2 Assume the random variable X is normally distributed with

2. Assume the random variable X is normally distributed with a mean = 50 and standard deviation = 4.5.

Find the P (x > 60) _________ (three-decimal accuracy)

Find the P (35 < x < 55)___________

Find x so that the area below x is .12.

3. assume the random variable X is normally distributed with a mean = 80 and standard deviation = 9.4.
For an x-value of 95, the z-score is_________. (two-decimal accuracy)
For a z-score of -2.3, x =___________

4. For a standard normal distribution, find the z-score so that the area below z is .21______

5. The cholesterol levels of men in the United States are normally distributed, with a mean = 215 milligrams per deciliter and a standard deviation = 25 milligrams per deciliter.

If a man is randomly selected, what is the probability that his cholesterol level is more than 245?

What cholesterol level cuts off the lowest 10% of levels?

6. If x follows a distribution not known to be normal, then to assume that the distribution of the sample mean is normal, what must be true about the sample size n?

Solution

Q1.
Mean ( u ) =150
Standard Deviation ( sd )= 15.3/ Sqrt ( 36 ) = 2.55

Q2.
Normal Distribution
Mean ( u ) =50
Standard Deviation ( sd )=4.5
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X > 60) = (60-50)/4.5
= 10/4.5 = 2.2222
= P ( Z >2.222) From Standard Normal Table
= 0.0131                  
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 35) = (35-50)/4.5
= -15/4.5 = -3.3333
= P ( Z <-3.3333) From Standard Normal Table
= 0.00043
P(X < 55) = (55-50)/4.5
= 5/4.5 = 1.1111
= P ( Z <1.1111) From Standard Normal Table
= 0.86674
P(35 < X < 55) = 0.86674-0.00043 = 0.8663                  
c)
P ( Z < x ) = 0.12
Value of z to the cumulative probability of 0.12 from normal table is -1.175
P( x-u/s.d < x - 50/4.5 ) = 0.12
That is, ( x - 50/4.5 ) = -1.17
--> x = -1.17 * 4.5 + 50 = 44.7125