What is the probability that a randomly selected student

Question

Assignment

Please answer all questions by working them out on a sheet of paper. You can earn partial credit only if you show your work. You will need to scan and upload this assignment by the due date (found on Canvas).

These homework assignments can be difficult. Feel free to work in groups and ask questions on Piazza. Each student will need to hand in his/her own assignment and list the names (first and last) of other students who worked together at the very end of the assignment.

I. Mountainback

The following data are from the Tussey Mountainback, a local 50 mile ultramarathon which is divided into 12 stages (aka "legs") so that it can also be run as a relay race. The race starts and finishes at the same point on Tussey Mountain and the course goes through the Rothrock State Forest.

Leg	Mileage	? Elev. (ft)	Tot. Climb (ft)	Tot.  Climb (mi)	Climb Grd (%)	Tot.  Desc (ft)	Tot.  Desc (mi)	Desc  Grd (%)
1	3.2	773	885	2.78	6.03	112	0.42	5.05
2	4	-887	0	0	0	887	4	4.2
3	3.8	-228	87	1.16	1.42	315	2.64	2.26
4	5.6	478	850	3.25	4.95	372	2.35	3
5	3.4	-498	85	0.56	2.87	583	2.84	3.89
6	4.1	1301	1330	3.8	6.63	29	0.3	1.83
7	3.7	-649	147	0.75	3.71	796	2.95	5.11
8	4.3	-470	189	1.3	2.75	659	3	4.16
9	2.9	472	553	2.55	4.11	83	0.35	4.49
10	5.5	-369	398	1.1	6.85	767	4.4	3.3
11	5.3	558	849	3.3	4.87	290	2	2.75
12	4.2	-481	112	0.9	2.36	592	3.3	3.4

a) Create a box plot and a histogram to display the distribution of the mileage and elevation change for each leg of the race.

b) For each variable above, what is something the histogram tells you that the boxplot does not? What is something the boxplot tells you that the histogram does not?

c) For each variable above, write whether you expect the mean to be be close to the median, much higher, or much lower. Why? Calculate the mean...were you right?

d) Identify the variable represented in the following boxplot and and the variable represented in the histogram.

II. Working with Sets

For this type of question, you may often see the sample space S defined as follows

S = [E₁, E₂, E₃, E₄, E₅, E₆, E₇, E₈, E₉, E₁₀]

I am providing another sample space with words instead of numbers. The idea of this problem is to demonstrate that we don't always have to use numbers.

T = [dogs, cats, gorillas, sharks, monkeys, hawks, turtles, mice, deer, pigs].

Note that there are 10 points in each sample space. I'm calling this one "T" because I do not want it to be confused with S above.

a) Using sample space S, Given that A = [E₁, E₃, E₆, E₉], define A'

b) Using sample space T, Given that X = [dogs, gorillas, hawks, deer], define X'

c) Using sample space S, Given that B = [E₃, E₅, E₆, E₁₀] and C = [E₃, E₄, E₆, E₉]

i) what is the intersection of B and C?
ii) what is the union of B and C?
iii) what is the complement of the union of A and B?

d) Using sample space T, Given that Y = [gorillas, monkeys, hawks, pigs] and Z=[gorillas, sharks, hawks, deer]

i) what is the intersection of Y and Z?
ii) what is the union of Y and Z?
iii) what is the complement of the union of X and Y?

e) What relationship do you see between S and T; A and X; B and Y; and C and Z? What insight does this provide about the sets?

III. Punctuality

Consider a population of 1000 students that arrive on time or late to class and the results regarding whether they pass or fail. These results are summarized in the table below. Answer the following questions:

            On Time       Late         Total
Pass         700           30           730
Fail           70             200         270
Total        770           230          1000

a) What is the probability that a randomly selected student passes?

b) What is the probability that a randomly selected student fails?

c) What is the probability that a randomly selected student both passed and arrived on time?

d) What is the probability that a randomly selected student either passed or arrived on time?

e) Given that a student was on time (i.e. only focus on students who are on time and ignore the other students), what is the probability that the student passed?

f) Given that a student was late (i.e. only focus on students who are late and ignore the other students), what is the probability that the student passed?

IV. Basketball Players

a) In a city of 110,000 people, there are 10,000 basketball players. What is the probability that a randomly selected person from the city will be a basketball player?

b) In a city of 130,000 people, there are 20,000 basketball players. What is the probability that two randomly selected people from the city will BOTH be basketball players?

V. New Machines

A company has just received new machinery that must be installed and checked before it becomes op- erational. The accompanying table shows a manager's probability assessment for the number of days required before the machinery becomes operational.

Number of Days     3       4       5       6       7
Probability           0.08  0.20   0.41  0.24   0.07

Let A be the event "It will be more than 4 days before the machinery becomes operational," and let B be the event "It will be less than 6 days before the machinery becomes operational."

a) find the probability of event A
b) find the probability of event B
c) find the probability of the complement of event A
d) find the probability of the intersection of events A and B
e) find the probability of the union of events A and B

VI. Venn Diagrams

Use Venn Diagrams to determine whether the following statements are true or false.

a) (A ∪ B)' ∩ C = (A' ∩ C) ∩ (B' ∩ C)
b) (A ∪ B)' ∩ C = A' ∩ B' ∩ C'

VII. Deck of Cards

Suppose you are dealt two cards from a standard deck of playing cards. There are 4 different suits (clubs, diamonds, hearts, spades) and 13 different denominations (Ace, 2, 3, ...10, Jack, Queen, King)

a) What is the probability of being dealt a pair of aces? (This means that both the cards are aces).

b) There are 13 pairs possible (Aces through Kings). What is the probability of being dealt a pair of any type (that is, the probability of being dealt a pair of aces OR 2s OR 3s, etc...)?

c) What is the probability of being dealt two cards of the same suit?

d) What is the probability of being dealt any two cards greater than a 10? This includes Jacks, Queens, Kings, and Aces.

VIII. Scatterplots: Height and Weight

The following scatterplots show height and weight data entered in the survey by students in a previous semester of this class. Which correlation do you believe more, r = .4379 or r = .5164? What is the difference between the two plots?