Maths Statistics

Match the formulae to the corresponding descriptor:

Population variance
Median
Sample variance
A standardised variable
Sample Mean
Zero

$~\frac{\sum\limits_{i=1}^n~(x_i-\bar{x})}{n}~$
$~\frac{\sum\limits_{i=1}^N~(x_i-\mu_x)^2}{N}~$
$~\frac{\sum\limits_{i=1}^n~x_i}{n}~$
$~\frac{x_i-\bar{x}}{s_x}~$
$~\frac{\sum\limits_{i=1}^n~(x_i-\bar{x})^2}{n-1}~$
$~x_{0.5(n+1)}~$ from an ordered series

Car sales in a week, for any given sales person, has the following distribution:
$\begin{tabular}{lllllll} No.~Cars~&~0~&~1~&~2~&~3~&~4~&~5~\\~ Pr~&~0.10~&~0.20~&~0.35~&~0.16~&~0.12~&~0.07% \end{tabular}$
If we take two sales people, what is the probability (3 decimal places) their total sales is greater than 6.

In a particular test individuals get marked on a scale 0 through to 5, any given individual, has the following distribution for their score:
$\begin{tabular}{lllllll} Mark~&~0~&~1~&~2~&~3~&~4~&~5~\\~ Pr~&~0.10~&~0.15~&~0.25~&~0.25~&~0.15~&~0.10% \end{tabular}$
If we take two individuals at random, what is the probability (3 decimal places) their scores differ by more than 2.

An individual who has automobile insurance from a certain company is randomly selected. Let Y be the number of moving violation for which the individual was cited during the last 3 years. The pmf of Y is:
$\begin{tabular}{llllll} &~y~&~~0~&~1~&~~2~&~3~\\ &~p~&~0.60~&~0.25~&~0.10~&~0.05~\\ \end{tabular}$
What is expected value of Y?

An applicance dealer sells three different models of upright freezers having 13.5, 5.9 and 19.1 cubic feet of storage space, respectively. Let X=the amount of storage space purchased by the next customer to buy a freezer. Suppose that X has probability mass function:
$\begin{tabular}{lllllll} &~x~&~13.5~&~15.9~&~19.1~\\ &~P(x)~&~0.2~&~0.5~&~0.3~\\ \end{tabular}$
What is the variance of X? (3 decimal places)

The jurisdiction of a rescue team includes emergencies on a stretch of river that is 4km long. Experience has shown that the distance along this stretch, at which an emergency occurs is equally likely over the range 0 to 4km. Then if X denotes the distance (in km) of an emergency , its probability density function is :
$f(x)=0.25,~0\leq~x\leq~4$
The rescue team’s base is at the mid point of the stretch of river. Find the probability (2 decimal places) that a given emergency arises more than 1.5km from this base.

A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X=the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is:
$f(x)=kx^2,~0\leq~x\leq~2$
What is the value of k? (3 decimal places)

In the year ended June 2003, roughly 20% of New Zealand’s $29,278 million exports were to the United States. Consider 12 randomly selected export units. What is the probability (3 decimal places) that seven of them were sent to countries other than the United States?

Consider the following bivariate distribution between the variable x and y:
$\begin{tabular}{lllll} &~&~~&~y~&~\\ &~~&~11~&~14~&~16~\\~ &~1~&~0.05~&~0.10~&~0.10~\\ x~&~2~&~0.00~&~0.10~&~0.20~\\ &~3~&~0.05~&~0.20~&~0.20~% \end{tabular}$
A random sample of 2 values is drawn from the distribution of Y. Match the values and associated probabilities of the sample mean from these 2 values of the random variable Y:

14.5
13.0
15.0
12.0
11.0
11.5
12.5
13.5
15.5
16.0
14.0

0.01
0.00
0.00
0.16
0.00
0.08
0.00
0.00
0.40
0.10
0.25

Consider the following bivariate distribution between the variable x and y:
$\begin{tabular}{lllll} &~&~~&~y~&~\\ &~~&~11~&~14~&~16~\\~ &~1~&~0.05~&~0.10~&~0.10~\\ x~&~2~&~0.00~&~0.10~&~0.20~\\ &~3~&~0.05~&~0.20~&~0.20~% \end{tabular}$
What is the expected value of the sample mean of Y based on the drawing of two observations (1 decimal place):

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Maths Statistics

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