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Statistical Power and
Sample Size Calculation

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Statistical Power and Sample Size Calculation
A sample size that is too small can compromise your study.  The sample size must be large enough to detect the differences set forth in the study plans.  As the sample size decreases, the power of the test also decreases, along with an increase in the probability of failing to reject a false null hypothesis. 

Contact us for assistance with your sample size and statistical power calculations.  Sample size justification is often required for grants and publications, and also is essential in survey planning.

Hypothesis Testing Basics
A hypothesis is a statement that sets forth what we believe to be true about the relationships among the characteristics being studied.   The null hypothesis, denoted by Ho, initially assumed to be true,  assumes that differences do not exist or relationships among the characteristics are due to chance.  The alternative hypothesis, denoted by HAassumes that there are real differences between the characteristics being studied, or there are factors that have a real influence on the study outcomes.

Hypothesis Test Example
A college would like to determine if the average GPA of their male and female freshmen students is the same.
Η0 : μmale = μfemale
ΗA : μmaleμfemale

The null hypothesis is that the GPA mean for male freshmen is equal to the female GPA mean (where μ represents the mean).  The alternative hypothesis  is that the average freshmen male and female GPAs are different.

Type I and Type II Errors
A Type I error occurs when a true null hypothesis is rejected.  The probability of a Type I error is denoted by α, and is the significance level of the hypothesis test, with 0.05 being a common value for α. On the other hand, a Type II error occurs when the  null hypothesis is false and it is not rejected.  A Type II error is denoted by β and is often set to 0.20.

Statistical Power
The power of a significance test is the probability of rejecting a false null hypothesis, and is equal to 1 -  β.  If  β is set to 0.20, the power = 0.80.

The power of a test is related to the magnitude of the difference between the null and alternative hypotheses, the sample size, standard deviation, and α, the significance level of the test.  Knowing the effect size, power, and α, the sample size can be calculated.  Since sample size and power are related, a small sample size results in less power, or reduced probability of rejecting a false null hypothesis.