Unit 2: Probability, Random Variables, and Probability Distributions
Statistics · Unit 2 · Paper 3

Probability, Random Variables, and Probability Distributions unit test

A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.

Paper 1Paper 2Paper 3
Each paper is built from this unit’s 70 terms and is the same for everyone, so a teacher can assign “Unit 2, Paper 3” and every student sits the identical test. Multiple choice is marked objectively; the written sections you mark yourself against the model answer and rubric.
Suggested time 31 min 28 points0/17 attempted
1

Law of large numbers and the gambler's fallacy

2

Geometric setting

3

Mean of a sum

4

Expected value is a long-run average

5

Complement rule

6

Conditional probability from a two-way table

7

Sampling distribution

8

General multiplication rule

9

Binomial conditions (BINS)

10

Sample space

11

Shape of the sampling distribution of p̂

12

Bias vs variability

Short answer 1. Define or explain: The 10% condition in sampling-distribution formulas

3 pts

Short answer 2. Define or explain: Variance of a difference

3 pts

Short answer 3. Define or explain: Tree diagrams

3 pts

Short answer 4. Define or explain: Requirements of a probability distribution

3 pts

Free response

4 pts

A researcher is investigating whether there is an association among professional athletes between age-group (in years) and type of sport played (basketball, football, baseball). The age-group and type of sport for all 4,193 professional athletes in these sports for a recent year: Age (years) Basketball Football Baseball Total Age < 25 232 807 259 1,298 25 ≤ Age < 30 175 1,326 620 2,121 30 ≤ Age < 35 90 287 276 653 35 ≤ Age 19 41 61 121 Total 516 2,461 1,216 4,193 A mosaic plot was constructed from the table: each sport is a column whose WIDTH is proportional to the share of all athletes playing that sport, and each column is divided vertically by age-group so that each band's HEIGHT is the share of that sport's athletes in the age-group. On the Football column, the width is labeled b, the height of the "25 ≤ Age < 30" band is labeled h, and that band's area is labeled x = b·h.

A. (i) What is the probability a randomly selected professional athlete is a football player? Show your work. (ii) What is the probability a randomly selected professional athlete is in the age-group "25 ≤ Age < 30" given they are a football player? Show your work.

B. Use the mosaic plot. (i) Which probability does b correspond to: the probability calculated in part A(i) or in part A(ii)? (ii) What probability does the x displayed in the mosaic plot represent in context?

C. (i) Are the events "Baseball" and "35 ≤ Age" mutually exclusive? Explain. (ii) Are the events "Baseball" and "35 ≤ Age" independent? Show your work.

D. Determine if it is appropriate to carry out a chi-square test for independence to investigate whether there is an association between age-group and sport played using these data. Explain your answer.