stuck on math Homework

Question

A population of values has a normal distribution with  and .

1. Find the probability that a single randomly selected value is less than 264.1. Round your answer to four decimal places.  
2. Find the probability that a randomly selected sample of size  has a mean less than 264.1. Round your answer to four decimal places.  

Answer 1

P(x < 264.1)

Transform above inequality to standard normal distribution for X = 264.1 using:

z = (X - mean)/std dev = (264.1 - 241.4)/69.7 = 0.325

so P(x < 264.1) = P(z < 0.325)

From standard normal probability table: P(z < 0.325) = 0.6274.

P(xbar < 264.1)

Transform above inequality to standard normal distribution for xbar = 264.1 using:

z = (xbar - mean)/(std dev/sqrt(sample size) = (264.1 - 241.4)/(69.7/sqrt(50)) = 2.3

so P(xbar > 264.1) = P(z > 2.3) = P(z < -2.3) because normal distribution is symmetric

From standard normal probability table: P(z < -2.3) = 0.0107.