UNIVERSITY OF THE PHILIPPINES DILIMAN

SCHOOL OF STATISTICS

Statistics 101: Elementary Statistics

Problem 1. (20 pts total)

I. Using your own words, answer the following questions as if you are talking to your friend or

relative who does not know basic concepts in probability. Answers should not exceed 20

words. (2 pts each)

1) What is probability?

2) What is conditional probability?

Bonus questions (2 pts total):

3) What is a probability distribution? (1 pt)

4) What does it mean when two events are not independent? (1 pt)

II. For each of the following experiments, describe the sample space. (2 pts each)

1) Toss a coin three times.

2) Count the number of insect-damaged leaves on a plant.

3) Measure the lifetime (in hours) of a particular light bulb.

4) Record the weights of first year student athletes.

5) Observe the proportion of households below poverty line in a barangay.

III. Given events and where , , if () = !

" and (#) = !

$, answer the following

questions.

Show all pertinent solutions. No points will be given to correct answers without

solutions, with incomplete solutions, or with wrong solutions, except otherwise

stated. Partial points will be given to wrong answers with correct solutions. No immediate

rounding. Round off final answers to four (4) decimal places. (6 pts)

1) What is ()? (1 pt)

2) Can and be disjoint? Yes or no? (No need to explain or show your solution.) (1 pt)

3) Explain your answer in #2. Show your complete solution. (3 pts)

4) Using your solution in #3, what is ( )? (1 pt)

Stat 101 Group Problem Set 2 | xjbilon

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Problem 2. (20 pts total)

I. Approximately one-third of all human twins are identical (one-egg) and two thirds are

fraternal (two-egg) twins. Identical twins are necessarily the same sex, with female and male

being equally likely. Among fraternal twins, approximately one-fourth are both female, onefourth

are both male, and half are one female and one male. Finally, among all births in the

Philippines, approximately 15 in 1000 are twin births (UNICEF, 2016).

We also define the following events.

= {a birth in the Philippines results in twin females}

= {a birth in the Philippines results in identical twins}

= {a birth in the Philippines results in twins}

Answer the following questions.

Show all pertinent solutions. No points will be given to correct answers without

solutions, with incomplete solutions, or with wrong solutions, except otherwise

stated. Partial points will be given to wrong answers with correct solutions. No immediate

rounding. Round off final answers to four (4) decimal places. (13 pts)

1) What is the value of ()? (1 pt)

2) What is the value of ()? (1 pt)

3) What is the value of ()? (1 pt)

4) State, in words, the event . (2 pts)

5) Find ( ). (2 pts)

6) Using your answers in #1-3 and #5, are events , , and mutually independent? Show

your solution. (2 pts) (Hint: Determine whether ( ) is equal or not equal to

()()().)

7) Which events are pairwise independent? Show your solution. (4 pts) (Hint: For each pair

of events !, % {,, } , determine whether (! %) is equal or not equal to

(!)(%).)

II. The Dunphys have three children. Alex, one of their children, is assigned female at birth.

Assuming that having a female child or a male child is equally likely, answer the following

questions.

Show all pertinent solutions. No points will be given to correct answers without

solutions, with incomplete solutions, or with wrong solutions, except otherwise

stated. Partial points will be given to wrong answers with correct solutions. No immediate

rounding. Round off final answers to four (4) decimal places. (7 pts)

1) What is the probability that at least two children are assigned females at birth? (2 pts)

2) What is the probability that only one of the children is assigned female at birth? (1 pt)

3) What is the probability that Alex is the middle child? (2 pts)

4) Suppose Claire and Phil, Alex's parents, told us that Alex is the middle child. What is

the probability that the eldest is assigned female at birth and the youngest is assigned

male at birth? (2 pts)

Stat 101 Group Problem Set 2 | xjbilon

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Problem 3. (20 pts total)

I. Given (|| > ) = &

%, where ~(, %), , % 0, and 0, solve the following

problems. Show all pertinent solutions. No points will be given to correct answers

without solutions, with incomplete solutions, or with wrong solutions, except

otherwise stated. Partial points will be given to wrong answers with correct solutions. No

immediate rounding. Round off final answers to four (4) decimal places. (20 pts)

1) What is || > equivalent to? Choose the letter of the best answer. (No need to show

your solution.) (1 pt)

a. > , >

b. > , <

c. > , >

d. < <

e. Both a and b

f. Both c and d

2) Using your answer in #1, sketch the area under the PDF curve that is equal to (|| >

). Show , , and . (1 pt)

3) Using your answer/s in #1 and/or #2, express (|| > ) in terms of ( ). (No

need to show your solution) (2 pts)

4) Using your answer in #3, express ( ) in terms of . (1 pt)

5) Find in terms of such that ( ) = ( ). (2 pts)

For questions 6 to 15, suppose = 0.05, = 12.43, and % = 1.012.

6) Using the given value of and your answer in #4, find such that ( ) = (

). (2 pts)

7) Using z expressed in terms of c you obtained in #5 and your answer in #6, what is the

value of ? (1 pt)

8) Using the value of given and/or the value of computed in #6, what is the probability

that || is greater than ? (1 pt)

9) Using your answer in #8, find (|| ). (1 pt)

10) Using the value of computed in #6, what is ( ||)? (1 pt)

11) Using your answer in #9, what is ( > ||)? (1 pt)

12) Using your answers in #7 and #10, is (|| > ) = ( > ||)? Yes or no? (No need to

explain or show your solution.) (1 pt)

For the succeeding questions, suppose = 2 + 2.

13) What are [] and ()? (No need to show your solution.) (1 pt)

14) What is []? (2 pts)

15) What is ()? (2 pts)

Stat 101 Group Problem Set 2 | xjbilon

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Problem 4. (20 pts total)

I. Solve the following problems. Show all pertinent solutions. No points will be given to

correct answers without solutions, with incomplete solutions, or with wrong

solutions, except otherwise stated. Partial points will be given to wrong answers with

correct solutions. No immediate rounding. Round off final answers to four (4) decimal

places. (4 pts each)

1) In a medical lab, a patient can have three (3) medical tests at most. The probabilities

that a patient will have 0, 1, 2, or 3 medical tests performed in the lab are '

!(, (

!(, "

!(, and

!

!(, respectively. What is the probability that a patient will have at most one medical test?

2) A cellular phone company kiosk always sells at least four (4) new phones per day, but

never more than ten (10) new phones per day. The probabilities that the kiosk sells

number of new phones per day are ( = 4) = 0.5, ( = 5) = 0.3, ( = 6) = 0.1,

( = 7) = 0.065, ( = 8) = 0.02, ( = 9) = 0.01, and ( = 10) = 0.005. What is the

probability that the kiosk will sell more than four (4) but less than nine (9) new phones

on a particular day?

3) Transportation officials reported that 8.25 out of every 1000 airline passengers lost

luggage during their travels last year. If we randomly select 350 airline passengers, what

is the probability that 6 lost some luggage?

The probability that = airline passengers lost their luggage out of randomly

selected airline passengers is given by

( = ) =

!

( )! !

)(1 )*+),

where is the probability that each airline passenger will lose their luggage.

4) For a medical study, a researcher wishes to select people in the middle two-thirds of the

population based on blood pressure. If the mean systolic blood pressure is 120 and the

standard deviation is 9, find the upper and lower readings that would qualify people to

participate in the study. Assume that blood pressure readings are normally distributed.

5) To help students improve their reading, a city decides to implement a reading program.

It is to be administered to the bottom 10% of the students in the district, based on the

scores on a reading achievement exam. If the average score for the students in the city is

132.6, find the cutoff score that will make a student eligible for the program. The

standard deviation is 20. Assume the score is normally distributed.

Bonus questions (2 pts total): Suppose, after the program, the initial bottom 10%

improved and the group had an average score of 122.6. The city also initiated a special

program for the top 10%, after which their average score is 190.2.

What is the average score of the middle 80%? (1 pt)

What is the new average score for all the students in the city? (1 pt)

Stat 101 Group Problem Set 2 | xjbilon

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