Question

1. Find the probabilities for the standard normal random variable z: (1 point)

P(-2.58<z<2.58)

2. x is a normal random variable with mean (μ) of 10 and standard deviation (σ) of 2. Find the following probabilities: (4 points)

1. P(x>13.5)               (1 point)
2. P(x<13.5)               (1 point)
3. P(9.4<x<10.6)     (2 points)

Answer 1

Solution:
We need to calculate P(-2.58<Z<2.58)
which can be calculated as
P(-2.58<Z<2.58) = P(Z<2.58) - P(Z<-2.58)
From Z table we found p-value
P(-2.58<Z<2.58) = P(Z<2.58) - P(Z<-2.58) = 0.9951 - 0.0049 = 0.9902
P(-2.58<Z<2.58) = 0.9902
Solution(b)
μ = 10
σ = 2
We need to calculate probability
P(X>13.5) = ?
We will use the standard distribution table, first, we will calculate Z-score which can be calculated as
Z-score = (X - μ )/σ = (13.5-10)/2 = 1.75
From Z table we found a p-value
P(X>13.5) = 1 - P(X<=13.5) = 1 - 0.9599 = 0.0401
So there is 4.01% probability that X is greater than 13.5
Solution(b)
We need to calculate probability
P(X<13.5) = ?
We will use the standard distribution table, first, we will calculate Z-score which can be calculated as
Z-score = (X - μ )/σ = (13.5-10)/2 = 1.75
From Z table we found a p-value
P(X<13.5) = 0.9599
So there is a 95.99% probability that X is less than 13.5
Solution(c)
We need to calculate the probability
P(9.4<X<10.6) = P(X<10.6) - P(X<9.4)
Z = (9.4-10)/2 = -0.3
Z = (10.6-10)/2 = 0.3
From Z table we found p-value
P(9.4<X<10.6) = P(X<10.6) - P(X<9.4) = 0.6179 - 0.3821 = 0.2358
So there is 23.58% probability that X is between 9.4 to 10.6