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Attention to Detail in Proportions Analysis

Understand the critical values, P-values, and significance levels in proportions analysis. Learn when to reject the null hypothesis and detect statistical significance with the right proportions. Master the essentials of testing proportions with precision.

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Attention to Detail in Proportions Analysis

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  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 -1.96 or greater than 1.96. (z is in the outer 5% of the standard normal distribution).

  21. Critical Values Reject Ho when z is less than -1.96 or greater than 1.96. (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 -1.96 or greater than 1.96. (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 = 1.0 - 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) normalcdf(-1EE99, -0.95) 0.171

  27. P-Value 2nd DISTR 2: normalcdf 2[normalcdf(-1EE99, z-score)] 2[normalcdf(-1EE99, -0.95)] 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 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* = 1.96 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 4 components:

  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

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