1. Welcome to .
Week 07 Thurs .
MAT135 Statistics

2. Probability
We’ve studied probability for things with outcomes like: H T Win Lose

3. Probability
These were all “discrete” outcomes

4. Probability
Now we will look at probabilities for things with an infinite number of outcomes that can be considered continuous

5. Probability
You can think of smooth quantitative data graphs as a series of skinnier and skinnier bars

6. Probability
When the width of the bars reach “zero” the graph is perfectly smooth

7. Probability
SO, a smooth quantitative (continuous) graph can be thought of as a bar chart where the bars have width zero

8. Probability
The probability for a continuous graph is the area of its bar: height x width

9. Probability
But… the width of the bars on a continuous graph are zero, so P = Bar Area = height x zero All the probabilities are P = 0 !

10. Probability
Yep. It’s true. The probability of any specific value on a continuous graph is: ZERO

11. Probability
So… Instead of a specific value, for continuous graphs we find the probability of a range of values – an area under the curve

12. Probability
Because this would require yucky calculus to find the probabilities, commonly-used continuous graphs are included in Excel Yay!

13. Questions?

14. Normal Probability
The most popular continuous graph in statistics is the NORMAL DISTRIBUTION

15. Normal Probability
Two descriptive statistics completely define the shape of a normal distribution: Mean µ Standard deviation σ

18. ? Suppose we have a normal distribution, µ = 10 Normal Probability
PROJECT QUESTION 1
Suppose we have a normal distribution, µ = 10 ?

19. ? ? ? 10 ? ? ? Suppose we have a normal distribution, µ = 10 σ = 5
Normal Probability
PROJECT QUESTION 1
Suppose we have a normal distribution, µ = 10 σ = 5
? ? ? ? ? ?

20. -5 0 5 10 15 20 25 Suppose we have a normal distribution, µ = 10 σ = 5
Normal Probability
PROJECT QUESTION 1
Suppose we have a normal distribution, µ = 10 σ = 5

21. Normal Probability
The standard normal distribution has a mean µ = 0 and a standard deviation σ = 1

22. ? ? ? ? ? ? ? For the standard normal distribution, µ = 0 σ = 1
Normal Probability
PROJECT QUESTION 2
For the standard normal distribution, µ = 0 σ = 1
? ? ? ? ? ? ?

23. -3 -2 -1 0 1 2 3 For the standard normal distribution, µ = 0 σ = 1
Normal Probability
PROJECT QUESTION 2
For the standard normal distribution, µ = 0 σ = 1

24. The standard normal is also called “z”
Normal Probability
The standard normal is also called “z”

25. Normal Probability
We can change any normally-distributed variable into a standard normal One with: mean = 0 standard deviation = 1

26. Normal Probability
To calculate a “z-score”: Take your value x Subtract the mean µ Divide by the standard deviation σ

27. Normal Probability
z = (x - µ)/σ

28. Normal Probability
IN-CLASS PROBLEMS 3,4,5
Suppose we have a normal distribution, µ = 10 σ = 2 z = (x - µ)/σ = (x-10)/2 Calculate the z values for x = 9, 10, 15

29. z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2
Normal Probability
IN-CLASS PROBLEMS 3,4,5
z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2

30. Normal Probability
IN-CLASS PROBLEMS 3,4,5
z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2 10 z = (10-10)/2 = 0

31. Normal Probability
IN-CLASS PROBLEMS 3,4,5
z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2 10 z = (10-10)/2 = 0 15 z = (15-10)/2 = 5/2

32. Normal Probability
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33. But… What about the probabilities??
Normal Probability
But… What about the probabilities??

34. Normal Probability
We use the properties of the normal distribution to calculate the probabilities

35. Normal Probability
IN-CLASS PROBLEMS 6,7,8
What is the probability of getting a z-score value between -1 and 1 -2 and 2 -3 and 3

36. Normal Probability
You can use sneaky logic to calculate other probabilities

37. Normal Probability
IN-CLASS PROBLEM 9
If the area under the entire curve is 100%, how much of the graph lies above “x”?

38. Normal Probability
P(x ≥ b) = 1 - P(x ≤ b) or = 100% - P(x ≤ b) or = 100% - 90% = 10%

39. What about probabilities between two “x” values?
Normal Probability
What about probabilities between two “x” values?

40. Normal Probability
P(a ≤ x ≤ b) equals P(x ≤ b) – P(x ≤ a) minus

41. Normal Probability
IN-CLASS PROBLEMS 10,11,12
What is the probability of getting a z-score value between -1 and 0 -2 and 1 -3 and -2

42. Normal Probability
You can use your calculators to calculate other normal probabilities!

43. Normal Probability
IN-CLASS PROBLEM 13
Find the probability that a z-score is between -1.5 and 2 P(-1.5 ≤ z ≤ 2)

44. Press 2nd VARS [DISTR] Scroll down to 2:normalcdf( Press ENTER
Normal Probability
IN-CLASS PROBLEM 13
Press 2nd VARS [DISTR] Scroll down to 2:normalcdf( Press ENTER

45. It will say: normalcdf( Type: -1.5,2) Press ENTER Answer: 0.91044
Normal Probability
IN-CLASS PROBLEM 13
It will say: normalcdf( Type: -1.5,2) Press ENTER Answer:

46. Normal Probability
IN-CLASS PROBLEM 14
What is the probability of getting a z-score value between –1 and 2.5 ?

47. Questions?

48. You survived! Turn in your homework! Don’t forget your homework
due next week!
Have a great
rest of the week!