Statistics Knowledge
One parameter
E: Estimate
μ: MeanV: Distributionc: constant
E{c} = c
E{y} = μ
E{cy} = cμ
V(c) = 0
Two parameters: y1, y2
Attention
Sample Mean
Sample Variance
Normal Distribution
The formal
The image
The related parameter
Sample Normal Distribution
Central Limit Theorem
If y1,y2,...yn is a sequence of n independent and identically distributed random variables with
Depending on these information, we generate the following formal
The formal
The image
t Distribution
The formal
The image
The related parameter
Two Sample t-test
Description: In general case, we have two data set, we try to figure out whether 
1. Assume that the variance of the two data sets are equal:
2. Form the following statistics
3. Assume to determine whether to reject H0, we would compare 
to the t distribution with n1+n2-1 degrees of freedom
to the t distribution with n1+n2-1 degrees of freedom
Get
,
two ways.
A. Define α, calculate n1+n2-2, Use table to find out the 
B. Use calculate
Input t0 
, V and use calculate to generate Min α.
, V and use calculate to generate Min α.
4. If
,
the assume should be rejected, since there is no chance that the assumption be proved.
If
,
the assume should be approved, since there is chance that the assumption be proved.
5. Find the confidence interval
CI = 100(1-α)
If
, the range L AND U should contain 0.
If
, the range L and U should not contain 0.
Example Question
Using t Distribution table
Find out that
Conclusion
reject H0
CI = 100(1-α) = 95%
=-0.55
=-0.01
One Sample Z-test
Description: Some experiments involve comparing only one population mean to a specified value.
1. Assume that the variance of the sets is 
2. Form the following statistics
If α=0.05,
1-0.5 α=0.975
Find in the following table, the Z=1.96
4. Compare the 
Z0 and
Z0 and
If Z0 >
,
the H0 should be rejected
If Z0 <
,
the H0 should be approved
Example Question
In the population, the average IQ is 100 with a standard deviation of 15. A team of Scientists wants to test a new medication to see if it has either a positive or negative effect on intelligence, or no effect at all. A sample of 30 participants who have taken the medication has a mean of 140. Did the medication affect intelligence, using α=0.05?
α=0.05, 
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