1. Wednesday, October 20
Sampling distribution of the mean.
Hypothesis testing using the normal Z-distribution
The t distribution

2. In reality, the sample mean is just one of many possible sample
SampleC XC _
SampleD XD sc _ n sd
Population n
SampleB XB _  sb n
SampleE XE
SampleA XA _ _ se sa n n
In reality, the sample mean is just one of many possible sample
means drawn from the population, and is rarely equal to µ.

3. In reality, the sample mean is just one of many possible sample
SampleC XC _
SampleD XD sc _ n sd
Population n
SampleB XB _  sb n
SampleE XE
SampleA XA _ _ se sa n n
In reality, the sample mean is just one of many possible sample
means drawn from the population, and is rarely equal to µ.

4. What’s the difference? SS
(N - 1) s2 = SS N 2 =

5. SS SS What’s the difference? s2 2 N (N - 1) ^ = =
(occasionally you will see this little “hat” on the symbol to clearly indicate that this is a variance estimate) – I like this because it is a reminder that we are usually just making estimates, and estimates are always accompanied by error and bias, and that’s one of the enduring lessons of statistics) SS
(N - 1) s2 = SS N 2 = ^

6. Standard deviation. SS
(N - 1) s =

7. In reality, the sample mean is just one of many possible sample
SampleC XC _
SampleD XD sc _ n sd
Population n
SampleB XB _  sb n
SampleE XE
SampleA XA _ _ se sa n n
In reality, the sample mean is just one of many possible sample
means drawn from the population, and is rarely equal to µ.

8. As sample size increases, the magnitude of the sampling error decreases; at a certain
point, there are diminishing returns of increasing sample size to decrease sampling error.

9. The sampling distribution of means from random samples
Central Limit Theorem
The sampling distribution of means from random samples
of n observations approaches a normal distribution
regardless of the shape of the parent population.
Just for fun, go check out the Khan Academy

10. Wow! We can use the z-distribution to test a hypothesis._
z =
X -  X -

11. Step 1. State the statistical hypothesis H0 to be tested (e. g
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 probability
of a Type I error.
Step 3. Assuming H0 to be correct, find the probability of obtaining a sample mean that
differs 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.

12. An Example
You draw a sample of 25 adopted children. You are interested in whether they
are different from the general population on an IQ test ( = 100,  = 15).
The mean from your sample is What is the null hypothesis?

13. An Example
You draw a sample of 25 adopted children. You are interested in whether they
are different from the general population on an IQ test ( = 100,  = 15).
The mean from your sample is What is the null hypothesis?
H0:  = 100

14. Test this hypothesis at  = .05
An Example
You draw a sample of 25 adopted children. You are interested in whether they
are different from the general population on an IQ test ( = 100,  = 15).
The mean from your sample is What is the null hypothesis?
H0:  = 100
Test this hypothesis at  = .05

15. Test this hypothesis at  = .05
An Example
You draw a sample of 25 adopted children. You are interested in whether they
are different from the general population on an IQ test ( = 100,  = 15).
The mean from your sample is 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 that
differs 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.

18. GOSSET, William Sealy

19. GOSSET, William Sealy

20. The t-distribution is a family of distributions varying by degrees of freedom (d.f., where
d.f.=n-1). At d.f. = , but at smaller than that, the tails are fatter.

21. _
z =
X -  X - _
t =
X -  sX - s -
sX =
 N

22. The t-distribution is a family of distributions varying by degrees of freedom (d.f., where
d.f.=n-1). At d.f. = , but at smaller than that, the tails are fatter.

23. Degrees of Freedom
df = N - 1

25. Problem
Sample:
Mean = 54.2
SD = 2.4
N = 16
Do you think that this sample could have been drawn from a population with  = 50?

26. Problem
Sample:
Mean = 54.2
SD = 2.4
N = 16
Do you think that this sample could have been drawn from a population with  = 50? _
t =
X -  sX -

27. The mean for the sample of 54. 2 (sd = 2
The mean for the sample of 54.2 (sd = 2.4) was significantly different from a hypothesized population mean of 50, t(15) = 7.0, p < .001.

28. The mean for the sample of 54. 2 (sd = 2
The mean for the sample of 54.2 (sd = 2.4) was significantly reliably different from a hypothesized population mean of 50, t(15) = 7.0, p < .001.

29. rXY rXY Population XY rXY _ rXY rXY SampleC SampleD SampleB SampleE
SampleA _
rXY
rXY

30. The t distribution, at N-2 degrees of freedom, can be used to test the probability that the statistic r was drawn from a population with  = 0. Table C.
H0 :  XY = 0
H1 :  XY  0
where
r N - 2
1 - r2
t =