## MATH1005 Quizzes

Quiz 7: Sampling distributions
Question 1 Questions
Suppose ${X}_{1},\dots ,{X}_{n}$ are independent and each ${X}_{i}$ has mean $\mu$ and variance ${\sigma }^{2}$. If $\overline{X}=\frac{1}{n}{\sum }_{i=1}^{n}{X}_{i}$, what is the distribution of $\overline{X}$ when $n$ is large ? Exactly one option must be correct)
 a) $N\left(n\mu ,\frac{{\sigma }^{2}}{n}\right)$ b) $N\left(n\mu ,{\sigma }^{2}\right)$ c) $N\left(\mu ,\frac{{\sigma }^{2}}{n}\right)$ d) $N\left(\mu ,{\sigma }^{2}\right)$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Note that for large $n$ we use the central limit theorem.
Choice (d) is incorrect
Let ${X}_{1},\dots ,{X}_{n}$ be a random sample from a population with mean $\mu$ and variance ${\sigma }^{2}$. If $\overline{X}=\frac{1}{n}{\sum }_{i=1}^{n}{X}_{i}$ is the sample mean, then what is the standard deviation of $\overline{X}$ ? Exactly one option must be correct)
 a) $\frac{\sigma }{\sqrt{n}}$ b) $\frac{{\sigma }^{2}}{n}$ c) $\frac{\sigma }{n}$ d) $\sigma$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Suppose ${X}_{1},\dots ,{X}_{n}$ are independent and ${X}_{i}\sim N\left(5,{3}^{2}\right)$ for $i=1,\dots ,n$. If $\overline{X}=\frac{1}{n}{\sum }_{i=1}^{n}{X}_{i}$, what is the distribution of $\overline{X}$ ? Exactly one option must be correct)
 a) $\overline{X}\sim N\left(5,{3}^{2}\right)$ b) $\overline{X}\sim N\left(5,\frac{{3}^{2}}{n}\right)$ c) $\overline{X}\sim N\left(5,{\left(\frac{3}{n}\right)}^{2}\right)$ d) $\overline{X}\sim N\left(5,\frac{3}{\sqrt{n}}\right)$

Choice (a) is incorrect
Choice (b) is correct!
Each ${X}_{i}$ has mean 5 and variance ${3}^{2}$ hence $\overline{X}$ has mean 5 and variance $\frac{{3}^{2}}{n}$ and so $\overline{X}\sim N\left(5,\frac{{3}^{2}}{n}\right)$.
Choice (c) is incorrect
Choice (d) is incorrect
Suppose ${X}_{1},\dots ,{X}_{n}$ are independent and ${X}_{i}\sim N\left(3,{5}^{2}\right)$ for $i=1,\dots ,n\phantom{\rule{0.3em}{0ex}}.$ If $\overline{X}=\frac{1}{n}{\sum }_{i=1}^{n}{X}_{i}$, what is the distribution of $\overline{X}$ when $n=25$ ? Exactly one option must be correct)
 a) $\overline{X}=N\left(3,1\right)$ b) $\overline{X}=N\left(3,5\right)$ c) $\overline{X}=N\left(3,25\right)$ d) $\overline{X}=N\left(5,1\right)$

Choice (a) is correct!
Each ${X}_{i}$ has mean 3 and variance ${5}^{2}$ so $\overline{X}\sim N\left(3,\frac{{5}^{2}}{25}\right)=N\left(3,1\right)\phantom{\rule{0.3em}{0ex}}.$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Consider the random variable $X$ with distribution $N\left(3,1\right)$. What is the value of $P\left(X>3.6\right)\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) 0.4522 b) 0.6058 c) 0.7258 d) 0.2742

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$\begin{array}{rcll}P\left(\overline{X}>3.6\right)& =& P\left(\frac{X-\mu }{\sigma }>\frac{3.6-\mu }{\sigma }\right)=P\left(Z>0.6\right)& \text{}\\ & =& 1-P\left(Z<0.6\right)& \text{}\\ & =& 1-0.7258.& \text{}\\ & =& 0.2742\phantom{\rule{0.3em}{0ex}}.& \text{}\end{array}$
Suppose ${X}_{1},\dots ,{X}_{n}$ are independent and ${X}_{i}\sim N\left(5,{3}^{2}\right)$ for $i=1,\dots ,n\phantom{\rule{0.3em}{0ex}}.$ If $\overline{X}=\frac{1}{n}{\sum }_{i=1}^{n}{X}_{i}$, what is the value of $P\left(\overline{X}>2\right)$ if $n=36\phantom{\rule{0.3em}{0ex}}$ ? Exactly one option must be correct)
 a) $\approx 0.99995$ b) $<0.0001$ c) 0.8413 d) 0.1587

Choice (a) is correct!
$\overline{X}\sim N\left(5,0.25\right)=N\left(5,{\left(0.5\right)}^{2}\right)\phantom{\rule{0.3em}{0ex}}.$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
A fair coin is tossed 30 times. The best approximation for the distribution of the number of heads is Exactly one option must be correct)
 a) $N\left(30,\frac{1}{2}\right)$ b) $N\left(15,7.5\right)$ c) $N\left(30,7.5\right)$ d) $N\left(15,\frac{7.5}{30}\right)$

Choice (a) is incorrect
Choice (b) is correct!
$X\sim B\left(30,0.5\right)\sim N\left(np,np\left(1-p\right)\right)=N\left(15,7.5\right)\phantom{\rule{0.3em}{0ex}}.$
Choice (c) is incorrect
Choice (d) is incorrect
Suppose $\overline{X}\sim N\left(15,\frac{7.5}{30}\right)$. What is $P\left(\overline{X}>18\right)\phantom{\rule{0.3em}{0ex}}$ ? Exactly one option must be correct)
 a) $\Phi \left(3\right)$ b) $1-\Phi \left(3\right)$ c) $\Phi \left(3\right)$ d) $1-\Phi \left(6\right)$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$\overline{X}\sim N\left(15,.{5}^{2}\right)$.
Hence, $P\left(\overline{X}>18\right)=P\left(Z>\frac{18-15}{0.5}\right)$
$=P\left(Z>6\right)=1-P\left(Z<6\right)=1-\Phi \left(6\right)\phantom{\rule{0.3em}{0ex}}.$
Note that there are no continuity corrections for $\overline{X}$.
Suppose $X\sim B\left(30,0.5\right)$. Use a normal approximation for $B$ to find $P\left(X>20\right)$. Exactly one option must be correct)
 a) $\Phi \left(2.008\right)$ b) $1-\Phi \left(1.643\right)$ c) $1-\Phi \left(2.008\right)$ d) $1-\Phi \left(1.826\right)$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$X\sim B\left(30,0.5\right)$, $Y\sim N\left(15,7.5\right)$ $\begin{array}{rcll}P\left(X>20\right)& =& P\left(X\ge 21\right)\phantom{\rule{1em}{0ex}}\left(use\phantom{\rule{1em}{0ex}}\ge \phantom{\rule{1em}{0ex}}for\phantom{\rule{1em}{0ex}}binomial\phantom{\rule{1em}{0ex}}distribution\right)& \text{}\\ & =& P\left(Y>20.5\right)\phantom{\rule{1em}{0ex}}\left(continuity\phantom{\rule{1em}{0ex}}correction\right)& \text{}\\ & =& P\left(Z>\frac{20.5-15}{\sqrt{7.5}}\right)& \text{}\\ & =& P\left(Z>2.008\right)& \text{}\\ & =& 1-P\left(Z<2.008\right)& \text{}\\ & =& 1-\Phi \left(2.008\right)\phantom{\rule{0.3em}{0ex}}.& \text{}\end{array}$
Choice (d) is incorrect
Suppose $X\sim B\left(30,0.5\right)$. Use a normal approximation for $B$ to find $P\left(X<13\right)$. Exactly one option must be correct)
 a) $\Phi \left(0.913\right)$ b) $\Phi \left(0.548\right)$ c) $1-\Phi \left(0.913\right)$ d) $1-\Phi \left(0.548\right)$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$X\sim B\left(30,0.5\right)$, $Y\sim N\left(15,7.5\right)$ $\begin{array}{rcll}P\left(X<13\right)& =& P\left(X\le 12\right)\phantom{\rule{1em}{0ex}}\left(we\phantom{\rule{1em}{0ex}}must\phantom{\rule{1em}{0ex}}use\phantom{\rule{1em}{0ex}}\le \phantom{\rule{1em}{0ex}}for\phantom{\rule{1em}{0ex}}a\phantom{\rule{1em}{0ex}}binomial\phantom{\rule{1em}{0ex}}distribution\right)& \text{}\\ & =& P\left(Y<12.5\right)\phantom{\rule{1em}{0ex}}\left(continuity\phantom{\rule{1em}{0ex}}correction\right)& \text{}\\ & =& P\left(Z<\frac{12.5-15}{\sqrt{7.5}}\right)=P\left(Z<-0.913\right)& \text{}\\ & =& P\left(Z>0.913\right)& \text{}\\ & =& 1-P\left(Z<0.913\right)& \text{}\\ & =& 1-\Phi \left(0.913\right)\phantom{\rule{0.3em}{0ex}}.& \text{}\end{array}$
Choice (d) is incorrect