A set of 11 (x, y) data points is least-squares fitted to a quadratic polynomial. If the sum of squares of error is 2.4, the variance of error is ________ (round off to 1decimal place).
Variance of Error Calculation
This solution explains how to find the variance of error when fitting a quadratic polynomial using the least-squares method.
Quadratic Fit Details
- Number of data points, $n = 11$.
- Model: Quadratic polynomial ($y = ax^2 + bx + c$). This implies $p = 3$ estimated parameters.
- Sum of Squares of Error (SSE) is given as 2.4.
Error Degrees of Freedom
Degrees of freedom (df) for error are calculated as the number of data points minus the number of parameters:
$ df = n - p $
With $n=11$ and $p=3$:
$ df = 11 - 3 = 8 $
Final Variance Calculation
The variance of the error (also known as Mean Squared Error or MSE) is SSE divided by the degrees of freedom:
$ \text{Variance of Error} = \frac{SSE}{df} $
Substituting the given values:
$ \text{Variance of Error} = \frac{2.4}{8} = 0.3 $
The variance of the error, rounded to one decimal place, is 0.3.
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)
