1. Let X be a discrete random variable that may assume values x = 0; 1; 2 respectively with probabilities p1; 0:5; p2. Which values of p1 and p2 define

a probability distribution?

A p1 = 0.5 and p2 = 0.5

B p1 = 0.4 and p2 = 0.1

C p1 = 0.2 and p2 = 0.8

D We do not have enough information to answer.

2. Let X be a discrete random variable. Which of the following sentences about the cumulative distribution function is false?

A limx->+F(x) = 1

B limx->- F(x) = -1

C F(x) is a not decreasing function.

D None of the previous sentences.

3. Let A and B be two events. If they are independent then

A P(AB)=P(A)+P(B)

B P(AB)= P(A) +P(B) - P(AB)

C P(A B) = P(A) P(B)

D None of the previous sentences.

4. Let X be a variable that may assume two possible values 0 and 1 with probabilities p and q. The variable X is a random variable if and only if

A p + q = 1

B p, q 0 and p + q = 1 and X is increasing with x.

C p, q 0 and p + q = 1

D None of the previous sentences.

5. Let A and B1, B2,...., Bk possible causes of A. The Bayes theorem states that:

A P(Bh A) = P(A Bh)P(Bh)/ P(A)

B P(Bh A) = P(A Bh)P(Bh)/ P(B)

C P(Bh A) = P(A Bh)P(Bh)

D None of the previous sentences.

6. Suppose we want to make inference about a population mean knowing its variance. According to the data, we obtain the following 95% confidence interval: [2; 5]. Using such interval, we would like to verify the following hypothesis: H0 = = 8 and H1 = > 8. What do you conclude?

A reject H0

B not reject H0

C accept H0

D we do not have enough information to answer.

7. Let X be a discrete random variable that may assume values x = 0; 1; 2 respectively with probabilities p1; 0:5; p2. We know that E[X] = 0.9 Which values of p1 and p2 define a probability distribution?

A p1 = 0.25 and p2 = 0.25

B p1 = 0.4 and p2 = 0.1

C p1 = 0.3 and p2 = 0.2

D We do not have enough information to answer.

8. Let A and B be two events. Then

A P(A B) = P(AB)P(B)

B P(A B) = P(A) + P(B) - P(A B)

C P(A B) = P(A)P(B)

D None of the previous sentences.

9. Suppose that we define a confidence interval for the population mean according to a sample of n = 100 statistical units. The obtained confidence interval is [10; 30]. From the same population, we sample n = 1000 statistical units. Which of the following confidence intervals is more appropriate?

A [10; 30]

B [8; 25]

C [8; 10]

10. Let A be the event "a randomly selected professor is a women" and B be the event "a randomly selected professor is a statisticians". We know that in the considered sample, the probability of selecting a statistician is 0.18 and the probability to select a women statistician is 0.04. A professor who is a women is selected from the sample. What's the probability she is a statistician?

A 0.22

B 0.18

C 0.04

D None of the previous sentences.

11. We want to study the variability monthly expenditure. We fit a first model having as explanatory variable the monthly income and a second model having as explanatory variables both the monthly income and the number of family components. Which of the following sentences is correct?

A R^2 of the rst model is equal to R^2 of the second model;

B R^2 of the rst model is larger to R^2 of the second model;

C R^2 of the rst model is smaller to R^2 of the second model;

D we do not have enough information to answer.

12. All of the following are characteristics of the Normal distribution , except

A The probability that x is exactly equal to any specific value is larger than 0.

B Bell-Shaped curve.

C Area under the curve is 1.

D It is a symmetric distribution.

13. When one refers to "how significant" the sample evidence is, he/she is referring to the:

A value of

B the sample size

C the p-value

D the distribution of the estimator

14. A p-value = 0.65 allows us:

A to reject H0

B not to reject H0

C it depends on

D none of the previous sentences.

15. The null and alternative hypotheses divide the parameter space into:

A two sets that overlap

B two non-overlapping sets

C two sets that may or may not overlap

D as many sets as necessary to cover all possibilities

16. Let X be a Bernoulli random variable with probability successes equal to0.8. What's the expected value and the variance of this variable?

A E[X] = 0.8; Var[X] = 0.8

B E[X] = 0.8; Var[X] = 0.5

C E[X] = 0.8; Var[X] = 0.32

D none of the previous sentences.

17. What does it mean to standardize a score?

A Make it so the mean of the scores is 10 and the standard deviation is 2.

B Make it so the mean of the distribution is 100 and the standard deviation is 10.

C Make it so the z-table can be used for percentiles.

D None of the previous sentences.

18. Which of the following is NOT an assumption of the Binomial distribution?

A The number of successes in the trials is counted.

B Each trial must be classied as a success or a failure.

C All trials are dependent on each other.

D None of the previous sentences.

19. A null hypothesis can only be rejected at the 5% significance level if an only if:

A a 95% confidence interval includes the hypothesized value of the parameter

B a 95% confidence interval does not include the hypothesized value of the parameter

C the null hypothesis is void

D the null hypotheses includes sampling error

20. Suppose you want to verify is the population mean is equal to 5. According to the 95% confidence interval, you reject the null hypothesis. What about your decision if = 0.1?

A reject the null hypothesis as well

B don't reject the null hypothesis

C it depends on the type II error

D we do not have enough information to draw a conclusion.

21. Let X be a random variable with a Normal distribution with mean equal to 3 e variance equal to 4.

A P(X < 3) = 0.250

B P(X < 3) = 0.159

C P(X < 3) = 0.01

D None of the previous sentences.

22. Let X be a random variable with Binomial distribution with parameters n = 10 and p = 0:2. Then

A P(X = 4) = 0.967

B P(X = 4) = 0.882

C P(X = 4) = 0.088

D None of the previous sentences.

23. Let X be a random variable with Poisson distribution with parameters = 3. Then the first quartile is:

A 10

B 5

C 3

D None of the previous sentences.

24. Let X be a random variable with a Binomial distribution with parameters n = 5 e p = 0.2. Compute the following probability P(0 X < 1)

A P(0 X < 1) = 0.327

B P(0 X < 1) = 0.410

C P(0 X < 1) = 0.737

D None of the previous sentences.

25. Let X be a random variable with Poisson distribution with parameters = 4. Then

A P(X > 3) = 0.11

B P(X > 3) = 0.25

C P(X > 3) = 0.57

D None of the previous sentences.