This test is a written exam that evaluates 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:

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.

Please write this article in **200 to 300 words**.

## 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 __________.

*Answer:* the bias

**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 _____.

*Answer:* 12 and 90

**4.** Which of the following statements about the X^{2} (Chi-square) distribution is always false?

- The X
^{2}distribution is asymmetrical. - The variance of a random variable having a X
^{2}distribution is twice its mean. - If X
_{1}and X_{2}are two independent random variables with a X^{2}distribution with n_{1}and n_{2}degrees of freedom respectively, then the variable Y = X_{1}+ X_{2}has an F (Fisher) distribution with n_{1}and n_{2}degrees of freedom. - If X
_{1}, …, X_{n}are independent random variables having a normal distribution N(0,1), then X_{1}^{2}+…+ X_{n}^{2}has a X^{2}distribution with n degrees of freedom. - The X
^{2}distribution is dependent only on a single parameter.

*Answer:* C

### Sampling

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

*Answer:* intra-cluster correlation

**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?

*Answer:* 3

**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:

- Divide a set of multivariate observations into classes.
- Assign a particular multivariate observation to one of several classes.
- Characterize the correlation structure between two sets of variables by replacing them by two smaller sets of variables which are highly correlated.
- Find the variables among a set of predictor variables that are the best predictors of a set of variables of interest.
- 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.

*Answer:* E

### Open Question

**9.** A basic question in planning a sample survey is the size of the sample that is required. In your opinion, what factors should be considered when determining the size of the sample and how does each one affect the sample size?