Improving Lectures in Statistics and the Use of
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Appendix D  Samples of question types

Question A function $ f:\mathbb{R} \to \mathbb{R}$ is given by
$\displaystyle f(x) = \left\{ \begin{array}{ll} 0 & x \in (-\infty, -1] \\ \alpha x^2 & x \in (-1,1) \\ 0 & x \in [1,\infty) \end{array} \right. $
What value must the parameter $ \alpha$ have so that $ f$ is a probability density function?
Answer 1 $ \frac{1}{2}$
Feedback No. Integrate the function on $ (-\infty,\infty)$ and then set the integral equal to 1.
Answer 2 1
Feedback No. Integrate the function on $ (-\infty,\infty)$ and then set the integral equal to 1.
Answer 3 $ \frac{3}{2}$
Feedback Correct.
Answer 4 2
Feedback No. Integrate the function on $ (-\infty,\infty)$ and then set the integral equal to 1.

Example 1: Recall a definition or formula and use it.

Question An optical character recognition software recognizes a character with an average error of 5%. What is the probability that it does not recognize correctly a word with 6 characters (i.e. it identifies at least one character incorrectly).
Answer 1 66.7%
Feedback No. Try to calculate the probability of identifying the word correctly and then take the probability of the complement.
Answer 2 73.5%
Feedback Correct.
Answer 3 60.0%
Feedback No. Try to calculate the probability of identifying the word correctly and then take the probability of the complement.
Answer 4 76.3%
Feedback No. Try to calculate the probability of identifying the word correctly and then take the probability of the complement.
Example 2: Substitute data into formula.

Question The height of an adult man is a random variable with normal distribution $ N(180,100)$. What is the probability that the average height in a randomly chosen group of 10 men will be between 175 cm and 180 cm?
Answer 1 41.2%
Feedback No. Remember $ \overline X \sim N(\mu, \frac{\sigma^2}{n})$.
Answer 2 44.3%
Feedback Correct.
Answer 3 47.6%
Feedback No. Remember $ \overline X \sim N(\mu, \frac{\sigma^2}{n})$.
Answer 4 51.2%
Feedback No. Remember $ \overline X \sim N(\mu, \frac{\sigma^2}{n})$.
Example 3: Carry out an algorithm.

Question The yield of a stock in a year is a random variable with normal distribution with mean $ \mu = 8\%$ and standard deviation $ \sigma = 10\%$ (the standard deviation is in this case called volatility of the stock). How many different stocks should we at least choose to our equally weighted portfolio, so that the average return is not negative, presuming the independence of the stock prices?
Answer 1 5
Feedback Correct.
Answer 2 6
Feedback No. We solve $ P(0<\overline X)=0.95$. Standardise $ \overline X$ and then solve for $ n$.
Answer 3 8
Feedback No. We solve $ P(0<\overline X)=0.95$. Standardise $ \overline X$ and then solve for $ n$.
Answer 4 9
Feedback No. We solve $ P(0<\overline X)=0.95$. Standardise $ \overline X$ and then solve for $ n$.
Example 4: Apply an algorithm to solve a word problem.

Question Which of the following is not an example of a linear regression model?
Answer 1 $ Y = \beta_0 + \beta_1X + \epsilon$
Feedback No. This is a linear model.
Answer 2 $ Y = \beta_0 + \beta_1X + \beta_2X^2 + \beta_3X^3 + \epsilon$
Feedback No. This is a linear model. The unknown parameters $ \beta_i$ are linear.
Answer 3 $ Y = \beta_0 + \beta_1X + \beta_2^2X^2 + \epsilon$
Feedback Correct.
Answer 4 $ Y = \beta_0 + \beta_1 e^X + \epsilon$
Feedback No. This is a linear model. The unknown parameters $ \beta_i$ are linear.
Example 5: Classify mathematical objects.

Question Choose the correct answer: A random variable...
Answer 1 ...assigns a real value to every outcome of the randomexperiment.
Feedback Correct.
Answer 2 ...shows the probabilities of every possible outcomeoftherandomexperiment.
Feedback No. That is a density or frequency function.
Answer 3 ...assigns a random number to every outcome of the randomexperiment.
Feedback No. Random is the outcome of the experiment, not the functionitself.
Answer 4 ...always has a density function.
Feedback No. For example, think of discrete random variables.
Example 6: Fill in the missing words, complete the sentence.

Question Which of the following statements is incorrect?
Answer 1 F-distribution has got two parameter, both called degrees offreedom.
Feedback No. This answer is correct.
Answer 2 Student distribution has got fatter tails for smaller $ \nu$.
Feedback No. This answer is correct.
Answer 3 The variance of a random variable with Student distributioncanbesmallerthan 1.
Feedback Correct.
Answer 4 A $ \chi^2$ distributed random variable cannot take negative values.
Feedback No. This answer is correct.
Example 7: Choose which of the given statements is true/false.

Question Which of the following distributions is symmetric about 0?
Answer 1 Poisson distribution, but only for some parametervalues.
Feedback No. Poisson distribution is defined only for nonnegativeintegers.
Answer 2 F-distribution, for any parameter values.
Feedback No. F-distribution is defined only for nonnegative values of$ x$.
Answer 3 Student distribution, for any parameter values.
Feedback Correct.
Answer 4 $ \chi^2$ distribution, but only for some parameter values.
Feedback No. $ \chi^2$ distribution is defined only for nonnegative values of $ x$.
Example 8: Choose the object that has a given property.

Question A psychological research examined the reaction times of people to exterior impulses. It was found out that thereaction time of people is normally distributed with mean $ \mu = 0.28 \textrm{s}$ and standard deviation $ \sigma = 0.05 \textrm{s}$. How many people should we at least select to a sample so that the 95% confidence interval for the average reaction time is not bigger than 0.03 s?
Answer 1 27
Feedback No. The confidence interval is given by $ [\mu - z_{0.975}\frac{\sigma}{\sqrt{n}}; \mu + z_{0.975}\frac{\sigma}{\sqrt{n}}]$. Solve for n.
Answer 2 36
Feedback No. The confidence interval is given by $ [\mu - z_{0.975}\frac{\sigma}{\sqrt{n}}; \mu + z_{0.975}\frac{\sigma}{\sqrt{n}}]$. Solve for n.
Answer 3 43
Feedback Correct.
Answer 4 48
Feedback No. The confidence interval is given by $ [\mu - z_{0.975}\frac{\sigma}{\sqrt{n}}; \mu + z_{0.975}\frac{\sigma}{\sqrt{n}}]$. Solve for n.
Example 9: Use an algorithm in a reversed way.

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© Jaroslav Borovička 2003