1. Statistics Normal Probability

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

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

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

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

6. Normal 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 !

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

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

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

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

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

14. Suppose we have a normal distribution, µ = 12 σ = 2
Normal Probability
Suppose we have a normal distribution, µ = 12 σ = 2

15. Normal Probability
If µ = 12 12

16. Normal Probability
If µ = 12 σ = 2

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

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

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

20. Questions?

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
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
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
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
z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2

30. Normal Probability
IN-CLASS PROBLEMS
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
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
What is the probability of getting a z-score value between -1 and 1 -2 and 2 -3 and 3

36. You also use symmetry to calculate probabilities
Normal Probability
You also use symmetry to calculate probabilities

37. Normal Probability
IN-CLASS PROBLEMS
What is the probability of getting a z-score value between -1 & 0 -2 & 0 -3 & 0
Percentage
of the curve
z-score

38. Normal Probability
IN-CLASS PROBLEMS
What is the probability of getting a z-score value between -1 & & & -1.5
Percentage
of the curve
z-score

39. Normal Probability
Note: these are not exact – more accurate values can be found using Excel
Percentage
of the curve
z-score

40. Questions?

41. Excel Probability
Normal probability values in Excel are given as cumulative values (the probability of getting an x or z value less than or equal to the value you want)

42. Excel Probability
Use the function: NORM.DIST

43. Excel Probability

44. Excel Probability
For a z prob, use: NORM.S.DIST

45. Excel Probability
Excel gives you the cumulative probability – the probability of getting a value UP to your x value: P (x ≤ your value)

46. Excel Probability
How do you calculate other probabilities using the cumulative probability Excel gives you?

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

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

49. What about probabilities between two “x” values?
Excel Probability
What about probabilities between two “x” values?

50. Excel Probability
P(a ≤ x ≤ b) equals P(x ≤ b) – P(x ≤ a) minus

51. Normal Probability
IN-CLASS PROBLEMS
Calculate the probabilities when µ = 0 σ = 1: P(-1 ≤ x ≤ 2) P(1 ≤ x ≤ 3) P(-2 ≤ x ≤ 1)

52. P(-1 ≤ x ≤ 2) = 34%+34%+13.5% = 81.5% Normal Probability
IN-CLASS PROBLEMS
P(-1 ≤ x ≤ 2) = 34%+34%+13.5% = 81.5%

53. P(-1 ≤ x ≤ 2) = 81.5% P(1 ≤ x ≤ 3) = 13.5%+2% = 15.5%
Normal Probability
IN-CLASS PROBLEMS
P(-1 ≤ x ≤ 2) = 81.5% P(1 ≤ x ≤ 3) = 13.5%+2% = 15.5%

54. Normal Probability
IN-CLASS PROBLEMS
P(-1 ≤ x ≤ 2) = 81.5% P(1 ≤ x ≤ 3) = 15.5% P(-3 ≤ x ≤ 1) = = 2%+13.5%+34%+34% = 83.5%

55. Questions?