Question

Complete the following statements by typing your answers in the spaces provided.  If you are asked to determine the probabilities, state your answers to 4 decimal places.  If you are asked to determine a value of Y, state your answers to 2 or 3 decimal places.  Because you will be typing in your answers,  I cannot ask you to draw diagrams as I have done in my previously posted practice finals and solutions to examples, I suggest when you are working on your answers on scrap pieces of paper (or a print out of the exam sheets), draw the diagrams.  You are more likely to get the right answer if you use diagrams.  For each of these statements I am asking you fill in more than one blank so that I can give you part marks if your intermediate calculation is correct but your final calculation is wrong.   

When answering each of the following, the random variable Y is normally distributed with a mean of 65 and a standard deviation of 4.

1. The value of z for being able to determine P(Y ≤ 60) is _____

The P(Y ≤ 60) is _____

2. The 2 values of z for being able to determine P( 70 ≤ Y ≤ 73) are _____ and _____

The P( 70 ≤ Y ≤ 73) is _____

3. The value of z for being able to determine P(Y ≥ 72) is _____

The P(Y ≥ 72) is _____

4. The closest value of z for being able to determine the value of y such that 10% of the values Y are less

than that y is ______

                  The approximate value of y such that 10% of the values Y are less than that y is ______

Answer 1

Y~Normal (Mean=65, SD=4)

Z is the standard normal distribution ~Normal (Mean =0, SD=1)

Z= (Y-Mean(Y))/SD(Y)

a. So for the value of z we have,

P(Y≤ 60) = P( Z ≤ (60-65)/4) =P (Z ≤ -5/4) =P(Z≤ -1.25)

The value of z is -1.25.

Now, P(Y≤ 60) =P( Z ≤ -1.25) = 1- P(Z≤1.25)=1-0.8944 (From Normal table) =0.1056

b.So for the 2 values of z we have,

P(70≤Y≤73)= P(Y≤73) - P(Y≥70) = P ( Z ≤ (73-65)/4) - P(Z ≥ (70-65)/4) = P(Z ≤ 2) - P( Z ≥ 1.25)

The values of z are 2 and 1.25

Now, P(70≤Y≤73)= P(Z ≤ 2) - P(Z ≥ 1.25)= P(Z ≤ 2) - P(Z≤1.25) (By symmetry)=P(Z ≤ 2) - P(Z≤1.25) -1 =0.9772-0.8944(From normal table) =0.0828

c.So for the value of z we have,

P(Y≥ 72) = P( Z ≤ (72-65)/4) =P (Z ≥ 1.75)

The value of z is 1.75

Now, P(Y≥ 72) = P (Z ≥ 1.75) = 1- P(Z≤1.75)=1-0.9599(From Normal table) =0.0401

d. Now, we need to find the value of z for the value of y such that 10% of the values of Y are less than that of y

For finding this we need to find z from P(Y≤y)=0.10

i.e, P (Z≤ (y-65)/4) =0.10 ............(1)

i.e, P(Z≤z)=0.10.

From the normal table we need to find z for P(Z≤z)=0.10.

We find the corresponding z value is -1.28.

The closest value of z for being able to determine the value of y such that 10% of the values Y are less than that y is -1.28

Now ,

z= -1.28

=> (y-65)/4 = -1.28 [From 1]

=> y-65 = -5.12

=> y =65-5.12 =59.88

The approximate value of y such that 10% of the values Y are less than that y is 59.88