2. Practice For an SAT test  = 500  = 100
What is the probability that a single person will have a SAT score of 550 or higher?

3. Step 1: Sketch out question

4. Step 2: Calculate the Z score
( ) / 100 = .50

5. Step 3: Look up Z score in Table
Z = .50; Column C =.3085
.3085

6. Practice
There is a probability that any single person will have an SAT score of 550 or higher.

7. Practice For an SAT test  = 500  = 100
What is the probability that a group of 10 people have a mean SAT score of 550 or higher?

8. Step 1: Sketch out question

9. Step 2: Calculate the standard error
100 / √10 = 31.62

10. Step 3: Calculate the Z score
( ) / = 1.58

11. Step 4: Look up Z score in Table
Z = 1.58; Column C =.0571
.0571

12. The next step
Based on a sample of participants how accurate is a given sample mean at estimating the population mean?
Example:
Opinion polls

13. Confidence Intervals
Using a sample of US citizens you ask if they approve of the president.
You find 43% approve
“The findings of the survey are accurate within 3 points, using a 95% level of confidence”

14. Confidence Intervals
Using a sample of US citizens you ask if they approve of the president.
You find 43% approve
Thus you are 95% confident the population values falls between 40% to 46%

15. Confidence Intervals
A recent survey found that on average a person each 10 pieces of fruit a week.
The findings of the survey are accurate within 5 pieces of fruit, using a 95% level of confidence
What does this mean?

16. Confidence Intervals
A recent survey found that on average a person each 10 pieces of fruit a week.
The findings of the survey are accurate within 5 pieces of fruit, using a 95% level of confidence
95% confident that on average a person eats between 5 and 15 pieces of fruit a week.

17. Note You can not use the Z formula to answer this type of question
You usually don’t know the  or the 
If you did you wouldn’t need a sample!
Better to use a “t distribution”

18. t distribution Different sample sizes have different t distributions
When using t distributions you must know the degrees of freedom (df)
df = N - 1
Thus, the df can range from 1 to 

19. t distribution

20. t distribution
There are an infinite number of t distributions; however, once df is greater than 120 it doesn’t change much
Table D page 392 (new book page 402)
First Column are df
For this chapter you will look at the top row

21. t distribution
95%
2.5%
2.5%
-1.96
1.96
df = 

22. t distribution 95% 2.5% 2.5% -1.96 1.96 df = 
NOTICE: Similarities to the z table

23. t distribution
95%
2.5%
2.5%
-2.26
2.26
df = 9

24. t distribution
95%
2.5%
2.5%
-2.26
2.26
df = 9

25. t distribution
95%
2.5%
2.5%
-4.30
4.30
df = 2

26. t distribution
95%
2.5%
2.5%
-4.30
4.30
df = 2

27. Confidence Intervals Around a Mean
Inferential statistic
Based on a sample you calculate a mean
With a certain degree of confidence (usually 95%) you can state that the population mean is in the interval

28. Confidence Intervals To find the lower and upper limits:
LL = X - t (sx)
UL = X + t(sx)
Where: X is the mean of the sample
t is a value from the t distribution table
sx is the standard error of the mean, calculated from a sample

29. Confidence Intervals Note: Similar to the backwards z LL = X - t (sx)
UL = X + t(sx)
X =  + (z)()

30. Confidence Intervals To find the lower and upper limits:
LL = X - t (sx)
UL = X + t(sx)
Where: X is the mean of the sample
t is a value from the t distribution table
sx is the standard error of the mean, calculated from a sample

31. Confidence Intervals
Sx = S / N

32. Remember
S = -1

33. Example
You are curious about how many hours a week students study. You sample 5 students:
20, 22, 19, 23, 17
What is the mean score? What are its 95% confidence intervals?

34. Step 1: Calculate S

35. Step 1: Calculate S
NOTE: In other examples you may be given X and X2. Make sure you understand where they came from!

36. 101
2.39 =
2063 5
5 - 1
N = 5
X = 101
 X2 = 2063

37. Step 2: Calculate Sx
Sx = S / N
1.07 = 2.39 / 5

38. Step 3: Look up t value df = N - 1 4 = 5 - 1 Want 95% confidence

39. Step 4: Calculate LL and UL
LL = X - t (sx)
UL = X + t(sx)
X = 20.2
Sx = 1.07
t = 2.776

40. Step 4: Calculate LL and UL
X = 20.2
Sx = 1.07
t = 2.776

41. Example
Thus, you are 95% confident that the average student studies between and hours a week

42. Example
Using the same data -- what if you want to be 99.9% confident.

43. Step 3: Look up t value df = N - 1 4 = 5 - 1 Want 99.9% confidence

44. Step 4: Calculate LL and UL
LL = X - t (sx)
UL = X + t(sx)
X = 20.2
Sx = 1.07
t = 8.610

45. Step 4: Calculate LL and UL
X = 20.2
Sx = 1.07
t = 2.776

46. Example
Thus, you are 95% confident that the average student studies between and hours a week
You are 99.9% confident that the average student studies between and hours a week.
As you get more confident your intervals will get larger

47. Practice
Old book
Page 168
8.24
New book
Page 172
8.23

48. Practice M = 1.20 S hat = .10 SE = .0577 t(2) = 31.598 LL = -.62
UL = 3.02
You can be 99.9% confident that the average burn rate of pots in this box is below 4 inches per minute.