1. Normal Distributions

2. OBJECTIVE
Find the area and probability of a standard normal distribution.

3. RELEVANCE
Can find probabilities and values of populations whose data can be represented with a normal distribution.

4. Definition……
Normal Distribution Curve – a symmetric distribution where the data values are evenly distributed about the mean.
2 other names for the normal distribution:
a. Guassian
b. Bell Curve

5. Skewness……
Three Possible Distributions where the tail indicates skewness:
Normal
Left (Negative)
Right (Positive)

6. Normal…… Normal – mean, median, and mode are all the same.
Mean, median, mode

7. Left Skew (Negative)……
Negative Skew – from left to right: mean, median, mode.
If you were looking at a coordinate graph, the tail is pointing toward the negative
mean
median
mode

8. Right Skew (Positive)……
If you are looking at a coordinate graph, the tail points to the positive. TAIL DETERMINES END BEHAVIOR!
mean
mode
median
Positive Skew- from right to left: mean, median, mode

9. Standard Normal Distributions…..
Any Questions about Skewness? Can anyone think of a better way to remember the different skews? Ok Now in this class we are going to focus on Normal Distributions. You still have to be able to identify the 3 types of Skewness, but we will just be working with normal.

10. Properties of a Standard Normal Distribution……
Total area under the curve = 1 (or 100%).
Mounded and symmetric; Never touches the x-axis.
Mean = 0; St. Deviation = 1
The mean divides the area in half → 0.50 on each side.
Percentages under the curve:

11. Does this look familiar?
68%
95%
Put picture of normal curve with
99.7%
THE EMPIRICAL RULE!

12. Z-score
All values can be transformed from a normal distribution to a standard normal distribution by using the z-score.
It represents how many standard deviations “x” is away from the mean.

13. Finding area under the curve……
Draw the distribution curve.
Shade the area in which you are interested.
Use the table to find the areas. You might have to add or subtract or both, depending on your interest.
****The table we are using gives areas from z = 0 to the other z-score.

14. Example……Find the area under the curve between z = 0 and z = 1.52.
Look up z = 1.52
The area between z = 0 and z = 1.52 is

15. You try…Find the area between z = 0 and z = 2.34.
Answer:
.4904

16. Find the area between z = 0 and z = -1.75.
Answer:
.4599

17. Find the area to the right of z = 1.11
Answer:
Half is 0.5.
Look up z = 1.11 to get an area of
Final Answer:
0.5 – =

18. Now you try…… Find the area to the left of z = -1.93.
Answer: 0.5 – =
Compare results with a neighbor, I walk around to keep on task.

19. Find the area between z = 2.00 and z = 2.47.
Look up z = 2.47 to get
Look up z = 2 to get
Subtract the two areas:
– =
How can we do this? What should I do first? (Draw & Shade)

20. You try…… Find the area between z = -2.48 and z = -0.83.
Answer: – =

21. Find the area between z = -1.37 and z = 1.68.
The area for z = 1.68 is
The area for z = =1.37 is
Add the two areas together for the final answer: =

22. Find the area to the left of z = 1.99.
The area for z = 1.99 is
The area for the entire left half of the distribution is
Add the two areas together for the final answer: =

23. Find the area to the right of z = -1.16.
The area for z = is
The area for the entire right half of the distribution is
Add the two areas together to get your final answer: =

24. Find the area to the right of z = 2.43 and to the left of z = -3.01.
The area for z = 2.43 is
The area for z = is
Add the two areas together and then subtract from 1: =
1 – =

25. Probability and the Normal Curve

26. Finding Probability Under the Curve……
Use the same procedure as finding the area under the curve; however, there is a different notation.
Today we are going to talk about finding probability under the curve. Yesterday, we found the area under the curve, today we are finding the probability. IT’S THE SAME THING!

27. Notations…… Area Notation: a. Between z=0 and z=2.32
b. To the left of z=1.65
c. To the right of z=1.91
d. Between z = and z=2.3
Probability Notation: a. b. c. d.

28. Find
The area for z = 2.32 is
The probability is

29. Find The probability for z = 1.65 is 0.4505.
The probability for the entire left half of the distribution is
The final probability is =

30. Find The probability for z = 1.91 is 0.4719.
The probability for the entire right half of the distribution is
Subtract the two answers to get your final answer: – =

31. Find The probability for z = -1.2 is 0.3849.
Add the two probabilities together to get your final answer: =