subject: Power Function Statistics [print this page] Statistical power definition help is important in math. In math statistical power means probability of reject a false null hypothesis. In statistics to test hypotheses and also test the null hypothesis. The power is equal to 1-beta. In odd position we want to reject our null hypothesis in favor of the alternative. It is more help for prepare an exam.
Statistics deal with frequency distribution. It is used to compare twoor more frequency distribution taken from different
population to see if there are any differences between them.The statistics uses the following measures for the comparisson. They are mean, median and mode. Mean is the average of all the observations.Median is the middle most value of the observations and mode has the maximum frequencies.
Mean can be used to see the average mark of the class obtained. This average helps to see how many students
are above average, how many are average students and how many are below averages. The teacher tries to help the
average and below average students to score more grades in future.
In a factory, the mean of the wages helps the authorities to know if the workers' welfare is maintained.
It also helps to compare the salaries of the employees of the different companies.
In sales, the average sales in the district helps the sales manager to plan for increasing the sales in the future.
The governemnt takes the average income and expense of the citizens to know whether the citizens rights are maintained.
The family finds the average of their expenses to balance their finance.
The average production of agricultural commodities, the industrial goods, the average exports and imports help the country to see their developments.
Formula for Power Function Statistics:
In the following formula for power function statistics
1- beta (before use this formula we need to find the following formula)
Formula to determine the mean critical value in power function statistics
'M_(cv) = mu + Z_(cv) (sigma_M)'
For example,
'mu' = 2.2 , 'Z_(cv)' = -1.332 and n = 10
'M_(cv) = 2.2 +(-1.332) (2/sqrt(10))'
= 0.868 x 0.632
= 0.548
This is a way to finding the statistical power.
Example Problem for Power Function Statistics:
Problem 1:
Find area under the curve 'mu' = 4.0 , 'sigma' = 1.0 , 'Z_(cv)' = -1.342 and n = 30 by using power function statistics calculation
Solution:
Given: 'mu' = 4.0 , 'sigma' = 1.0 , 'Z_(cv)' = -1.342 and n = 30
We know the mean critical value formula.
'M_(cv) = mu + Z_(cv) (sigma_M)'
= '4.0 +(-1.342) (4/sqrt(30))'
= 3.0199
Then use standard error of the mean formula to finding the area under the curve
Standard error of the mean formula
Z = '(M-mu)/ (sigma_M)'
= '(3.0199 - 4.0) / (2/sqrt(30))'
= -2.684
Here use statistical power calculation
= '1 - beta'
= 1- 0.0037 (Use Z table to find the value of -2.68 = 0.0037)
= 0.9963
Problem 2:
Find area under the curve 'mu' = 7.0 , 'sigma' = 2.0 , 'Z_(cv)' = -1.533 and n = 40 by using power function statistics calculation
Solution:
Given: 'mu' = 7.0 , 'sigma' = 2.0 , 'Z_(cv)' = -1.533 and n = 40
We know the mean critical value formula.
'M_(cv) = mu + Z_(cv) (sigma_M)'
= '7.0 +(-1.533) (2/sqrt(40))'
= 6.515
Then use standard error of the mean formula to finding the area under the curve
Standard error of the mean formula
Z = '(M-mu)/ (sigma_M)'
= '(6.515- 7.0) / (2/sqrt(40))'
= -1.533
Here use statistical power calculation
= '1 - beta'
= 1- 0.0630 (Use Z table to find the value of -1.53 = 0.0630)
