Find the mean of the dataset $(2, 4, 6, 12)$.
Answer:Solution.
$$ \mu=\dfrac{2+4+6+12}{4}=\dfrac{24}{4}=6 $$
Find the population variance of the dataset $\vec x=(2, 4, 6, 12)$.
Answer:Solution.
Since $\mu=\dfrac{2+4+6+12}{4}=6$, the population variance is $$ \begin{align*} \sigma^2 &= \frac{1}{4}\|(2,4,6,12)-(6,6,6,6)\|^2\\[1em] &= \frac{1}{4}\|(-4,-2,0,6)\|^2\\[1em] &= \frac{1}{4}\left[(-4)^2+(-2)^2+0^2+6^2\right]\\[1em] &= \frac{1}{4}\left[16+4+0+36\right]\\[1em] &= \frac{56}{4}= 14\\[1em] \end{align*} $$
Find the median of the dataset $(2, 4, 6, 12)$.
Answer:Solution.
Since the middle numbers are $4$ and $6$, we average their results to get the median $\dfrac{4+6}{2}=5$.
The dataset $(2, 3, x, 5)$ has a mean of $7$. What is $x$?
Answer:Solution.
$$ 7=\dfrac{2+3+x+5}{4}=\dfrac{x+10}{4}\quad\Rightarrow\quad 7\times 4-10=x $$ so $x=18$.
The dataset $(2, 3, 3, 5, 6)$ has a population variance of $3.2$. What is the population variance of the dataset $(3, 4, 4, 6, 7)$?
Answer:Solution.
The new dataset is an additive shift of the old dataset; this does not change the variance.
The dataset $(2, 3, 3, 5, 6)$ has a population variance of $3.2$. What is the population variance of the dataset $(10, 15, 15, 25, 30)$?
Answer:Solution.
The new dataset $\vec y = (10, 15, 15, 25, 30)$ is 5x the old dataset $\vec x=(2, 3, 3, 5, 6)$, namely $\vec y = 5\vec x$; therefore $\text{var}(\vec y)=\text{var}(5\vec x)=5^2\cdot\text{var}(\vec x)=25\cdot 3.2=80$.
TODO: Celsius to Farenheit via reverse formula $F = (9/5)C+32$
Answer:Solution.
TODO.