Find the z-score corresponding to the given area. Remember, z is distributed as the standard normal distribution with mean µ = 0 and standard deviation σ = 1. (12 pts.)a. The area to the left of z is 15%.b. The area to the right of z is 65%.c. The area to the left of z is 10%.d. The area to the right of z is 5%e. The area between –z and z is 95%. (Hint: draw a picture and figure out the area to the left of the –z)f. The area between –z and z is 99%

a)

b)

c)

d)  

e)  

f)  

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let Z the random variable that represent the variable of a population, and for this case we know the distribution for Z is given by:

Where and

We want 15% of the area in the left so we can use z table or the following code in excel in order to find the value required:

"=NORM.INV(0.15,0,1)"

And we got

Part b

We want 65% of the area in the right and that implies 100-65=35% of the area in the left so we can use z table or the following code in excel in order to find the value required:

"=NORM.INV(0.35,0,1)"

And we got

Part c

We want 10% of the area in the left so we can use z table or the following code in excel in order to find the value required:

"=NORM.INV(0.10,0,1)"

And we got

Part d

We want 5% of the area in the right and that implies 100-5=95% of the area in the left so we can use z table or the following code in excel in order to find the value required:

"=NORM.INV(0.95,0,1)"

And we got

Part e

For this case we waant two values that accumulates 95% of the area in the middle. That means we need to have 100-95 % = 5% of the area in the tails and since the distribution is symmetric we need to have 5/2 = 2.5% of the area on each tail. So we can use the following codes to find the two values:

"=NORM.INV(0.025,0,1)"

"=NORM.INV(1-0.025,0,1)"

Part f

For this case we waant two values that accumulates 99% of the area in the middle. That means we need to have 100-99 % = 1% of the area in the tails and since the distribution is symmetric we need to have 1/2 = 0.5% of the area on each tail. So we can use the following codes to find the two values:

"=NORM.INV(0.005,0,1)"

"=NORM.INV(1-0.005,0,1)"