Understanding the F Ratio Calculation
The F ratio is commonly used in statistical tests (like ANOVA) to compare the variances of two samples. It is calculated as the ratio of the larger variance to the smaller variance.
Identifying Sample Variances
Given two yarn samples with standard deviations $\sigma_1$ and $\sigma_2$, their respective variances are:
- Variance 1: ${\sigma_1}^2$
- Variance 2: ${\sigma_2}^2$
Applying the Condition for F Ratio
We are given the condition that $\sigma_1 < \sigma_2$. Squaring both sides of this inequality preserves the direction:
${\sigma_1}^2 < {\sigma_2}^2$
This means ${\sigma_2}^2$ is the larger variance and ${\sigma_1}^2$ is the smaller variance.
Calculating the F Ratio
Following the definition, the F ratio is the larger variance divided by the smaller variance:
F Ratio = $\frac{\text{Larger Variance}}{\text{Smaller Variance}}$
Substituting the identified variances:
F Ratio = $\frac{{\sigma_2}^2}{{\sigma_1}^2}$
Therefore, the correct F ratio is ${\sigma_2}^2 / {\sigma_1}^2$.
Important Questions from Variance
- Consider a random variable $X$ with mean $\mu_X = 0.1$ and variance $\sigma_X^2 = 0.2$. A new random variable $Y = 2X + 1$ is defined. The variance of the random variable $Y$ (rounded off to one decimal place) is ________________.
- Variance of the sum of two statistically independent random variables $X$ and $Y$, $\sigma_{X+Y}^2$, is
A continuous random variable $x$ has a probability density function given by
$f(x) = e^{-a|x|} \text{ } (-\infty < x < \infty)$
where $a$ is a real constant. The variance of $x$ is __________ (correct up to one decimal place).
- Two yarns have variance of strength as $V_1$ and $V_2$. If $V_1 < V_2$, the variance ratio 'F' would be
People were prohibited ________ their vehicles near the entrance of the main administrative building.
