## MATH1005 Quizzes

Quiz 3: Mean, Variance, Correlation and Regression
Question 1 Questions
A set of values, ${x}_{i}$, has ${\sum }_{i=1}^{6}{x}_{i}=40$ and ${\sum }_{i=1}^{6}{x}_{i}^{2}=1600$. What are the sample mean, $\overline{x}$, and sample variance, ${s}^{2}$ ? Exactly one option must be correct)
 a) $\overline{x}=6.6\stackrel{̇}{6}$, ${s}^{2}=0$ b) $\overline{x}=6.6\stackrel{̇}{6}$, ${s}^{2}=1333.\stackrel{̇}{3}$ c) $\overline{x}=6.6\stackrel{̇}{6}$, ${s}^{2}=266.6\stackrel{̇}{6}$ d) $\overline{x}=266.6\stackrel{̇}{6}$, ${s}^{2}=16.32$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$\begin{array}{rcll}\overline{x}& =& \frac{1}{6}\sum _{i=1}^{6}{x}_{i}=\frac{40}{6}=6.6\stackrel{̇}{6}& \text{}\\ {s}^{2}& =& \frac{1}{n-1}\left[\left(\sum _{i=1}^{6}{x}_{i}^{2}\right)-\frac{1}{n}{\left(\sum _{i=1}^{6}{x}_{i}\right)}^{2}\right]& \text{}\\ & =& \frac{1}{5}\left(1600-\frac{1}{6}{\left(40\right)}^{2}\right)& \text{}\\ & =& 266.6\stackrel{̇}{6}& \text{}\end{array}$
Choice (d) is incorrect
Suppose ${\sum }_{i=1}^{7}{x}_{i}=21$ and ${\sum }_{i=1}^{9}{x}_{i}=24\phantom{\rule{0.3em}{0ex}}.$ Find ${x}_{8}+{x}_{9}\phantom{\rule{0.3em}{0ex}}.$ Exactly one option must be correct)
 a) 3 b) -3 c) 45 d) There is not enough information provided.

Choice (a) is correct!
${x}_{8}+{x}_{9}={\sum }_{i=1}^{9}{x}_{i}-{\sum }_{i=1}^{7}{x}_{i}=24-21=3.$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Consider the following frequency table ($i=1,\dots ,4$)

$\begin{array}{ccccc}\hfill {x}_{i}\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill {f}_{i}\hfill & \hfill 3\hfill & \hfill 6\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}$ Find the mean of this data set. Exactly one option must be correct)
 a) 6.25 b) 6 c) 2 d) $2.08\stackrel{̇}{3}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$\overline{x}=\frac{1}{12}{\sum }_{i=1}^{4}{f}_{i}{x}_{i}\phantom{\rule{0.3em}{0ex}}.$
$\overline{x}=\frac{1}{12}\left(3+12+6+4\right)=\frac{25}{12}=2.083\phantom{\rule{0.3em}{0ex}}.$
The following table gives the relation between pairs of data values $\left({x}_{i},{y}_{i}\right)$ for $i=1,\dots ,5\phantom{\rule{0.3em}{0ex}}.$

$\begin{array}{cccccc}\hfill {x}_{i}\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill {y}_{i}\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 6\hfill & \hfill 8\hfill & \hfill 10\hfill \end{array}$ All the points lie on the line $y=bx$ with slope $b$ and correlation coefficient $r$. Find $b$ and $r$. Exactly one option must be correct)
 a) $r=2$, $b=1$ b) $r=2$, $b=2$ c) $r=1$, $b=2$ d) $r=1$, $b=\frac{1}{2}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The line is clearly $y=2x$, so $b=2$. Since the relationship is linear and the slope is positive, $r=1$.
Choice (d) is incorrect
The correlation coefficient $r$ satisfies
$0\le {r}^{2}\le 1.$
Which of the following statements is true ? Exactly one option must be correct)
 a) $-1\le r\le 1$ b) $r\le -1$ c) $r\ge 1$ d) $r\ge 1$ or $r\le -1$

Choice (a) is correct!
In general if ${x}^{2} then $-\sqrt{a}.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Calculate the correlation coefficient, $r$, (to 2dp) for the bivariate data $\left({x}_{i},{y}_{i}\right):\left(0,0\right)\phantom{\rule{1em}{0ex}}\left(3,1.4\right)\phantom{\rule{1em}{0ex}}\left(6,2.6\right)\phantom{\rule{1em}{0ex}}\left(7,3.8\right)\phantom{\rule{1em}{0ex}}\left(9,7.2\right).$ You may use the totals: ${\sum }_{i}{x}_{i}^{2}=175$ and ${\sum }_{i}{y}_{i}^{2}=75$. Exactly one option must be correct)
 a) $0.93$ b) $0.09$ c) $0.97$ d) None of these.

Choice (a) is correct!
${\sum }_{i}{x}_{i}=25,{\sum }_{i}{y}_{i}=15,{\sum }_{i}{x}_{i}{y}_{i}=111.2$ so ${S}_{xx}=50,{S}_{yy}=30$ and ${S}_{xy}=36.2$ and $r=\frac{36.2}{\sqrt{50×30}}=0.93$ to 2dp.
Choice (b) is incorrect
Check your calculations for ${S}_{xx}$, ${S}_{yy}$ and ${S}_{xy}$.
Choice (c) is incorrect
Check your calculations for ${S}_{xx}$, ${S}_{yy}$ and ${S}_{xy}$.
Choice (d) is incorrect
Check your calculations for ${S}_{xx}$, ${S}_{yy}$ and ${S}_{xy}$.
The correlation coefficient for a set of bivariate data $\left({x}_{i},{y}_{i}\right)$ is $r=0.87$, where the ${x}_{i}$ are measured in inches and the ${y}_{i}$ are measured in lbs. A second analyst records the ${x}_{i}$ values in cm. (1 inch $\approx$ 2.5 cm). What is the second analyst’s value of the correlation coefficient (to 2dp)? Exactly one option must be correct)
 a) $0.35$ b) $0.87$ c) $2.18$ d) Unable to determine without knowing the ${y}_{i}$ units.

Choice (a) is incorrect
How do you adjust $r$ for a change of units?
Choice (b) is correct!
$r$ is not affected by a change of units.
Choice (c) is incorrect
$r$ cannot exceed 1.
Choice (d) is incorrect
How do you adjust $r$ for a change of units?
Find the correlation coefficient for 6 pairs of observations if the LSR line is $y=0.5-0.05x$ and if $81%$ of the variation in $y$ is explained by regression on $x$. Exactly one option must be correct)
 a) $0.9$ b) $0.81$ c) $-0.05$ d) None of these.

Choice (a) is incorrect
Check the slope of the LSR line.
Choice (b) is incorrect
This is ${r}^{2}$.
Choice (c) is incorrect
Try again.
Choice (d) is correct!
The slope is negative and ${r}^{2}=0.81$, so $r=-0.9$.
For the bivariate data $\left({x}_{1},{y}_{1}\right)\phantom{\rule{1em}{0ex}}\left({x}_{2},{y}_{2}\right)\phantom{\rule{1em}{0ex}}\cdots \left({x}_{n},{y}_{n}\right)$, the least squares regression line is fitted. The line is $y=2.51-4.1x$. You know that the first data point is $\left({x}_{1},{y}_{1}\right)=\left(0.1,2.0\right)$, so the residual at this point is: Exactly one option must be correct)
 a) $2.1$ b) $-0.1$ c) $0.1$ d) $2.0$

Choice (a) is incorrect
You have found the $y$ value on the LSR line at $x=0.1$.
Choice (b) is correct!
At $x=0.1$, the value of $y$ on the LSR line is $ŷ=2.51-0.41=2.1$. Therefore the residual at $x=0.1$ is $2.0-2.1=-0.1$.
Choice (c) is incorrect
Try again.
Choice (d) is incorrect
Try again.
A correlation coefficient of $r=0.8$ is reported for a sample of pairs $\left({x}_{i},{y}_{i}\right)$. Without any further information, this implies that: Exactly one option must be correct)
 a) as the $x$ values decrease, the $y$ values increase. b) $80%$ of the variation in $y$ is due to regression on $x$. c) the $\left({x}_{i},{y}_{i}\right)$ are scattered about a straight line of unknown positive slope. d) the $\left({x}_{i},{y}_{i}\right)$ are scattered about a straight line of slope $0.8$.

Choice (a) is incorrect
This would be the answer for a negative value of $r$.
Choice (b) is incorrect
This is the interpretation of ${r}^{2}$ not of $r$.
Choice (c) is correct!
Choice (d) is incorrect
$r$ is not the slope of the LSR line.