This calculator will compute the critical values of F-statistics corresponding to n_N (numerator) and n_D (denominator) degrees of freedom, at the desired probability level. See for example Table 3-1 on page 44 of Stanton A. Glanz "Primer of Biostatistics", 3^rd Edition, McGraw-Hill, New York, 1992. The numerator and denominator degrees of freedom must be whole numbers correponding to the sample sizes.

Example

Problem

The following is Problem 3-2 from Glanz (1992). The effect of "environmental smoke" on lung disease was studied by measuring the mean forced mid-respiratory flow for five groups of people. Each group contained 200 subjects.

group	respiratory flow	standard deviation
nonsmokers working in clean environment	3.17	0.74
nonsmokers working in smoky environment	2.72	0.71
light smokers	2.63	0.73
moderate smokers	2.29	0.70
heavy smokers	2.12	0.72

Is there any evidence that the mean respiratory flow is any different among the five different experimental groups?

Solution

First we estimate the "within groups" variance as s²_wit = ( 0.74² + 0.71² + 0.73² + 0.70² + 0.72² )/5 = 0.5186.

Next we compute the standard deviation of the sample means:

From these results, the "between groups" estimate of variance is s²_bet = 200 × (0.408²) = 33.29. The F-statistics is defined as F = s²_bet / s²_wit = 33.29 / 0.5186 = 64.19.

We are now ready to test the null hypothesis, which assumes that there is no difference between aerial from in the five different groups of test subjects. If the hypothesis is true, the critical value of F at (say) 95% confidence level (α = 0.05) should be larger than 64.19.

The numerator degrees of freedom are equal to the number of groups minus one: n_N = 5 - 1 = 4. The denominator degrees of freedom are number of groups × (number of subjects minus one: n_D = 4 × (200 - 1) = 796.

Type the "4" and "796" into the editable fields in the form above, select the confidence level (0.05) and clisk on the Calculate button. The results are:

Conclusion

The critical value of F at 95% probability level is much lower (2.38) than the observed value of F (64.19), which means that the null hypothesis is false. The data does suggest that the differenes between aerial flow seen within different groups (smokers, nonsmokers) are significant.