² / 10 = 1.6
std dev of X = 1.6^.5 = 1.26

The Attempt at a Solution

To convert to standard normal distribution to use z table to solve I get..

P(X-10/1.26 < (8-10)/1.26) = P(Z < -1.58) = 0

I really doubt that I have done this problem correctly because it is on a sample exam and seems too easy. Have I done anything incorrectly or am I just second guessing myself needlessly?

 

[Billy Bob]

Have I done anything incorrectly

Just the typo in the last line where you say the probability is 0.

am I just second guessing myself needlessly?

Yes!

 

[Shackman]

Ah. A silly mental error as I was using an applet to find the value for z that matched the probability but with a curve that had the mean and standard deviation of the original variable. Thanks for helping me again Billy Bob!