MATH1005 Quizzes

Quiz 6: Normal distributions
Question 1 Questions
Which of the following random variables would you expect to be discrete? Exactly one option must be correct)
 a) The weights of mechanically produced items. b) The number of children at a Christmas party. c) The lifetimes of resistors. d) The distance between Centre Point tower and any point on the Australian continent. e) The times, in seconds, for a 100m sprint.

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Let $X\sim N\left(3,{2}^{2}\right)$. What does this tell us about the distribution of $X$ ? Exactly one option must be correct)
 a) $X$ is binomial with $n=3$ and $p=2$. b) $X$ is normal with mean 3 and variance 4. c) $X$ is normal with mean 3 and variance 2. d) $X$ is binomial with mean 2 and variance 9.

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
$X$ is a random normal variable, with mean $\mu$ and variance ${\sigma }^{2}$. The “standardised form” of $X$ is $Z=\frac{X-\mu }{\sigma }$.
What are the mean and variance, respectively, of $Z$ ? Exactly one option must be correct)
 a) 0$,$1 b) 2$,$1 c) 1$,$0 d) 2$,$0

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Let $X\sim N\left(5,{3}^{2}\right)$. Which of the following is a standard normal variable ? Exactly one option must be correct)
 a) $Z=\frac{X-5}{5}$ b) $Z=\frac{X-3}{5}$ c) $Z=\frac{X-5}{3}$ d) $Z=\frac{X-3}{3}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Suppose $X\sim N\left(5,{3}^{2}\right)$. What is $P\left(X\le 8\right)$ in terms of the standard normal variable $Z$ ? Exactly one option must be correct)
 a) $P\left(Z\le 1\right)$ b) $P\left(Z\le -1\right)$ c) $P\left(Z\le 0.6\right)$ d) $P\left(Z\le -0.6\right)$ e) $P\left(Z\le 1.67\right)$

Choice (a) is correct!
$P\left(X\le 8\right)=P\left(\frac{X-\mu }{\sigma }\le \frac{8-\mu }{\sigma }\right)=P\left(Z\le \frac{3}{3}\right)=P\left(Z\le 1\right)$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Suppose $X\sim N\left(5,{3}^{2}\right)$. What is the value of $P\left(X\le 2\right)$ ? Exactly one option must be correct)
 a) 0.8413 b) 0.1587 c) 0.7258 d) 0.2742

Choice (a) is incorrect
Choice (b) is correct!
$P\left(X\le 2\right)=P\left(Z\le \frac{2-5}{3}\right)=P\left(Z\le -1\right)=1-P\left(Z\le 1\right)$
$=1-0.8413=0.1587\phantom{\rule{0.3em}{0ex}}.$
Choice (c) is incorrect
Choice (d) is incorrect
Suppose $X$ is normally distributed with mean 5 and standard deviation 0.4 . Using the standard transformation $Z=\frac{X-\mu }{\sigma }$ we find $P\left(X\le {X}_{0}\right)=P\left(Z\le 1.3\right)$. What is the value of ${X}_{0}$ ? Exactly one option must be correct)
 a) 6.9 b) 4.48 c) 2 d) 5.52

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$\begin{array}{rcll}\frac{{X}_{0}-5}{0.4}& =& 1.3,& \text{}\\ {X}_{0}-5& =& 0.52,& \text{}\\ {X}_{0}& =& 5.52.& \text{}\end{array}$
Suppose $X$ is normally distributed with mean 5. If $P\left(X\le 3\right)=0.2$ what is the standard deviation of $X$ ? Exactly one option must be correct)
 a) $\sigma =0.42$ b) $\sigma =-0.42$ c) $\sigma =0.38$ d) $\sigma =2.38$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$P\left(X\le 3\right)=P\left(Z\le \frac{3-5}{\sigma }\right)=0.2$
$\text{therefore,}P\left(Z\le \frac{5-3}{\sigma }\right)=0.8\phantom{\rule{0.3em}{0ex}}.$
But $P\left(Z\le 0.84\right)=0.8⇒\frac{5-3}{\sigma }=0.84⇒\sigma =2.38\phantom{\rule{0.3em}{0ex}}.$
Suppose that $X\sim N\left(2,1\right)$ and $Y\sim N\left(3,2\right)$. Assuming $X$ and $Y$ are independent what is the distribution of $X+Y$ ? Exactly one option must be correct)
 a) $N\left(3,5\right)$ b) $N\left(5,3\right)$ c) $N\left(3,3\right)$ d) $N\left(5,5\right)$

Choice (a) is incorrect
Choice (b) is correct!
The mean of $X+Y$ is $2+3=5\phantom{\rule{0.3em}{0ex}}.$ The variance of $X+Y$ is $1+2=3\phantom{\rule{0.3em}{0ex}}.$
Choice (c) is incorrect
Choice (d) is incorrect
$X$ and $Y$ are independent random variables. The mean and variance of $X$ are 2 and 1 respectively. The mean and variance of $Y$ are 3 and 2 respectively.

Which of the statements below about the random variable $X-Y$ is true ? Exactly one option must be correct)
 a) $X-Y\sim N\left(-1,1\right)$ b) $X-Y$ has mean 5 and variance 3. c) $X-Y\sim N\left(-1,3\right)$ d) $X-Y$ has mean $-1$ and variance 3.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
We do not know that $X$ and $Y$ are normally distributed. The mean of $X-Y$ is $2-3=-1$. The variance of $X-Y$ is $1+{\left(-1\right)}^{2}\phantom{\rule{0.3em}{0ex}}2=3\phantom{\rule{0.3em}{0ex}}.$