Download - Calculating a single sample z test by hand
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Calculating a Single-Sample Z Test
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In this presentation you will be shown how to calculate a single sample z-test by hand.
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We first determine the z-critical for our question.
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We will use the following problem:
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A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82 indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
![Page 6: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/6.jpg)
First, we determine that our decision rule is that we will reject the null hypothesis. In this case we will reject the null hypothesis if the p value is less than .05.
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First, we determine that our decision rule is that we will reject the null hypothesis. In this case we will reject the null hypothesis if the p value is less than .05. Basically, we are saying that we are willing live with the probability of being wrong 5 times out of 100 (.05) or 1 time out of 20.
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With a cut off of .05, if we hypothesize that sample has a higher value than the population then our cut off z-score would be 1.64 (this can be located in a z-table)
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With a cut off of .05, if we hypothesize that sample has a higher value than the population then our cut off z-score would be 1.64 (this can be located in a z-table)
95%
mean-1σ +1σ-2σ +2σ
Common
+1.64
rare
![Page 10: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/10.jpg)
With a cut off of .05, if we hypothesize that sample has a lower value than the population then our cut off z-score would be -1.64 (this can be located in a z-table)
![Page 11: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/11.jpg)
With a cut off of .05, if we hypothesize that sample has a lower value than the population then our cut off z-score would be -1.64 (this can be located in a z-table)
95%
mean-1σ +1σ-2σ +2σ
Common
+1.64
rare
![Page 12: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/12.jpg)
With a cut off of .05, if we hypothesize that sample could have either a lower or higher value than the population then our cut off z-scores would be -1.96 and +1.96
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With a cut off of .05, if we hypothesize that sample could have either a lower or higher value than the population then our cut off z-scores would be -1.96 and +1.96
rarerare
95%
mean-1σ +1σ-2σ +2σ
Common
-1.96 +1.96
![Page 14: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/14.jpg)
So if the z statistic we calculate is less than -1.96 (e.g., -1.99) or greater than +1.96 (e.g., +2.30) then we will consider this to be a rare event and reject the null hypothesis and state that there is a statistically significant difference between .9 (population) and .82 (the sample).
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So if the z statistic we calculate is less than -1.96 (e.g., -1.99) or greater than +1.96 (e.g., +2.30) then we will consider this to be a rare event and reject the null hypothesis and state that there is a statistically significant difference between .9 (population) and .82 (the sample).
Let’s calculate the z statistic and see where if falls!
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So if the z statistic we calculate is less than -1.96 (e.g., -1.99) or greater than +1.96 (e.g., +2.30) then we will consider this to be a rare event and reject the null hypothesis and state that there is a statistically significant difference between .9 (population) and .82 (the sample).
Let’s calculate the z statistic and see where if falls!
We do this by using the following equation:
![Page 17: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/17.jpg)
So if the z statistic we calculate is less than -1.96 (e.g., -1.99) or greater than +1.96 (e.g., +2.30) then we will consider this to be a rare event and reject the null hypothesis and state that there is a statistically significant difference between .9 (population) and .82 (the sample).
Let’s calculate the z statistic and see where if falls!
We do this by using the following equation:𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=
�̂�−𝑝
√𝑝 (1−𝑝)𝑛
![Page 18: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/18.jpg)
So if the z statistic we calculate is less than -1.96 (e.g., -1.99) or greater than +1.96 (e.g., +2.30) then we will consider this to be a rare event and reject the null hypothesis and state that there is a statistically significant difference between .9 (population) and .82 (the sample).
Let’s calculate the z statistic and see where if falls!
We do this by using the following equation:
Zstatistic is what we are trying to find to see if it is outside or inside the z critical values (-1.96 and +1.96).
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=�̂�−𝑝
√𝑝 (1−𝑝)𝑛
![Page 19: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/19.jpg)
Here’s the problem again:
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A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82 indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
![Page 21: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/21.jpg)
A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82 indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
![Page 22: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/22.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=�̂�−𝑝
√𝑝 (1−𝑝)𝑛
![Page 23: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/23.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=�̂�−𝑝
√𝑝 (1−𝑝)𝑛
Note – this little hat () over the p means that this proportion is an
estimate of a population
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(.90)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=�̂�−𝑝
√𝑝 (1−𝑝)𝑛
![Page 25: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/25.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=�̂�−𝑝
√𝑝 (1−𝑝)𝑛
(100)
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=�̂�−𝑝
√𝑝 (1−𝑝)𝑛
A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82
indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
(100)
![Page 27: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/27.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=�̂�−𝑝
√𝑝 (1−𝑝)𝑛
A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82
indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
(100)
![Page 28: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/28.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=�̂�−𝑝
√𝑝 (1−𝑝)𝑛
Let’s plug in the numbers
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=.82−𝑝
√𝑝 (1−𝑝)𝑛
Sample Proportion
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=.82−𝑝
√𝑝 (1−𝑝)𝑛
A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82
indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
Sample Proportion
![Page 31: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/31.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=.82−𝑝
√𝑝 (1−𝑝)𝑛
A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82
indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
Sample Proportion
![Page 32: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/32.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=.82− .90
√ .90(1− .90)𝑛
Population Proportion
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=.82− .90
√ .90(1− .90)𝑛
A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82
indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
Population Proportion
![Page 34: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/34.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=.82− .90
√ .90(1− .90)𝑛
A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82
indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
Population Proportion
![Page 35: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/35.jpg)
The difference
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=.82− .90
√ .90(1− .90)𝑛
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08
√ .90(1− .90)𝑛
The difference
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Now for the denominator which is the estimated standard error. This value will help us know how many standard error units .82 and .90 are apart from one another (we already know they are .08 raw units apart)
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08
√ .90(1− .90)𝑛
Now for the denominator which is the estimated standard error. This value will help us know how many standard error units .82 and .90 are apart from one another (we already know they are .08 raw units apart)
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08
√ .90(1− .90)𝑛
Note - If the standard error is small then the z statistic will be larger. The larger the z statistics the more likely that it will exceed the -1.96 or +1.96 boundaries, compelling us to reject the null hypothesis. If it is smaller than we will not.
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08
√ .90(1− .90)𝑛
Let’s continue our calculations and find out:
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08
√ .90(1− .90)𝑛
A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82
indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
Let’s continue our calculations and find out:
![Page 42: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/42.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08
√ .90( .10)𝑛
Let’s continue our calculations and find out:
![Page 43: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/43.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08
√ .09𝑛
Let’s continue our calculations and find out:
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08
√ .09100
Sample Size:
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08
√ .09100
A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82
indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05
Sample Size:
![Page 46: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/46.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08
√.0009
Let‘s continue our calculations:
![Page 47: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/47.jpg)
𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=− .08.03
Let‘s continue our calculations:
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=−2.67
Let‘s continue our calculations:
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𝒛 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=−2.67
Let‘s continue our calculations:
Now we have our z statistic.
![Page 50: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/50.jpg)
Let’s go back to our distribution:
rarerare
95%
mean-1σ +1σ-2σ +2σ
Common
-1.96 +1.96
![Page 51: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/51.jpg)
Let’s go back to our distribution: So, is this result rare or common?
rarerare
95%
mean-1σ +1σ-2σ +2σ
Common
-1.96 +1.96-2.67
![Page 52: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/52.jpg)
Let’s go back to our distribution: So, is this result rare or common?
rarerare
95%
mean-1σ +1σ-2σ +2σ
Common
-1.96 +1.96
This is the Z-Statistic we
calculated
-2.67
![Page 53: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/53.jpg)
Let’s go back to our distribution: So, is this result rare or common?
rarerare
95%
mean-1σ +1σ-2σ +2σ
Common
-1.96 +1.96-2.67
This is the Z – Critical
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Looks like it is a rare event therefore we will reject the null hypothesis in favor of the alternative hypothesis:
![Page 55: Calculating a single sample z test by hand](https://reader036.vdocuments.us/reader036/viewer/2022062419/55847a66d8b42a15768b4f1f/html5/thumbnails/55.jpg)
Looks like it is a rare event therefore we will reject the null hypothesis in favor of the alternative hypothesis:
The proportion of a sample of 100 medical doctors who recommend aspirin for their patients with headaches IS statistically significantly different from the claim that 9 out of 10 doctors recommend aspirin for their patients with headaches.