1. The Standard Normal Distribution
Lecture 20
Section 6.3.1
Wed, Oct 13, 2004

2. The Standard Normal Distribution
The standard normal distribution – The normal distribution with mean 0 and standard deviation 1.
It is denoted by the letter Z.
Therefore, Z is N(0, 1).

3. The Standard Normal Distribution1 2 3 -1 -2 -3
N(0, 1)

4. Areas Under the Standard Normal Curve
What proportion of values of Z will fall below 0?
What proportion of values of Z will fall below +1?
What proportion of values of Z will fall above +1?
What proportion of values of Z will fall below –1?

5. Areas Under the Standard Normal Curve
How do we find the area under the curve to the left of +1? -3 -2 -1 1 2 3

6. Areas Under the Standard Normal Curve
This is too hard to calculate by hand.
We will use two methods.
Standard normal table.
The TI-83 function normalcdf.

7. The Standard Normal Table
See pages 372 – 373 or pages 942 – 943.
The entries in the table are the areas to the left of the z-value.
To find the area to the left of +1, locate 1.00 in the table and read the entry.

8. The Standard Normal Tablez
.00
.01
.02 … :
0.9
0.8159
0.8186
0.8212
1.0
0.8413
0.8438
0.8461
1.1
0.8643
0.8665
0.8686

9. The Standard Normal Table
The area to the left of 1.00 is
That means that 84.13% of that population is below 1.00.
0.8413 -3 -2 -1 1 2 3

10. Standard Normal Areas What is the area to the right of +1?
What is the area to the left of –1?
What is the area to the right of –1?
What is the area between –1 and +1?

11. Let’s Do It! Let’s Do It! 6.1, p. 332 – More Standard Normal Areas.
Use the standard normal table.

12. TI-83 – Standard Normal Areas
Press 2nd DISTR.
Select normalcdf (Item #2).
Enter the lower and upper bounds of the interval.
If the interval is infinite to the left, enter -99 as the lower bound.
If the interval is infinite to the right, enter 99 as the upper bound.

13. TI-83 – Standard Normal Areas
Press ENTER.
Examples:
normalcdf(-99, 1) =
normalcdf(1, 99) =
normalcdf(-99, -1) =
normalcdf(-1, 99) =
normalcdf(-1, 1) =

14. Let’s Do It Again!
Let’s Do It! 6.1, p. 332 – More Standard Normal Areas.
Use the TI-83.

15. The “ Rule”
The Rule: For any normal distribution N(, ),
68% of the values lie within  of .
95% of the values lie within 2 of .
99.7% of the values lie within 3 of .

16. The “68-95-99.7 Rule” Equivalently,
68% of the values lie in the interval
[ – ,  + ], or   .
95% of the values lie in the interval
[ – 2,  + 2], or   2.
99.7% of the values lie in the interval
[ – 3,  + 3], or   3.

17. The Empirical Rule
The well-known Empirical Rule is similar, but more general.
If X has a “mound-shaped” distribution, then
Approximately 68% lie within  of .
Approximately 95% lie within 2 of .
Approximately 99.7% lie within 3 of .

18. Let’s Do It! Let’s Do It! 6.4, p. 335 – Pine Needles.
Let’s Do It! 6.5, p. 335 – Last Longer?