What is a two sample z test?
Download as PPTX, PDF2 likes1,315 views
Researchers tested a new anti-anxiety medication on 200 people and a placebo on another 200 people. 64 of those on the medication and 92 of those on the placebo reported anxiety symptoms. The researchers want to determine if there is a statistically significant difference in reported anxiety between the two groups using a two-sample z-test with an alpha of 0.05. A two-sample z-test is used to compare differences between two sample proportions and determines if any observed difference is likely due to chance or not.
![Since the test uses a .05 alpha value, that is what we
will use to determine if the probability of a meaningful
difference is rare or common.
The alpha value makes it possible to determine what is
called the z critical. If the z statistic that we are about
to calculate from the data in the question is outside of
the z critical [for a one-tailed test (e.g., +1.64) or a two-tailed
test (e.g., -1.96 or +1.96]](https://image.slidesharecdn.com/whatisatwo-sampleztest-140913011359-phpapp02/85/What-is-a-two-sample-z-test-12-320.jpg)
What is a two sample z test?
- 1. Two-Sample Z Test Explanation
- 2. A two-sample Z test is used to compare the differences statistically between two sample proportions.
- 3. A two-sample Z test is used to compare the differences statistically between two sample proportions. Consider the following question:
- 4. A two-sample Z test is used to compare the differences statistically between two sample proportions. Consider the following question: Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 5. What are the researchers asking here?
- 6. What are the researchers asking here? They are trying to determine if there is a difference in reported anxiety symptoms between those taking the new anti-anxiety drug and those taking the placebo.
- 7. What are the researchers asking here? They are trying to determine if there is a difference in reported anxiety symptoms between those taking the new anti-anxiety drug and those taking the placebo. What makes this question one that can be answered using a two-sample Z test is the fact that we are examining the difference between proportions (64 out of 200 or .32 compared to 92 out 200 or .46).
- 8. What are the researchers asking here? They are trying to determine if there is a difference in reported anxiety symptoms between those taking the new anti-anxiety drug and those taking the placebo. What makes this question one that can be answered using a two-sample Z test is the fact that we are examining the difference between proportions (64 out of 200 or .32 compared to 92 out 200 or .46). Are these differences similar enough to make any differences we would find if we were to repeat the experiment to be due to chance or not?
- 9. With that in mind, we are able to state the null-hypothesis and then determine if we will reject or fail to reject it:
- 10. With that in mind, we are able to state the null-hypothesis and then determine if we will reject or fail to reject it: There is NO significant proportional difference in reported anxiety symptoms between a sample of participants who took a new anti-anxiety drug and a sample who took a placebo.
- 11. Since the test uses a .05 alpha value, that is what we will use to determine if the probability of a meaningful difference is rare or common.
- 12. Since the test uses a .05 alpha value, that is what we will use to determine if the probability of a meaningful difference is rare or common. The alpha value makes it possible to determine what is called the z critical. If the z statistic that we are about to calculate from the data in the question is outside of the z critical [for a one-tailed test (e.g., +1.64) or a two-tailed test (e.g., -1.96 or +1.96]
- 13. One tailed test visual depiction: If the z value we are about to calculate lands above this point, we will reject the null hypothesis common rare +1.64
- 14. One tailed test visual depiction: common rare +1.64 If the z value lands below this point we will fail to reject the null hypothesis
- 15. A one tailed test could also go the other direction if we are testing the probability of one sample having a smaller proportion than another. rare common -1.64
- 16. A two-tailed test implies that we are not sure as to which direction it will go. We don’t know if the placebo or the new anxiety medicine will have better results.
- 17. Here is a visual depiction of the two-tailed test. rare -1.96 Common rare +1.96
- 18. All we have left to do now is calculate the z statistic.
- 19. All we have left to do now is calculate the z statistic. Here is the formula for the z statistic for a Two-Sample Z-Test: 풁풔풕풂풕풊풔풕풊풄 = (푝 1 − 푝 2) 푝 1 − 푝 1 푛1 + 1 푛2
- 20. In the numerator we are subtracting one sample proportion from another sample proportion: 풁풔풕풂풕풊풔풕풊풄 = (푝 1 − 푝 2) 푝 1 − 푝 1 푛1 + 1 푛2
- 21. In the numerator we are subtracting one sample proportion from another sample proportion: 풁풔풕풂풕풊풔풕풊풄 = (푝 1 − 푝 2) 푝 1 − 푝 1 푛1 + 1 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 22. In the numerator we are subtracting one sample proportion from another sample proportion: 풁풔풕풂풕풊풔풕풊풄 = (푝 1 − 푝 2) 푝 1 − 푝 1 푛1 + 1 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 23. In the numerator we are subtracting one sample proportion from another sample proportion: 풁풔풕풂풕풊풔풕풊풄 = (푝 1 − 푝 2) 푝 1 − 푝 1 푛1 + 1 푛2 92/200=.46 64/200=.32 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 24. In the numerator we are subtracting one sample proportion from another sample proportion: 풁풔풕풂풕풊풔풕풊풄 = (.32 − .46) 푝 1 − 푝 1 푛1 + 1 푛2 92/200=.46 64/200=.32 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 25. In the numerator we are subtracting one sample proportion from another sample proportion: 풁풔풕풂풕풊풔풕풊풄 = (−. ퟏퟐ) 푝 1 − 푝 1 푛1 + 1 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 26. The denominator is the estimated standard error: 풁풔풕풂풕풊풔풕풊풄 = (−.12) 푝 1 − 푝 1 푛1 + 1 푛2 This proportion is called the pooled standard deviation. It is the same value we use with independent sample t-tests. Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 27. The denominator is the estimated standard error: 풁풔풕풂풕풊풔풕풊풄 = (−.12) 푝 1 − 푝 1 푛1 + 1 푛2 It is interesting to note that when you multiply a proportion (.80) by its complement (1-.8 = .20) you get the variance (.80*.20 = .16). If you square that amount it comes to .04 and that is the standard deviation.
- 28. Why is this important? Because the standard deviation divided by the square root of the sample size is the standard error. 풁풔풕풂풕풊풔풕풊풄 = (−.12) 푝 1 − 푝 1 푛1 + 1 푛2 It is interesting to note that when you multiply a proportion (.80) by its complement (1-.8 = .20) you get the variance (.80*.20 = .16). If you square that amount it comes to .04 and that is the standard deviation.
- 29. And the larger the standard error the less likely the two groups will be statistically significantly different. 풁풔풕풂풕풊풔풕풊풄 = (−.12) 푝 1 − 푝 1 푛1 + 1 푛2 It is interesting to note that when you multiply a proportion (.80) by its complement (1-.8 = .20) you get the variance (.80*.20 = .16). If you square that amount it comes to .04 and that is the standard deviation.
- 30. Conversely, the smaller the standard error the more likely the two groups will be statistically significantly different. 풁풔풕풂풕풊풔풕풊풄 = (−.12) 푝 1 − 푝 1 푛1 + 1 푛2 It is interesting to note that when you multiply a proportion (.80) by its complement (1-.8 = .20) you get the variance (.80*.20 = .16). If you square that amount it comes to .04 and that is the standard deviation.
- 31. So, let’s compute the pooled standard deviation: 풑 1 − 풑
- 32. So, let’s compute the pooled standard deviation: 풑 1 − 풑 풑 = 푥1 + 푥2 푛1 + 푛2
- 33. So, let’s compute the pooled standard deviation: 풑 1 − 풑 풑 = 푥1 + 푥2 푛1 + 푛2 Here’s the problem again: Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 34. So, let’s compute the pooled standard deviation: 풑 1 − 풑 풑 = 푥1 + 푥2 푛1 + 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 35. So, let’s compute the pooled standard deviation: 풑 1 − 풑 풑 = 64 + 푥2 푛1 + 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 36. So, let’s compute the pooled standard deviation: 풑 1 − 풑 풑 = 64 + 푥2 푛1 + 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 37. So, let’s compute the pooled standard deviation: 풑 1 − 풑 풑 = 64 + 푥2 200 + 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 38. So, let’s compute the pooled standard deviation: 풑 1 − 풑 풑 = 64 + 푥2 200 + 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 39. So, let’s compute the pooled standard deviation: 풑 1 − 풑 풑 = 64 + 92 200 + 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 40. So, let’s compute the pooled standard deviation: 풑 1 − 풑 풑 = 64 + 92 200 + 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 41. So, let’s compute the pooled standard deviation: 풑 1 − 풑 풑 = 64 + 92 200 + 200 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 42. Add the fractions: 풑 1 − 풑 풑 = 64 + 92 200 + 200
- 43. Add the fractions: 풑 1 − 풑 풑 = 156 200 + 200
- 44. Add the fractions: 풑 1 − 풑 풑 = 156 200 + 200
- 45. Add the fractions: 풑 1 − 풑 풑 = 156 200
- 46. Add the fractions: 풑 1 − 풑 풑 = .78
- 47. Now we plug this pooled proportion: 풑 1 − 풑 풑 = .78
- 48. Now we plug this pooled proportion: 풑 1 − 풑 풑 = .78 into the standard error formula in the denominator
- 49. Now we plug this pooled proportion: 풑 1 − 풑 풑 = .78 into the standard error formula in the denominator
- 50. Now we plug this pooled proportion: .78 1−. ퟕퟖ 풑 = .78 into the standard error formula in the denominator
- 51. Now we plug this pooled proportion: .78 .22 into the standard error formula in the denominator
- 52. Now we plug this pooled proportion: . ퟏퟕퟐ into the standard error formula in the denominator
- 53. Now we plug this pooled proportion: 푍푠푡푎푡푖푠푡푖푐 = (−. ퟏퟒ) . ퟏퟕퟐ 1 풏ퟏ + 1 푛2 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 54. Now we need to calculate n or the sample size: 푍푠푡푎푡푖푠푡푖푐 = (−. ퟏퟒ) .172 1 200 + 1 푛 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 55. Now we need to calculate n or the sample size: 푍푠푡푎푡푖푠푡푖푐 = (−. ퟏퟒ) .172 1 200 + 1 200 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 56. Now we need to calculate the Zstatistic 푍푠푡푎푡푖푠푡푖푐 = (−. ퟏퟒ) .172 1 200 + 1 200 Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the placebo? Test this claim using alpha = 0.05.
- 57. Now we need to calculate the Zstatistic 푍푠푡푎푡푖푠푡푖푐 = (−. ퟏퟒ) .172 2 200
- 58. Now we need to calculate the Zstatistic 푍푠푡푎푡푖푠푡푖푐 = (−. ퟏퟒ) .172 1 100
- 59. Now we need to calculate the Zstatistic 푍푠푡푎푡푖푠푡푖푐 = (−. ퟏퟒ) .172 .01
- 60. Now we need to calculate the Zstatistic 푍푠푡푎푡푖푠푡푖푐 = (−. ퟏퟒ) .172 (.1)
- 61. Now we need to calculate the Zstatistic 푍푠푡푎푡푖푠푡푖푐 = (−. ퟏퟒ) (.414) (.1)
- 62. Now we need to calculate the Zstatistic 푍푠푡푎푡푖푠푡푖푐 = (−. ퟏퟒ) .0414
- 63. Now we need to calculate the Zstatistic 푍푠푡푎푡푖푠푡푖푐 = −3.379
- 64. Let’s see where this lies in the distribution: rare -1.96 Common rare +1.96
- 65. Let’s see where this lies in the distribution: rare -1.96 Common rare -3.38 +1.96
- 66. Because -3.38 is outside the -1.96 range we consider it to be a rare occurrence and therefore we will reject the null hypothesis and accept the alternative hypothesis:
- 67. Because -3.38 is outside the -1.96 range we consider it to be a rare occurrence and therefore we will reject the null hypothesis and accept the alternative hypothesis: There was a significant difference in effectiveness between the medication group and the placebo group, z = -3.379, p < 0.05.
- 68. Because -3.38 is outside the -1.96 range we consider it to be a rare occurrence and therefore we will reject the null hypothesis and accept the alternative hypothesis: There was a significant difference in effectiveness between the medication group and the placebo group, z = -3.379, p < 0.05 In summary A two-sample Z test is used to compare the differences statistically between two sample proportions.