Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal...

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Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distributi

Transcript of Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal...

Page 1: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

Wednesday, October 17

Sampling distribution of the mean.Hypothesis testing using the normal Z-distribution.

Page 2: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

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 3: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

X =

N

_

What is the relationship between the population standard deviation and the standard error of the mean?

Page 4: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

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 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

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 6: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

_

z = X -

X-

Wow! We can use the z-distribution to test a hypothesis.

Page 7: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

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 8: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

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 9: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

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 10: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

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 11: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

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 12: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.
Page 13: Wednesday, October 17 Sampling distribution of the mean. Hypothesis testing using the normal Z-distribution.

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.