# Examples of questions for the written test for Mathematical Statisticians

The written test is evaluating knowledge AND ability to communicate in writing. It consists of two parts. Part A contains one question to test writing ability. Part B tests knowledge and contains 15 multiple choice questions, 15 fill-in-the-blank questions and 2 open questions. There is no break period between the two parts of the test.

Examples of questions similar to those found on the test are given below.

Please note that tests from previous years are not available.

## Part A - Writing Ability

Example 1
Prepare a letter of approximately 200 to 300 words to the director of recruitment for Statistics Canada in which you explain how your training, work experience and interpersonal skills make you a strong candidate for a position as a mathematical statistician.

Example 2
In 2002 and 2007, public health officials conducted a survey on the lifestyle choices of members of your community. The following table shows an extract of the official results of that survey:

Survey on the lifestyle choices of members of your community
Year Population (Number of Adults) Estimated number of "smoking" adults Estimated number of adults with "hypertension" Estimated number of "smoking adults with "hypertension"
2007 5,000 300 350 250
2002 4,000 400 400 200

As a journalist involved in community affairs, you have followed this story closely from the beginning. Therefore, your editor-in-chief has asked you to write an article for your newspaper which explains the survey results to your readers.

## Part B - Knowledge

### Probability and Statistics

1. In a batch of 10 items, we wish to extract a sample of 3 without replacement. How many different samples can we extract?

Answer: 10! / (7!*3!) = 10*9*8 / (3*2*1) = 120

2. The difference between the parameter we wish to estimate and the expected value of its estimator is __________.

3. Let X and Y be independent random variables. Suppose the respective expected values are E(X) = 8 and E(Y) = 3 and the respective variances are V(X) = 9 and V(Y) = 6. Let Z be defined as Z = 2X – 3Y +5. Based on these data, the value of E(Z) is _____ and the value of V(Z) is _____.

4. Which of the following statements about the X2 (Chi-square) distribution is always false?

1. The X2 distribution is asymmetrical.
2. The variance of a random variable having a X2 distribution is twice its mean.
3. If X1 and X2 are two independent random variables with a X2 distribution with n1 and n2 degrees of freedom respectively, then the variable Y = X1 + X2 has an F (Fisher) distribution with n1 and n2 degrees of freedom.
4. If X1 , …, Xn are independent random variables having a normal distribution N(0,1), then X12 +…+ Xn2 has a X2 distribution with n degrees of freedom.
5. The X2 distribution is dependent only on a single parameter.

### Sampling

5. Single stage cluster sampling is more precise than simple random sampling when the ___________ is negative.

6. In order to estimate the total for a variable of interest, a simple random sample without replacement of size N/4 from a population of size N is sought. After some thought, it is decided that a simple random sample without replacement of size N/2, instead, will be drawn from the same population. By what factor is the variance of this estimate reduced with this increased sample size?

### Mathematics

7. The inverse of the matrix $X=5\left[\begin{array}{cc}2& 5\\ 4& -1\end{array}\right]$ is ___________.

Answer: ${X}^{-1}=\frac{1}{5}\left[\begin{array}{cc}\frac{1}{22}& \frac{5}{22}\\ \frac{2}{11}& \frac{-1}{11}\end{array}\right]$ or ${X}^{-1}=\left[\begin{array}{cc}\frac{1}{110}& \frac{1}{22}\\ \frac{2}{55}& \frac{-1}{55}\end{array}\right]$

### Data Analysis

8. The primary goal of principal component analysis is to:

1. Divide a set of multivariate observations into classes.
2. Assign a particular multivariate observation to one of several classes.
3. Characterize the correlation structure between two sets of variables by replacing them by two smaller sets of variables which are highly correlated.
4. Find the variables among a set of predictor variables that are the best predictors of a set of variables of interest.
5. Explain the variability in a large set of variables by replacing it by a smaller set of transformed variables that explains a large portion of the total variability.