Q. 22 - Elementary Statistics

Question

Determine and sketch the area under the standard normal curve that lies

a. to the left of -3.02.

b. to the right of 0.61.

c. between 1.11 and 2.75.

d. between -2.06 and 5.02.

e. between -4.11  and  -1.5.

f. either to the left of I or to the right of 3.

Step-by-Step Solution

Answer

a): Area under the standard normal curve that lies to the left of - 3.02 is 0.0013 

b): The area under the standard normal curve that lies to the right of 0.61 is

1-0.7291  =  0.2709.

c):  Area we see is 0.9970-0.8665= 0.1305 

d):  Area we seek is 1.0000-0.0197 = 0.9803 

e): Area we seek is 0.0668-00000 = 0.0668 

f): Area under the standard normal curve that lies to the right of 3 is

1-0.9987= 0.0013.

Step 1. Part ( A )

 We use the standard normal table of negative z scores because-3.02 is negative. First, we go down to the right-hand column, labeled z, to -3.0. Then, going across that row to the column labeled "0.02", we reach 0.0013

Therefore, the area under the standard normal curve that lies to the left of - 3.02 is 0.0013

Step 2. Part ( b )

Because the total area under the standard normal curve is 1, the area to the right of 0.61 equals 1 minus the area to the left of 0.61.

We use the standard normal table of positive z scores because 0.61 is positive. First, we go down to the left-hand column, labeled z, to 0.6. Then, going across that row to the column labeled "0.01, we reach 0.7291, which the area under the standard normal curve that lies to the left of 0.61.

Thus the area under the standard normal curve that lies to the right of 0.61 is

1-0.7291  =  0.2709.

Step 3. Part ( c )

The area under the standard normal curve that lies between 1.11 and 2.75 equals the area to the left of 2.75 minus the area to the left of 1.11.

Standard normal table shows that these latter two areas are 0.9970 and 0.8665, respectively.

So the area we see is 0.9970-0.8665= 0.1305

Step 4. Part ( d )

The area under the standard normal curve that lies between-2.06 and 5.02 equals the area to the left of 5.02 minus the area to the left of -2.06.

Standard normal table shows that these latter two areas are 1.0000 and 0.0197, respectively.

So the area we seek is 1.0000-0.0197 = 0.9803

Step 5. Part ( e )

The area under the standard normal curve that lies between - 4.11 and -1.5 equals the area to the left of - 1.5 minus the area to the left of - 4.11.

Standard normal table shows that these latter two areas are 0.0668 and 0.0000, respectively

So the area we seek is 0.0668-00000 = 0.0668

Step 6. Part ( f )

First find the area to the left of 1:

We use the standard normal table of positive z scores because 1.00 is positive. First, we go down to the left-hand column, labeled z, to 1.0. Then, going across that row to the column labeled "0.00, we reach 0.8413.

Now, find the area to the right of 3:

Because the total area under the standard normal curve is 1, the area to the right of 3 equals 1 minus the area to the left of 3.

We use the standard normal table of positive z scores because 3 is positive. First, we go down to the left-hand column, labeled z, to 3.0. Then, going across that row to the column labeled "0.00," we reach 0.9987, which is the area under the standard normal curve that lies to the left of 3.

Step 7. Part ( f ) graph

Therefore, the area under the standard normal curve that lies to the left of 1 or to the right of 3 is

0.8413 +0.0013 = 0.8426.