1. Section 8.2
Testing a Proportion

2. Significance Tests
Goal: test significance of difference between sample and standard

3. Significance Tests
Used to make decisions from data by comparing results from a sample to some predetermined standard

4. What does variation in sampling mean?

5. Variation in sampling: you do not get exactly the same result each time you take a random sample from a population

6. Variation in sampling: you do not get exactly the same result each time you take a random sample from a population
Example: If you draw five cards from a standard deck of cards, you will not get the same five cards each time you repeat this sampling.

7. Statistically Significant
When difference between sample and predetermined standard is large enough that difference can not be reasonably attributed to chance,

8. Statistically Significant
When difference between sample and predetermined standard is large enough that difference can not be reasonably attributed to chance, you can conclude standard no longer holds.

9. Statistically Significant
Thus, large difference between sample data and standard is statistically significant.

10. Statistically Significant
Thus, large difference between sample data and standard is statistically significant.
Note: Just because there is a difference does not mean it is a statistically significant difference.

11. Statistically Significant
Sample proportion is statistically significant if it is not a reasonably likely outcome when the proposed standard is true.

12. Statistically Significant
Sample proportion is statistically significant if it is not a reasonably likely outcome when the proposed standard is true.
In other words, sample proportion is statistically significant if it is a rare outcome when the proposed standard is true.

13. Statistically Significant
The statement the proposed standard is true is called the null hypothesis, Ho

14. Page 490
Turn to page Read through the example on the barn swallows at Chernobyl.
Determine what the reasonably likely interval would be if overall percentage was still 2%.

15. 2%(266) = 5.32
1-PropZInt
x: 6
n: 266
C-level: .95

16. (.005, .04)
So, 16% would be a ______ _____.

17. (.005, .04)
So, 16% would be a rare event.
Therefore, there was an increased probability of genetic mutations in the Chernobyl area.

18. Informal Significance Testing
Decision to reject a standard is same as constructing a confidence interval

19. Informal Significance Testing
If your sample falls within the reasonably likely interval, do not reject the standard.

20. Informal Significance Testing
If your sample falls within the reasonably likely interval, do not reject the standard.
If your sample does not fall within the reasonably likely interval, reject the standard.

21. Attention to Detail Needed
Three different proportions to keep straight

22. Attention to Detail Needed
Three different proportions to keep straight p:

23. Attention to Detail Needed
Three different proportions to keep straight
p: population proportion

24. Attention to Detail Needed
Three different proportions to keep straight
p: population proportion :

25. Attention to Detail Needed
Three different proportions to keep straight
p: population proportion
: sample proportion

26. Attention to Detail Needed
Three different proportions to keep straight
p: population proportion
: sample proportion
po:

27. Attention to Detail Needed
Three different proportions to keep straight
p: population proportion
: sample proportion
po: hypothesized value of the population proportion

28. Test Statistic
To determine if p is reasonably likely or a rare event for a given standard po, you need to check the value of the test statistic,

29. Test Statistic
To determine if p is reasonably likely or a rare event for a given standard po, you need to check the value of the test statistic,

30. Test Statistic
To determine if p is reasonably likely or a rare event for a given standard po, you need to check the value of the test statistic,
What does the value of z tell us?

31. Test Statistic
To determine if p is reasonably likely or a rare event for a given standard po, you need to check the value of the test statistic,
This tells you how many standard errors the sample proportion lies from the hypothesized standard, po.

32. Test Statistic

33. Page 494, D23

34. Page 494, D23
A z-score (test statistic) of 0 tells you the sample proportion lies 0 standard errors from the hypothesized standard.
In other words, = po

35. Page 494, D24

36. Page 494, D24
(a) sample size increases?

37. Page 494, D24 (a) sample size increases?
z increases because denominator gets smaller

38. Page 494, D24
(b) gets further from po?

39. Page 494, D24 (b) gets further from po?
z increases because numerator gets larger

40. Page 510, P20

41. Page 510, P20
(a) Decide to test if two-thirds of today’s teens want to study more about medical research.
So, the standard, hypothesized value is
p0 =

42. Page 510, P20
(a) Decide to test if two-thirds of today’s teens want to study more about medical research.
So, the standard, hypothesized value is
p0 =

43. Page 510, P20
(b) Sample proportion is:

44. Page 510, P20
(b) Sample proportion is:

45. Page 510, P20
p = ⅔

46. Page 510, P20 (c) No, this result is not statistically
significant because is a reasonably
likely result when the proportion of
successes in the population is between 65%
and 70% (see the chart).
Conclude that there is no evidence that the
proportion has changed from two-thirds.

47. Questions?

48. Page 490, Activity 8.2a We need for the class:
(a) total number of heads out of 40 total spins
(b) total number of heads out of 40 total flips

49. Page 490, Activity 8.2a
You have 3 minutes to write your answers to questions # 2 and 4.

50. Page 493, D21

51. Page 493, D21
(a) the process of spinning a penny results in heads half the time

52. Page 493, D21
(a) the process of spinning a penny results in heads half the time
(b)

53. Page 493, D21
(a) the process of spinning a penny results in heads half the time
(b)
(c) reasonably likely sample proportions for p = 0.5 are in the interval (0.35, 0.65)

54. Page 493, D22

55. Page 493, D22
(a) the process of flipping a penny results in heads half the time

56. Page 493, D22
(a) the process of flipping a penny results in heads half the time
(b)

57. Page 493, D22
(a) the process of flipping a penny results in heads half the time
(b)
(c) reasonably likely sample proportions for p = 0.5 are in the interval (0.35, 0.65)