MATH1015 Quizzes

Question 1 Questions
Summary statistics for two samples of data are
 Sample 1: mean$=19$ variance$=10$ Sample 2: mean$=10$ variance$=19$

Which sample has the larger spread of observations? Exactly one option must be correct)
 a) Sample 2. b) Sample 1. c) Neither; they have the same spread. d) There is not enough information to answer the question.

Choice (a) is correct!
The variance of sample 2 is larger.
Choice (b) is incorrect
You should compare variances, not means.
Choice (c) is incorrect
Try again.
Choice (d) is incorrect
Try again.
Consider the following ordered set of data

$44\phantom{\rule{2em}{0ex}}49\phantom{\rule{2em}{0ex}}50\phantom{\rule{2em}{0ex}}51\phantom{\rule{2em}{0ex}}53\phantom{\rule{2em}{0ex}}57\phantom{\rule{2em}{0ex}}58\phantom{\rule{2em}{0ex}}62$
$66\phantom{\rule{2em}{0ex}}66\phantom{\rule{2em}{0ex}}68\phantom{\rule{2em}{0ex}}71\phantom{\rule{2em}{0ex}}75\phantom{\rule{2em}{0ex}}77\phantom{\rule{2em}{0ex}}80\phantom{\rule{2em}{0ex}}85$

What is the IQR?

Correct!
${Q}_{1}=52$ and ${Q}_{3}=73$ so IQR$=21$.
Check your values for the first and third quartiles.
Put your calculator into statistics mode and input the following data,

$15,21,24,16,13,18,$

treating the data as a sample.

What are the sample mean and the sample standard deviation (to 2 decimal places)? (Note that we use ${\sigma }_{n-1}$ when calculating the sample standard deviation.) Exactly one option must be correct)
 a) The mean is 17.83 and the standard deviation is 4.07 . b) The mean is 17.83 and the standard deviation is 3.72 . c) The mean is 17.67 and the standard deviation is 4.07 . d) The mean is 17.67 and the standard deviation is 3.72 .

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Consider the data set. $\begin{array}{cccc}\hfill i\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ ̲& ̲& ̲& ̲\\ \hfill {x}_{i}\hfill & \hfill 1.3\hfill & \hfill 1.7\hfill & \hfill 2.7\hfill \end{array}$ Write down $1.{3}^{2}+1.{7}^{2}+2.{7}^{2}$ in summation notation. Exactly one option must be correct)
 a) ${\sum }_{i=1}^{3}{i}^{2}$ b) ${\left({\sum }_{i=1}^{3}{x}_{i}\right)}^{2}$ c) ${\sum }_{i=1}^{3}{x}_{i}^{2}$ d) ${\left({\sum }_{i=1}^{3}i\right)}^{2}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
${\sum }_{i=1}^{3}{x}_{i}^{2}={x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}=1.{3}^{2}+1.{7}^{2}+2.{7}^{2}\phantom{\rule{0.3em}{0ex}}.$
Choice (d) is incorrect
The table below gives 5 pairs of numbers $\left({x}_{i},{y}_{i}\right)$. $\begin{array}{cccccc}\hfill i\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ ̲& ̲& ̲& ̲& ̲& ̲\\ \hfill {x}_{i}\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill {y}_{i}\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 6\hfill & \hfill 8\hfill & \hfill 10\hfill \end{array}$ What is the value of ${\sum }_{i=1}^{5}{x}_{i}{y}_{i}$ ? Exactly one option must be correct)
 a) 450 b) 45 c) 36 d) 110

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
${\sum }_{i=1}^{3}{x}_{i}{y}_{i}={x}_{1}{y}_{1}+{x}_{2}{y}_{2}+{x}_{3}{y}_{3}+{x}_{4}{y}_{4}+{x}_{5}{y}_{5}=2+8+18+32+50=110$.
Consider the following table: $\begin{array}{cccc}\hfill i\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ ̲& ̲& ̲& ̲\\ \hfill {x}_{i}\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill {y}_{i}\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}$ Which of the following set of statements is true ? Exactly one option must be correct)
 a) $\begin{array}{ccc}\sum _{i=1}^{3}{x}_{i}^{2}=14,\hfill & \sum _{i=1}^{3}{y}_{i}^{2}=21,\hfill & \sum _{i=1}^{3}{x}_{i}{y}_{i}=16,\hfill \\ \sum _{i=1}^{3}{\left({x}_{i}-\overline{x}\right)}^{2}=2,\hfill & \sum _{i=1}^{3}{\left({y}_{i}-\overline{y}\right)}^{2}=2,\hfill & \sum _{i=1}^{3}\left({x}_{i}-\overline{x}\right)\left({y}_{i}-\overline{y}\right)=4\frac{2}{3}.\hfill \end{array}$ b) $\begin{array}{ccc}\sum _{i=1}^{3}{x}_{i}^{2}=21,\hfill & \sum _{i=1}^{3}{y}_{i}^{2}=14,\hfill & \sum _{i=1}^{3}{x}_{i}{y}_{i}=16,\hfill \\ \sum _{i=1}^{3}{\left({x}_{i}-\overline{x}\right)}^{2}=2,\hfill & \sum _{i=1}^{3}{\left({y}_{i}-\overline{y}\right)}^{2}=\frac{14}{3},\hfill & \sum _{i=1}^{3}\left({x}_{i}-\overline{x}\right)\left({y}_{i}-\overline{y}\right)=2.\hfill \end{array}$ c) $\begin{array}{ccc}\sum _{i=1}^{3}{x}_{i}^{2}=14,\hfill & \sum _{i=1}^{3}{y}_{i}^{2}=21,\hfill & \sum _{i=1}^{3}{x}_{i}{y}_{i}=16,\hfill \\ \sum _{i=1}^{3}{\left({x}_{i}-\overline{x}\right)}^{2}=2,\hfill & \sum _{i=1}^{3}{\left({y}_{i}-\overline{y}\right)}^{2}=\frac{14}{3},\hfill & \sum _{i=1}^{3}\left({x}_{i}-\overline{x}\right)\left({y}_{i}-\overline{y}\right)=2.\hfill \end{array}$ d) $\begin{array}{ccc}\sum _{i=1}^{3}{x}_{i}^{2}=14,\hfill & \sum _{i=1}^{3}{y}_{i}^{2}=14,\hfill & \sum _{i=1}^{3}{x}_{i}{y}_{i}=2,\hfill \\ \sum _{i=1}^{3}{\left({x}_{i}-\overline{x}\right)}^{2}=2,\hfill & \sum _{i=1}^{3}{\left({y}_{i}-\overline{y}\right)}^{2}=2,\hfill & \sum _{i=1}^{3}\left({x}_{i}-\overline{x}\right)\left({y}_{i}-\overline{y}\right)=\frac{14}{3}.\hfill \end{array}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Consider the 3 observations
$3,7,11$
for which the mean is 7 and the standard deviation is 4.

If we add 2 to each value, what are the new mean and standard deviation? Exactly one option must be correct)
 a) The mean is 9 and the standard deviation is 2. b) The mean is 9 and the standard deviation is 4. c) The mean is 7 and the standard deviation is 4. d) The mean is 7 and the standard deviation is 6.

Choice (a) is incorrect
Choice (b) is correct!
The mean increases by 2 but the standard deviation remains the same, as it is not affected by a shift of location.
Choice (c) is incorrect
Choice (d) is incorrect
The mean of a sample of $n$ values is $\overline{x}$ and the standard deviation is $s$. Suppose we add a constant value $a$, to each observation so that the new data values are
${x}_{1}+a,{x}_{2}+a,\dots ,{x}_{n}+a.$
Find the new mean and the new standard deviation. Exactly one option must be correct)
 a) The new mean is $\overline{x}+a$ and the new standard deviation is $s+a$. b) The new mean is $\overline{x}+a$ and the new standard deviation is $s$. c) The new mean is $\overline{x}$ and the new standard deviation is $s+a$. d) The new mean is $\overline{x}$ and the new standard deviation is $s$.

Choice (a) is incorrect
Choice (b) is correct!
The mean has $a$ added to it and the standard deviation remains the same. Question 7 gives a specific example of this.
Choice (c) is incorrect
Choice (d) is incorrect
The mean of a sample of $n$ values is $\overline{x}$ and the standard deviation is $s$. Suppose that the observations are multiplied by a constant value $c$, so that the new data values are
$c{x}_{1},c{x}_{2},\dots ,c{x}_{n}.$
Find the new mean and the new standard deviation. Exactly one option must be correct)
 a) The new mean is $\overline{x}$ and the new standard deviation is $cs$. b) The new mean is $c\overline{x}$ and the new standard deviation is $s$. c) The new mean is $\overline{x}$ and the new standard deviation is $s$. d) The new mean is $c\overline{x}$ and the new standard deviation is $cs$.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Both the mean and the standard deviation are affected by a change of scale and are multiplied by $c$.
Consider the following table: $\begin{array}{cccc}\hfill i\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ ̲& ̲& ̲& ̲\\ \hfill {x}_{i}\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill {y}_{i}\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \end{array}$ Using the formula
${S}_{xy}=\sum _{i=1}^{n}{x}_{i}{y}_{i}-\frac{1}{n}\left(\sum _{i=1}^{n}{x}_{i}\right)\left(\sum _{i=1}^{n}{y}_{i}\right),$
find ${S}_{xy}$. Exactly one option must be correct)
 a) ${S}_{xy}=-1$ b) ${S}_{xy}=0$ c) ${S}_{xy}=-2$ d) ${S}_{xy}=2$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
${\sum }_{i=1}^{n}{x}_{i}{y}_{i}=10$, ${\sum }_{i=1}^{n}{x}_{i}=6$, ${\sum }_{i=1}^{n}{y}_{i}=6$
so ${S}_{xy}=10-\frac{1}{3}×36=-2$.
Choice (d) is incorrect