# Confidence interval for ratio of variances

Confidence Interval for ratio of variances

## Confidence Interval for ratio of variances

Let $X_1, X_2, \cdots , X_{n_1}$ be a random sample of size $n_1$ from $N(\mu_1, \sigma_1^2)$ and $Y_1, Y_2, \cdots , Y_{n_2}$ be a random sample of size $n_2$ from $N(\mu_2, \sigma_2^2)$. Moreover, $X$ and $Y$ are independently distributed.

$100(1-\alpha)$% confidence interval estimate of ratio of variances is

 \begin{aligned} \bigg(\frac{s_1^2}{s_2^2}\frac{1}{F_{(\alpha/2,n_1-1,n_2-1)}}, \frac{s_1^2}{s_2^2}\frac{1}{F_{(1-\alpha/2,n_1-1,n_2-1)}}\bigg) \end{aligned}

## Assumptions

a. The two populations are independent.

b. The two samples are simple random samples.

c. The two populations are normally distributed.

## Step by step procedure

Step by step procedure to estimate the confidence interval for the ratio of two population variances is as follows:

#### Step 1 Specify the confidence level

Specify the confidence level $(1-\alpha)$.

#### Step 2 Given information

Specify the given information, sample sizes $n_1$, $n_2$, sample standard deviations $s_1$ and $s_2$.

#### Step 3 Specify the formula

$100(1-\alpha)$% confidence interval for the ratio of variances $\sigma^2_1/\sigma^2_2$ is

 \begin{aligned} \frac{s_1^2}{s_2^2}\cdot\frac{1}{F_{(\alpha/2, n_1-1, n_2-1)}} \leq \frac{\sigma^2_1}{\sigma^2_2} \leq \frac{s_1^2}{s_2^2}\cdot\frac{1}{F_{(1-\alpha/2, n_1-1, n_2-1)}}. \end{aligned}

#### Step 4 Determine the critical value

Find the critical values $F_{(\alpha/2, n_1-1, n_2-1)}$ and $F_{(1-\alpha/2, n_1-1, n_2-1)}$ for desired confidence level and degrees of freedoms.

#### Step 5 Determine the confidence interval

$100(1-\alpha)$% confidence interval estimate for the mean of the difference is

 \begin{aligned} \bigg(\frac{s_1^2}{s_2^2}\frac{1}{F_{(\alpha/2,n_1-1,n_2-1)}}, \frac{s_1^2}{s_2^2}\frac{1}{F_{(1-\alpha/2,n_1-1,n_2-1)}}\bigg) \end{aligned}

Hope you enjoyed the step by step procedure of finding confidence interval for ratio of variances.

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VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.