find crucial values that the F-distribution check hypotheses concerning two populace standard deviations

The critical parameters we need to compare in between two populations are the variance and standard deviation. Before we can develop a hypothesis test to compare two populace parameters, us need an additional distribution.

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Fisher"s F-distribution

Unlike the mean, the standard deviation is extremely susceptible to too much values, and also consequently does a an extremely poor project of measuring spread out for distributions that room not symmetric. So prior to we do any kind of inference regarding population standard deviations, we must an initial verify the the samples come from normally dispersed populations.

Fisher"s F-distribution

If and also

and also
room sample variances indigenous independent basic random samples of dimension n1and n2, respectively, attracted from typical populations, then

follows the F-distribution through n1-1 degrees of liberty in the numerator and n2-1 levels of flexibility in the denominator.

properties of the F-distribution like the Χ2 distribution, the is not symmetric. That is skewed right The shape depends on the levels of freedom in the numerator and also denominator. F≥0

Notice right here that the samples must come from normally dispersed populations.

Finding vital Values

Find critical values in the F-distribution using a table is excellent in a similar manner to the t and Χ2 tables, though with some differences. The values in the table still represent values v the indicidated α area come the right, but because the F distribution has two degreees of freedom rather than one, it requires a separate table because that each α.

Before we start the section, you require a copy the the table. You can download a printable copy the this table, or use Table VII starting on page A-14 in the earlier of your textbook. The table will look various than the printable version linked over because the publisher go not provide a digital version.


So the values in the table over are the an important values:


You might wonder how we find crucial values because that left-tailed tests. To perform that, we use the very same table and the adhering to formula:


Let"s shot a couple examples.

Example 1

Find the value of the F-distribution that has α=0.05 area to the right, v 10 degrees of liberty in the numerator, and also 15 degrees of liberty in the denominator.

< reveal answer >


So F0.05, 10, 15 = 2.54

Example 2

Find the value of the F-distribution that has α=0.05 area come the left, v 20 levels of freedom in the numerator, and 8 degrees of liberty in the denominator.

< expose answer >

F0.95, 20, 8 = 1 1 ≈ 0.41
F0.05, 8, 20 2.45

Finding vital Values making use of StatCrunch

Click ~ above Stat > Calculators > F

Enter the numerator and denominator levels of freedom, the direction of the inequality, and also the probability (leave X blank). Then push Compute.

Example 3

Use the modern technology of your selection to uncover the value from the F-distribution v α=0.01 area to the best if samples of size 15 and also 20 room taken.

< expose answer >

Using StatCrunch:


So F0.01, 14, 19 = 3.19

Performing a hypothesis Test concerning Two populace Standard Deviations

Step 1: State the null and different hypotheses.

Two-Tailed H0: H1: Left-Tailed H0: H1:
Right-Tailed H0: H1:

Step 2: decision on a level that significance, α.

Step 3: Compute the test statistic, .

Step 4: recognize the P-value.

Step 5: reject the null hypothesis if the P-value is less than the level of significance, α.

Step 6: State the conclusion.

Hypothesis Testing regarding Two populace Standard Deviations utilizing StatCrunch

go to Stat > Variance > Two Sample > data/summary get in the sample variances or choose the proper column pick Next. Collection the null variance ratio (standard is 1) and also the different hypothesis. Click Calculate.

The results should appear.

Example 4

Problem:In instance 1 in section 11.2, we contrasted the average scores that men and women top top a Mth096 exam. In that test, we assumed the the typical deviations of the two teams were equal. Check the presumption at the α=0.1level the significance.


First, we need to check the conditions. We understand from instance 1 the neither sample has outliers, however we carry out not understand if castle come indigenous normally dispersed populations. We"ll use StatCrunch to carry out Q-Q plots.


While the two plots aren"t exactly linear, the does appear that the samples can come from normally spread populations, for this reason our conditions are satisfied.

Step 1: H0: H1:

Step 2: α = 0.1 (given)

Step 3: (we"ll use StatCrunch)

Step 4: using StatCrunch:


Step 5: due to the fact that the P-value > α, we carry out not refuse the null hypothesis.

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Step 6: based upon these results, over there is no proof to assistance the insurance claim that the traditional deviations room not equal.