Test of Significance for Two Normal Population Variances: Procedure Steps, Example Solved Problems

**TEST OF SIGNIFICANCE FOR TWO NORMAL POPULATION VARIANCES**

*Test procedure:*

This test compares the variances of two independent normal
populations, *viz., N*(*μ _{X}*,

and *N*(*μ _{Y}*,

**Step 1** **: ****Null Hypothesis**** ***H*_{0}** **:** ***σ _{X}*

That is, there is no significant difference between the variances
of the two normal populations.

The alternative hypothesis can be chosen suitably from any one of
the following

(i) H_{1} : σ_{X}^{2} < σ_{Y}^{2}
(ii) *H*_{1} : σ_{X}^{2}
> σ_{Y}^{2} (iii) *H*_{1}
: σ_{X}^{2} ≠ σ_{Y}^{2}

**Step 2 : ****Data**

Let *X*_{1}, *X*_{2},. . ., *X _{m}*
and

**Step 3 : ****Level of significance**** **

*α*

**Step 4 : ****The test Statistic**

under *H*_{0} and its sampling
distribution under *H*_{0} is *F*_{(m-1,} _{n}_{-1)}.

**Step 5 : ****Calculation of the Test Statistic**

The test statistic

**Step 6 : ****Critical values**

**Step 7 : ****Decision**

**Note 1: **Since** ***f**(m–*1*, n*–1)*,1-α *** **is not avilable in the
given** ***F*-table, it is computed
as the reciprocal of *f _{(n–}*

**Note 2: **A** ***F*-test is based on the ratio of variances, it is also known as
Variance Ratio Test.

**Note 3: **When** ***μ _{X}*

and follows* F _{m}*

**Example 3.1**

Two samples of sizes 9 and 8 give the sum of squares of deviations
from their respective means as 160 inches square and 91 inches square
respectively. Test the hypothesis that the variances of the two populations
from which the samples are drawn are equal at 10% level of significance.

*Solution:*

**Step 1 : ****Null Hypothesis:**** ***H*_{0}** **:** ***σ _{X}*

That is there is no significant difference between the two
population variances.

**Alternative Hypothesis: ***H*_{1}** **:** ***σ _{X}*

That is there is significant difference between the two population
variances.

**Step 2 : ****Data**

*m *= 9,* n *= 8

**Step 3 : ****Level of ****significance**

α = 10%

**Step 4 :** **Test Statistic**

, under *H*_{0}.

**Step 5 : ****Calculation**

**Step 6 : ****Critical values**

Since *H*_{1} is a two-sided alternative hypothesis
the corresponding critical values are:

**Step 7 : ****Decision**

Since *f* _{(8, 7),0.95} = 0.286 < *F*_{0}
= 1.54 < *f* _{(8, 7),0.05} = 3.73, the null hypothesis is not
rejected and we conclude that there is no significant difference between the
two population variances.

**Note 4: **The critical values of** ***F*** **corresponding to** ***α*** **= 0.05 requires table
values at 0.025 and 0.975** **which are not provided. Hence *α* is taken as 0.1 in this
example.

**Example 3.2**

A medical researcher claims that the variance of the heart rates
(in beats per minute) of smokers is greater than the variance of heart rates of
people who do not smoke. Samples from two groups are selected and the data is
given below. Using = 0.05, test whether there is enough evidence to support the
claim.

*Solution:*

**Step 1 : ****Null Hypothesis:**** ***H*_{0}** **:** ***σ*_{1}^{2}** **=**
***σ*_{2}^{2}

That is there is no significant difference between the two
population variances.

*H*_{1}* *:* σ*_{1}^{2}* *>* σ*_{2}^{2}

That is, the variance of heart rates of smokers is greater than
that of non-smokers.

**Step 2 : ****Data**

**Step 3 : ****Level of significance**** ***α*** **= 5%

**Step 4 : ****Test statistic**

**Step 5 : ****Calculation**

**Step 6 : ****Critical value**

*f* _{(m-1,n-1),0.05} = *f* _{(24,17),0.05}
= 2.19

**Step 7 : ****Decision**

Since *F*_{0} = 3.6 > *f* _{(24,17),0.05}
= 2.19, the null hypothesis is rejected and we conclude that the variance of
heart beats for smokers seems to be considerably higher compared to that of
the non-smokers.

**Example 3.3**

The following table gives the random sample of marks scored by
students in two schools, A and B.

Is the variance of the marks of students in school A is less than
that of those in school B?

Test at 5% level of significance.

*Solution:*

Let *X*_{1}, *X*_{2} , …, *X _{m}*
represent sample values for school A and let

**Step 1 : ****Null Hypothesis:**** ***H*_{1}** **:** ***σ _{X}*

That is, there is no significant difference between the two
population variances.

**Alternative Hypothesis: ***H*_{1}** **:** ***σ _{X}*

That is, the variance of marks in school A is significantly less
than that of school B.

**Step 2 : ****Data**

*X*_{1}*, X*_{2}*,…, X _{m} *are sample from school A

*Y*_{1},* Y*_{2}, ...,* Y _{n} *are sample from
school B

**Step 3 : ****Test statistic**

**Step 4 : ****Calculations**

**Step 5 : ****Level of significance**

= 5%

**Step 6 : ****Critical value**

*f _{(}*

**Step 7 : ****Decision**

Since *F* _{0} *=* 0.609 > *f _{(}*

Tags : Procedure Steps, Example Solved Problems | Statistics , 12th Statistics : Chapter 3 : Tests Based on Sampling Distributions II

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12th Statistics : Chapter 3 : Tests Based on Sampling Distributions II : Test of Significance for Two Normal Population Variances | Procedure Steps, Example Solved Problems | Statistics

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