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)

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is correct!*

$$\begin{array}{rcll}\overline{x}& =& \frac{1}{6}\sum _{i=1}^{6}{x}_{i}=\frac{40}{6}=6.6\stackrel{\u0307}{6}& \text{}\\ {s}^{2}& =& \frac{1}{n-1}\left[\left(\sum _{i=1}^{6}{x}_{i}^{2}\right)-\frac{1}{n}{\left(\sum _{i=1}^{6}{x}_{i}\right)}^{2}\right]& \text{}\\ & =& \frac{1}{5}\left(1600-\frac{1}{6}{\left(40\right)}^{2}\right)& \text{}\\ & =& 266.6\stackrel{\u0307}{6}& \text{}\end{array}$$

*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)

*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)

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

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is correct!*

$\overline{x}=\frac{1}{12}{\sum}_{i=1}^{4}{f}_{i}{x}_{i}\phantom{\rule{0.3em}{0ex}}.$

$\overline{x}=\frac{1}{12}\left(3+12+6+4\right)=\frac{25}{12}=2.083\phantom{\rule{0.3em}{0ex}}.$

$\overline{x}=\frac{1}{12}\left(3+12+6+4\right)=\frac{25}{12}=2.083\phantom{\rule{0.3em}{0ex}}.$

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)

$$\begin{array}{cccccc}\hfill {x}_{i}\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill {y}_{i}\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 6\hfill & \hfill 8\hfill & \hfill 10\hfill \end{array}$$ 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)

*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)
*Choice (a) is correct!*

In general if ${x}^{2}<a$
then $-\sqrt{a}<x<\sqrt{a}$.

*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)

*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
$r=\frac{36.2}{\sqrt{50\times 30}}=0.93$ 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)

*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)

*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)

*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 $\u0177=2.51-0.41=2.1$. 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)

*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.