![Page 1: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/1.jpg)
![Page 2: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/2.jpg)
Wednesday, October 16
Sampling distribution of the mean.Hypothesis testing using the normal Z-distribution.
![Page 3: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/3.jpg)
Population
SampleA XA
µ
_
SampleB XB
SampleE XE
SampleD XD
SampleC XC
_
_
_
_
In reality, the sample mean is just one of many possible samplemeans drawn from the population, and is rarely equal to µ.
sa
sb
sc
sd
se
n
n
n
n n
![Page 4: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/4.jpg)
As sample size increases, the magnitude of the sampling error decreases; at a certainpoint, there are diminishing returns of increasing sample size to decrease sampling error.
![Page 5: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/5.jpg)
X =
n
_
What is the relationship between the population standard deviation and the standard error of the mean?
![Page 6: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/6.jpg)
Central Limit Theorem
The sampling distribution of means from random samplesof n observations approaches a normal distribution regardless of the shape of the parent population.
![Page 7: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/7.jpg)
_
z = X -
X-
Wow! We can use the z-distribution to test a hypothesis.
![Page 8: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/8.jpg)
Step 1. State the statistical hypothesis H0 to be tested (e.g., H0: = 100)
Step 2. Specify the degree of risk of a type-I error, that is, the risk of incorrectly concluding that H0 is false when it is true. This risk, stated as a probability, is denoted by , the probabilityof a Type I error.
Step 3. Assuming H0 to be correct, find the probability of obtaining a sample mean thatdiffers from by an amount as large or larger than what was observed.
Step 4. Make a decision regarding H0, whether to reject or not to reject it.
![Page 9: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/9.jpg)
An Example
You draw a sample of 25 adopted children. You are interested in whether theyare different from the general population on an IQ test ( = 100, = 15).
The mean from your sample is 108. What is the null hypothesis?
![Page 10: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/10.jpg)
An Example
You draw a sample of 25 adopted children. You are interested in whether theyare different from the general population on an IQ test ( = 100, = 15).
The mean from your sample is 108. What is the null hypothesis?
H0: = 100
![Page 11: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/11.jpg)
An Example
You draw a sample of 25 adopted children. You are interested in whether theyare different from the general population on an IQ test ( = 100, = 15).
The mean from your sample is 108. What is the null hypothesis?
H0: = 100
Test this hypothesis at = .05
![Page 12: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/12.jpg)
An Example
You draw a sample of 25 adopted children. You are interested in whether theyare different from the general population on an IQ test ( = 100, = 15).
The mean from your sample is 108. What is the null hypothesis?
H0: = 100
Test this hypothesis at = .05
Step 3. Assuming H0 to be correct, find the probability of obtaining a sample mean thatdiffers from by an amount as large or larger than what was observed.
Step 4. Make a decision regarding H0, whether to reject or not to reject it.
![Page 13: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/13.jpg)
![Page 14: Wednesday, October 16 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution](https://reader035.vdocuments.mx/reader035/viewer/2022062805/5697bff91a28abf838cbf92c/html5/thumbnails/14.jpg)
An Example
You draw a sample of 25 adopted children. You are interested in whether theyare different from the general population on an IQ test ( = 100, = 15).
The mean from your sample is 108. What is the null hypothesis?
H0: = 100
Test this hypothesis at = .01
Step 3. Assuming H0 to be correct, find the probability of obtaining a sample mean thatdiffers from by an amount as large or larger than what was observed.
Step 4. Make a decision regarding H0, whether to reject or not to reject it.