1. Section 8.2
Day 2

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

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

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

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

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

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

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

9. Page 510, P23

10. Page 510, P23

11. Page 510, P23

12. Critical Values
Critical value, z*: value against which a test statistic, z, is compared in order to decide whether or not to reject the null hypothesis, Ho
Denoted as z*

13. Critical Values
Critical value, z*: value against which a test statistic, z, is compared in order to decide whether or not to reject the null hypothesis, Ho

14. Critical Values
Critical value, z*: value against which a test statistic, z, is compared in order to decide whether or not to reject the null hypothesis, Ho
Do not reject Ho when z is inside interval (-z*, z*)

15. Critical Values
Critical value, z*: value against which a test statistic, z, is compared in order to decide whether or not to reject the null hypothesis, Ho
Reject Ho when z is outside interval (-z*, z*) z

16. When do you accept the null hypothesis, Ho?

17. When do you accept the null hypothesis, Ho?
NEVER!!

18. Critical Values
Denoted as z*.
For 95% CI, the critical values are?

19. Critical Values Denoted as z*. For 95% CI, critical values are 1.96
When would we reject Ho?

20. Critical Values
Reject Ho when z is less than or greater than (z is in the outer 5% of the standard normal distribution).

21. Critical Values
Reject Ho when z is less than or greater than (z is in the outer 5% of the standard normal distribution).
The corresponding proportion for rejecting Ho, 0.05 in this case, is the level of significance, denoted α

22. Critical Values
Reject Ho when z is less than or greater than (z is in the outer 5% of the standard normal distribution).
The corresponding proportion for rejecting Ho, 0.05 in this case, is the level of significance, denoted α
α = confidence level proportion

23. P-Value
P-value weighs the evidence found from the data.

24. P-Value
The P-value for a test is the probability of seeing a result from a random sample that is as extreme or more extreme than the result you got from your random sample if the null hypothesis, Ho, is true.

25. P-Value

26. P-Value 2nd DISTR 2: normalcdf normalcdf(-1EE99, z-score)
0.171

27. P-Value 2nd DISTR 2: normalcdf 2[normalcdf(-1EE99, z-score)]
0.342

28. P-Value
Very small P-value tells you the sample proportion is quite far away from po
--Not reasonable to assume null hypothesis is true so reject Ho

29. P-Value
Large P-value means null hypothesis is plausible but still might not give exact value for p, population proportion.

30. P-Value
Large P-value means null hypothesis is plausible but still might not give exact value for p, population proportion.
Therefore, do not reject Ho.

31. P-Value Note: “Do not reject” is not the same as “accept”
Large P-value means null hypothesis is plausible but still might not give exact value for p, population proportion.
Therefore, do not reject Ho.
Note: “Do not reject” is not the same as “accept”

32. Using Critical Values and Level of Significance
If the value of the test statistic z is more extreme than the critical values, z*, you have chosen

33. Using Critical Values and Level of Significance
If the value of the test statistic z is more extreme than the critical values, z*, you have chosen (or equivalently, the P-value is less than α, the level of significance),

34. Using Critical Values and Level of Significance
If the value of the test statistic z is more extreme than the critical values, z*, you have chosen (or equivalently, the P-value is less than α, the level of significance), you have evidence against the null hypothesis.

35. Using Critical Values and Level of Significance
If the value of the test statistic z is more extreme than the critical values, z*, you have chosen (or equivalently, the P-value is less than α, the level of significance), you have evidence against the null hypothesis.
Reject the null hypothesis and say that the result is statistically significant.

36. Using Critical Values and Level of Significance
If the value of the test statistic z is less extreme than the critical values, z*, you have chosen (or equivalently, the P-value is greater than α, the level of significance), you do not have evidence against the null hypothesis.
In this case, you do not reject the null hypothesis

37. Using Critical Values and Level of Significance
If a level of significance is not specified, it is usually safe to assume that
z* = and α = 0.05
[This is a 95% confidence level]

38. Tests of Significance

39. Tests of Significance
Since we can reject Ho because of a significant difference in either direction, this represents a two-sided test.

40. One-Sided Tests of Significance
Tests of significance can be one-sided if investigator has an indication of which way any change from the standard should go.
This must be decided before looking at the data.

41. Tests of Significance

42. Components of a Significance Test for a Proportion

43. Components of a Significance Test for a Proportion
1. Give name of the test and check conditions for its use.

44. Components of a Significance Test for a Proportion
1. Give name of the test and check conditions for its use.
Name: Two-sided (or one-sided as appropriate) significance test for a proportion

45. Components of a Significance Test for a Proportion
1. Give name of the test and check conditions for its use.
Name: Two-sided (or one-sided) significance test for a proportion
Conditions: three conditions must be met

46. Components of a Significance Test for a Proportion
1. Give the name of the test and check the conditions for its use.
Sample is a simple random sample from a binomial population

47. Components of a Significance Test for a Proportion
1. Give the name of the test and check the conditions for its use.
Sample is a simple random sample from a binomial population
Both npo and n(1 – po) are at least 10

48. Components of a Significance Test for a Proportion
1. Give the name of the test and check the conditions for its use.
Sample is a simple random sample from a binomial population
Both npo and n(1 – po) are at least 10
Population size at least 10 times sample size

49. Components of a Significance Test for a Proportion
2. State the hypotheses, defining any symbols.

50. Components of a Significance Test for a Proportion
2. State the hypotheses, defining any symbols.
When testing a proportion, generically the null hypothesis, Ho, is:
Ho: the proportion of success, p, in the population from which the sample came is equal to po
Ho: p = po

51. 2. State Hypotheses
Alternative hypothesis, Ha, can be of 3 forms.

52. 2. State Hypotheses
Ha: the proportion of successes, p, in the population from which the sample came is not equal to po
Ha: p po

53. 2. State Hypotheses
Ha: the proportion of successes, p, in the population from which the sample came is greater than po
Ha: p > po

54. 2. State Hypotheses
Ha: the proportion of successes, p, in the population from which the sample came is less than po
Ha: p < po

55. Components of Significance Test
3. Compute the test statistic, z, find the critical values, z*, and the P-value.
Include a sketch that illustrates the situation.

56. Components of Significance Test
3. Compute the test statistic, z, find the critical values, z*, and the P-value.
Include a sketch that illustrates the situation.

57. Components of Significance Test
4. Write a conclusion. Two parts:

58. Components of Significance Test
4. Write a conclusion. Two parts:
Compare value of z to predetermined critical values (z*), or compare P-value to α. Then say whether you reject null hypothesis or do not reject it, linking your reason to the critical values or P-value.

59. Components of Significance Test
4. Write a conclusion. Two parts:
Compare value of z to predetermined critical values (z*), or compare P-value to α. Then say whether you reject null hypothesis or do not reject it, linking your reason to the critical values or P-value.
Tell what your conclusion means in context of the situation.

60. Questions?