1. The Standard Normal Distribution
Lecture 25
Section 6.3.1
Mon, Mar 1, 2004

2. The Standard Normal Distribution
The normal distribution with mean 0 and standard deviation 1 is called the standard normal distribution.
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
We need to 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
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 is
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. The TI-83 and 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. Let’s Do It Again!
Let’s do it! 6.1, p. 332 – More Standard Normal Areas.
Use the TI-83.

14. Standard Normal Areas on the TI-83
Press ENTER.
Examples:
normalcdf(-99, 1) =
normalcdf(1, 99) =
normalcdf(-99, -1) =
normalcdf(-1, 99) =
normalcdf(-1, 1) =

15. Assignment
Page 341: 1, 2, 7a, 7b, 7c, 7d, 7e, 7f, 7g.

16. Areas Under Other Normal Curves
Let X be N(, ).
Then (X – )/ is N(0, 1).
That is, Z = (X – )/.
The value of Z is called the z-score or standard score of X.

17. Example
Let X be N(30, 5).
What proportion of values of X are below 38?
Compute z = (38 – 30)/5 = 8/5 = 1.6.
Find the area to the left of 1.6 under the standard normal curve.
Answer:

18. Let’s Do It!
Let’s do it! 6.2, p. 333 – IQ Scores.

19. 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 .

20. 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.

21. 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 .

22. Let’s Do It!
Let’s do it! 6.5, p. 335 – Last Longer?

23. Assignment
Page 341: Exercises 3, 4, 5, 9, 11, 14, 24, 25, 29, 31.