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 distribution with the following probability density function $$f(x) = \begin{cases} 0.5, & 0 < x < 2 \\ 0.0, & Otherwise \end{cases}$$ Given that the mean of the above probability distribution is 1, the variance (rounded off to two decimal places) is _______________.
- Let $X$ and $Y$ be two independent random variables. $X$ follows $Bernoulli(p = 0.3)$ distribution and $Y$ follows $Normal(\mu = 0, \sigma^2 = 100)$ distribution.
Which of the following options is the variance of $(2X - 1)Y$? - For a given data set $\{x_1, x_2, \ldots, x_n\}$, where $n = 100$, it is known that
$$ \frac{1}{2000} \sum_{i=1}^{n} \sum_{j=1}^{n} (x_i - x_j)^2 = 99 $$
Let us denote $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$.
The value of $\frac{1}{99} \sum_{i=1}^{n} (x_i - \bar{x})^2$ is __________ . (Answer in integer) - A random variable $X$ has the sample space $\{0,1\}$. The probability $P(X = 0) = 1/4$ and $P(X = 1) = 3/4$.
What is the variance of the random variable?
Hint: $\text{Mean } (\mu) = \sum_{i=1}^{n} x_i p(x_i) ; \text{Variance } (\sigma^2) = \sum_{i=1}^{n} (x_i - \mu)^2 p(x_i)$ The unbiased sample variance for the set of numbers: $S = \{40,45,50,55,60\}$ is_____. (write answer with one decimal place)
