*17*

To determine ifÂ the variances of two populations are equal, we can calculate the variance ratioÂ **Ïƒ ^{2}_{1} / Ïƒ^{2}_{2}**, whereÂ Ïƒ

^{2}

_{1}Â is the variance of population 1 andÂ Ïƒ

^{2}

_{2}Â is the variance of population 2.

To estimate the true population variance ratio, we typically take a simple random sample from each population and calculate the sample variance ratio,Â **s _{1}^{2}Â / s_{2}^{2}**, whereÂ s

_{1}

^{2}Â and s

_{2}

^{2}Â are the sample variances for sample 1 and sample 2, respectively.

This test assumes that bothÂ s_{1}^{2}Â and s_{2}^{2}Â are computed from independent samples of size n_{1} and n_{2}, both drawn from normally distributed populations.

The further this ratio is from one, the stronger the evidence for unequal population variances.Â

**The (1-Î±)100% confidence interval** for Ïƒ^{2}_{1} / Ïƒ^{2}_{2}Â is defined as:

(s_{1}^{2}Â / s_{2}^{2}) * F_{n1-1, n2-1, Î±/2Â Â }â‰¤Â Â Ïƒ^{2}_{1} / Ïƒ^{2}_{2}Â â‰¤Â (s_{1}^{2}Â / s_{2}^{2}) * F_{n2-1, n1-1,Â }_{Î±/2}

where F_{n2-1, n1-1, Î±/2Â }and F_{n1-1, n2-1,Â }_{Î±/2}_{Â }are the critical values from the F distribution for the chosen significance level Î±.

The following examples illustrate how to create a confidence interval forÂ Ïƒ^{2}_{1} / Ïƒ^{2}_{2}Â using three different methods:

- By hand
- Using Microsoft Excel
- Using the statistical software
*R*

For each of the following examples, we will use the following information:

**Î±**= 0.05**n**= 16_{1}**n**= 11_{2}**s**Â =28.2_{1}^{2}**s**= 19.3_{2}^{2}Â

**Creating a Confidence Interval By Hand**

To calculate a confidence interval forÂ **Ïƒ ^{2}_{1} / Ïƒ^{2}_{2}**Â by hand, weâ€™ll simply plug in the numbers we have into the confidence interval formula:

(s_{1}^{2}Â / s_{2}^{2}) * F_{n1-1, n2-1,Î±/2Â Â }â‰¤Â Â Ïƒ^{2}_{1} / Ïƒ^{2}_{2}Â â‰¤Â (s_{1}^{2}Â / s_{2}^{2}) * F_{n2-1, n1-1,Â }_{Î±/2}

The only numbers weâ€™re missing are the critical values.Â Luckily, we can locate these critical values in the F distribution table:

F_{n2-1, n1-1, Î±/2Â Â }=Â F_{10, 15, 0.025Â Â }=Â **3.0602**

F_{n1-1, n2-1,Â }_{Î±/2Â Â }=Â 1/ F_{15, 10, 0.025} = 1 /Â 3.5217 =Â **0.2839**

*(Click to zoom in on the table)*

Now we can plug all of the numbers into the confidence interval formula:

(s_{1}^{2}Â / s_{2}^{2}) * F_{n1-1, n2-1,Î±/2Â Â }â‰¤Â Â Ïƒ^{2}_{1} / Ïƒ^{2}_{2}Â â‰¤Â (s_{1}^{2}Â / s_{2}^{2}) * F_{n2-1, n1-1,Â }_{Î±/2}

(28.2 / 19.3) * (0.2839)Â â‰¤Â Â Ïƒ^{2}_{1} / Ïƒ^{2}_{2}Â â‰¤Â (28.2 / 19.3) * (3.0602)

0.4148 â‰¤Â Â Ïƒ^{2}_{1} / Ïƒ^{2}_{2}Â â‰¤Â 4.4714

Thus, the 95% confidence interval for the ratio of the population variances isÂ **(0.4148, 4.4714)**.

**Creating a Confidence Interval Using Excel**

The following image shows how to calculate a 95% confidence interval for the ratio of population variances in Excel. The lower and upper bounds of the confidence interval are displayed in column E and the formula used to find the lower and upper bounds are displayed in column F:

Thus, the 95% confidence interval for the ratio of the population variances isÂ **(0.4148, 4.4714)**. This matches what we got when we calculated the confidence interval by hand.

**Creating a Confidence Interval Using R**

The following code illustrates how to calculate a 95% confidence interval for the ratio of population variances in R:

#define significance level, sample sizes, and sample variances alpha #define F critical values upper_crit #find confidence interval lower_bound #output confidence interval paste0("(", lower_bound, ", ", upper_bound, " )") #[1] "(0.414899337980266, 4.47137571035219 )"

Thus, the 95% confidence interval for the ratio of the population variances isÂ **(0.4148, 4.4714)**. This matches what we got when we calculated the confidence interval by hand.

**Additional Resources**

**How to Read the F-Distribution Table How to Find the F Critical Value in Excel**