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Testing Differences in Population Variances. QSCI 381 – Lecture 42 (Larson and Farber, Sect 10.3). Recap – Testing Means. We used different tests when comparing means depending on whether we could assume that the population variances for the two populations were the same.

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Testing Differences in Population Variances

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## Testing Differences in Population Variances

QSCI 381 – Lecture 42

(Larson and Farber, Sect 10.3)

### Recap – Testing Means

• We used different tests when comparing means depending on whether we could assume that the population variances for the two populations were the same.

• Today we identify a test which can be used to test for differences between two population variances.

### The F-distribution-I

• Let and represent the sample variances of two populations. If both populations are normal and the population variances and are equal, then the sampling distribution of:

is called an .

F-distribution

### The F-distribution-II

• Properties of the F-distribution:

• Positively skewed.

• The curve is determined by the degrees of freedom for the numerator and that for the denominator.

• The area under the curve is 1.

• The mean value is approximately 1.

• F-values are always larger than 0.

### The Two-Sample F-test for Variances-I

• A two-sample F-test is used to compare two population variances and when a sample is randomly selected from each population. The populations must be independent and normally distributed. The test statistic is:

• where and represent the sample variances with . The degrees of freedom for the numerator is d.f.N=n1-1 and the degrees of freedom for the denominator is d.f.D=n2-1.

### The Two-Sample F-test for Variances-II(Finding the rejection region for the test)

• Specify the level of significance .

• Determine the degrees of freedom for the numerator, d.f.N.

• Determine the degrees of freedom for the denominator, d.f.D.

• Determine whether this a one-tailed or a two-tailed test.

• One-tailed – look up the  F-table for d.f.N and d.f.D.

• Two-tailed – look up the /2 F-table for d.f.N and d.f.D.

### The Two-Sample F-test for Variances-III(Finding the rejection region for the test)

One-tailed

Two-tailed

F0=2.901

F0=3.576

In EXCEL:

FINV(prob,dfN,dfD)

d.f.N=5; d.f.D=15; =0.05

### Example-A-I

• We sample two populations. The sample variances for the two populations are 9.622 (n1=46) and 10.352 (n2=51). Test the claim that the two variances are equal (=0.1).

• H0: ; Ha:

• Determine the critical value and the rejection region.

• This is a two-tailed test.

• We reverse the order of samples 1 and 2 so that . Therefore d.f.N=50; d.f.N=45.

• The critical value is F(0.05,50,45) = 1.626.

### Example-A-II

• The test statistic is:

• We fail to reject the null hypothesis

30 fish are sampled from a Marine Reserve and a fished area. Test the claim that the lengths in the Reserve are more variable than those in the fished area (assume =0.05). The data are:

### Example-B-II

• H0: ; Ha:

• =0.05; d.f.N=29; d.f.D=29. This is a one-sided test so the rejection region is F>1.861 = FINV(0.05,29,29)

• The test statistic is:

• We reject the null hypothesis (the data provide support for the claim)

### Confidence intervals for

• When and are the variances of randomly selected, independent samples from normally distributed populations, a confidence interval for is:

where FL is the left-tailed critical F-value and FR is the right-tailed critical F-value (based on probabilities of /2).

### Confidence intervals for(Example-1)

• Find a 95% confidence interval for for example A.

• The lower and upper critical points for the F-distribution are computed:

• FINV(0.975,50,45)=0.564

• FINV(0.025,50,45)=1.788

### Confidence intervals for(Example-2)

• The 95% confidence interval is given by: