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 $\text{var}(\vec y)=\text{var}(5\vec x)=5^2\cdot\text{var}(\vec x)=25\cdot 3.2=80$.
A random sample of 25 households from the Mountainview School District was surveyed. In this survey, data were collected on the age of the youngest child living in each household. The histogram below displays the data collected in the survey.
In which of the following intervals is the median of these data located?
Solution. The median of 25 data points is at the 13th position from the minimum (left). Since $4+1+2+5=12$ and $4+1+2+5+3=15$, the median is in the bar between age 8 and age 10.
The distribution of the (historical) daily maximum temperatures in Glassboro in February has a mean $5^{\circ}C$ of and a population standard deviation of $10^{\circ}C$. Express this population standard deviation in terms of Fahrenheit. Use the formula $F = (9/5)C+32$.
Answer:Solution.
The additive part of $F = (9/5)C+32$ does not affect the population standard deviation, so the population standard deviation in Fahrenheit is $\frac{9}{5}\times 10=18^{\circ}F$.