Calculating a two sample z test by hand
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![Page 1: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/1.jpg)
Calculating a Two-Sample Z Test by Hand
![Page 2: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/2.jpg)
Here is the problem you just saw in a previous presentation. Follow along doing the calculations on your own problem as you view this one.
![Page 3: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/3.jpg)
Here is the problem you just saw in a previous presentation. Follow along doing the calculations on your own problem.
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.
![Page 4: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/4.jpg)
What are the researchers asking here?
![Page 5: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/5.jpg)
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.
![Page 6: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/6.jpg)
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).
![Page 7: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/7.jpg)
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?
![Page 8: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/8.jpg)
With that in mind, we are able to state the null-hypothesis and then determine if we will reject or fail to reject it:
![Page 9: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/9.jpg)
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.
![Page 10: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/10.jpg)
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.
![Page 11: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/11.jpg)
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]
![Page 12: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/12.jpg)
One tailed test visual depiction:
rarecommon
+1.64
If the z value we are about to
calculate lands above this point, we will reject the null hypothesis
![Page 13: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/13.jpg)
One tailed test visual depiction:
rarecommon
+1.64
If the z value lands below this point
we will fail to reject the null
hypothesis
![Page 14: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/14.jpg)
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
![Page 15: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/15.jpg)
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.
![Page 16: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/16.jpg)
Here is a visual depiction of the two-tailed test.
rare
-1.96
Common rare
+1.96
![Page 17: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/17.jpg)
All we have left to do now is calculate the z statistic.
![Page 18: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/18.jpg)
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:
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(�̂�1− �̂�2)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2
![Page 19: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/19.jpg)
In the numerator we are subtracting one sample proportion from another sample proportion:
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(�̂�1− �̂�2)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2
![Page 20: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/20.jpg)
In the numerator we are subtracting one sample proportion from another sample proportion:
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(�̂�1− �̂�2)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers 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.
![Page 21: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/21.jpg)
In the numerator we are subtracting one sample proportion from another sample proportion:
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(�̂�1− �̂�2)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers 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.
![Page 22: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/22.jpg)
In the numerator we are subtracting one sample proportion from another sample proportion:
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(�̂�1− �̂�2)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers 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.
92/200=.4664/200=.32
![Page 23: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/23.jpg)
In the numerator we are subtracting one sample proportion from another sample proportion:
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=( .32− .46 )
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers 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.
92/200=.4664/200=.32
![Page 24: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/24.jpg)
In the numerator we are subtracting one sample proportion from another sample proportion:
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(−.𝟏𝟐)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers 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.
![Page 25: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/25.jpg)
The denominator is the estimated standard error:
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(− .12)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers 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.
This proportion is called the pooled
standard deviation. It is the same value
we use with independent
sample t-tests.
![Page 26: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/26.jpg)
The denominator is the estimated standard error:
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(− .12)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2
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.
![Page 27: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/27.jpg)
Why is this important? Because the standard deviation divided by the square root of the sample size is the standard error.
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(− .12)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2
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.
![Page 28: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/28.jpg)
And the larger the standard error the less likely the two groups will be statistically significantly different.
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(− .12)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2
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.
![Page 29: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/29.jpg)
Conversely, the smaller the standard error the more likely the two groups will be statistically significantly different.
𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(− .12)
√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2
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.
![Page 30: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/30.jpg)
So, let’s compute the pooled standard deviation:
![Page 31: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/31.jpg)
So, let’s compute the pooled standard deviation:
�̂�=𝑥1+𝑥2𝑛1+𝑛2
![Page 32: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/32.jpg)
So, let’s compute the pooled standard deviation:
�̂�=𝑥1+𝑥2𝑛1+𝑛2
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.
Here’s the problem again:
![Page 33: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/33.jpg)
So, let’s compute the pooled standard deviation:
�̂�=𝑥1+𝑥2𝑛1+𝑛2
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.
![Page 34: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/34.jpg)
So, let’s compute the pooled standard deviation:
�̂�=64+𝑥2𝑛1+𝑛2
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.
![Page 35: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/35.jpg)
So, let’s compute the pooled standard deviation:
�̂�=64+𝑥2𝑛1+𝑛2
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.
![Page 36: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/36.jpg)
So, let’s compute the pooled standard deviation:
�̂�=64+𝑥2200+𝑛2
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.
![Page 37: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/37.jpg)
So, let’s compute the pooled standard deviation:
�̂�=64+𝑥2200+𝑛2
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.
![Page 38: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/38.jpg)
So, let’s compute the pooled standard deviation:
�̂�=64+92200+𝑛2
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.
![Page 39: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/39.jpg)
So, let’s compute the pooled standard deviation:
�̂�=64+92200+𝑛2
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.
![Page 40: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/40.jpg)
So, let’s compute the pooled standard deviation:
�̂�=64+92200+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.
![Page 41: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/41.jpg)
Add the fractions:
�̂�=64+92200+200
![Page 42: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/42.jpg)
Add the fractions:
�̂�=156
200+200
![Page 43: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/43.jpg)
Add the fractions:
�̂�=156
200+200
![Page 44: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/44.jpg)
Add the fractions:
�̂�=156200
![Page 45: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/45.jpg)
Add the fractions:
�̂�=.78
![Page 46: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/46.jpg)
Now we plug this pooled proportion:
�̂�=.78
![Page 47: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/47.jpg)
Now we plug this pooled proportion:
into the standard error formula in the denominator
�̂�=.78
![Page 48: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/48.jpg)
Now we plug this pooled proportion:
into the standard error formula in the denominator
�̂�=.78
![Page 49: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/49.jpg)
Now we plug this pooled proportion:
into the standard error formula in the denominator
�̂�=.78
![Page 50: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/50.jpg)
Now we plug this pooled proportion:
into the standard error formula in the denominator
![Page 51: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/51.jpg)
Now we plug this pooled proportion:
into the standard error formula in the denominator
![Page 52: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/52.jpg)
Now we plug this pooled proportion:
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.
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒)
√ .𝟏𝟕𝟐√ 1𝒏𝟏+ 1𝑛2
![Page 53: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/53.jpg)
Now we need to calculate n or the sample size:
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.
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒)
√ .172√ 1200
+ 1𝑛
![Page 54: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/54.jpg)
Now we need to calculate n or the sample size:
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.
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒)
√ .172√ 1200
+ 1200
![Page 55: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/55.jpg)
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.
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒)
√ .172√ 1200
+ 1200
Now we need to calculate the Zstatistic
![Page 56: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/56.jpg)
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒 )
√ .172√ 2200
Now we need to calculate the Zstatistic
![Page 57: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/57.jpg)
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒 )
√ .172√ 1100
Now we need to calculate the Zstatistic
![Page 58: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/58.jpg)
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒)
√ .172√ .01
Now we need to calculate the Zstatistic
![Page 59: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/59.jpg)
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(− .𝟏𝟒 )√ .172(.1)
Now we need to calculate the Zstatistic
![Page 60: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/60.jpg)
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(− .𝟏𝟒 )
(.414)(.1)
Now we need to calculate the Zstatistic
![Page 61: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/61.jpg)
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒).0414
Now we need to calculate the Zstatistic
![Page 62: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/62.jpg)
Now we need to calculate the Zstatistic
𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=−3 .379
![Page 63: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/63.jpg)
Let’s see where this lies in the distribution:
rare
-1.96
Common rare
+1.96
![Page 64: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/64.jpg)
Let’s see where this lies in the distribution:
rare
-1.96
Common rare
+1.96-3.38
![Page 65: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/65.jpg)
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:
![Page 66: Calculating a two sample z test by hand](https://reader035.fdocuments.net/reader035/viewer/2022062307/55847a74d8b42a15768b4f31/html5/thumbnails/66.jpg)
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.