STATISTICS for Prelim Students
Hilary Term, 2005
These tutorials are designed to provide you with an introduction to basic concepts in
statistics, namely the presentation of data, measures of central tendency and
dispersion), statistical inference (elementary probability theory and probability
distributions, sampling estimation and hypothesis testing), correlation and regression
(correlation coefficients, correlation and causality, ordinary least squares regression
and statistical inference of simple regression equations.
We will meet for 2 hour long classes in Weeks 1, 2, 3 and 4. The classes for Week 1
are in the Seminar Room, Nuffield College and are from 2 p.m. till 4 p.m. The tutorial
outline and questions for each week will be available at the following website:
http://www.nuff.ox.ac.uk/users/khan/.
For the classes I will ask a few of you to make 5 minute presentations in class on each
of the topics listed. Questions will be discussed and some answer solutions will be
handed out during class. Some additional questions may be handed out in the tutorial
for students to attempt on the board.
Please make sure that you understand the key concepts that you have been asked to
make notes on. They are aimed at developing your understanding and will also help
you to answer the questions set. Please feel free to bring along the key textbooks. I am
going to primarily make use of Wonnacott and Wonnacott and of Aczel (see details
on:
http://www.economics.ox.ac.uk/intra/Under/Prelims/PreStats/IntrotoStats.htm).
Both are reasonably good but I prefer Aczel.
Week 1
Reading List
Aczel, Chapters 1, 2
Wonnacott and Wonnacott: Ch 1, 2, 3
Permutations, combinations and basic probability rules
Make notes on the following and, where appropriate, demonstrate your
understanding with the use of an example:
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Experiment, Random Experiment
Event
Sample space
The Probability of an Event
Law of Large Numbers
Random Sample
Mutually exclusive events
Factorial rules
Permutations and combinations
The multiplicative and additive rules of probability
Independent events
Complementary events
Conditional Probability
Bayes Theorem
Note the following fundamental counting rules in probability of multiplication
and addition
Addition Rule.
If event A can occur in m possible ways and event B can occur in n possible ways,
there are m+n possible ways for either event A or event B to occur,
but only if there are no events in common between them.
n(A or B)=n(A)+n(B)-n(A B)
Multiplication Rule (can only be applied if the events are independent)
If event A can occur in m possible ways and event B can occur in n possible ways,
there are m•n possible ways for both events to occur.
n(A and then B)=n(A)×n(B)
Now do the following questions
Q1 Calculate by hand, showing your work, the following combinations:
a.
b.
c.
d.
e.
5C3.
6C4.
9C3.
10C4.
9C9
(explain why the answer is one)
Q2 Calculate by hand, showing your work, the following arrangements:
f.
g.
h.
i.
j.
5P3.
6P4.
9P3.
10P4.
9P9.
Q3 Assume you have 20 M&M candies as follows: 5 orange, 6 yellow, 5 red, and 4
green. In one selection, how many ways can you select 1 orange or 1 yellow M&M?
What is the probability of selecting 1 orange or 1 yellow M&M?
Q4 Assume you have 20 M&M's colour distributed as above. If selected without
replacement, in how many ways can you select two red ones in two selections? What
is the corresponding probability?
Q5 How many different ways (i.e. is it a boy or a girl) can parents have three
children?
Q6 You need to study, practice football, fix dinner, phone your parents, and go buy a
notebook. How many different ways can you arrange your schedule?
Q7 You have five places left for stamps in your stamp book and you have eight
stamps. How many different ways can you select (combine) five? How many different
ways can you arrange them (permutation) ?
Q8 How many permutations are possible of the word STATISTICS
Q9 How many permutations are possible with the word TRINITY
Q10 Imelda Marcos has 2003 pairs of shoes. In how many different ways can she
select a left shoe then a right shoe?
Q11 Scanti Lee Clad has 15 short shorts and 11 tiny blouses. In how many different
ways could she select a shorts-blouse combination?
Q12 The PPE reading list contains 12 Philosophy books and 15 Politics books.
a. In how many different ways could a student select a Phil or Pol book?
b. In how many different ways could a student select a Phil and then a Pol
book?
c. In how many different ways could a student select a Pol and then
another Pol book?
Q13 In how many ways could you arrange the following?
a.
b.
c.
d.
Four notebook sections from a set of six sections?
Ten homeworks from a set of twelve homeworks?
Five tests from a set of eight tests?
All 20 questions from a set of 20 questions?
Q 14 Six girls start playing a volleyball game.
a. In how many ways could the six positions be filled?
b. If the positions are selected at random, what is the probability that
Nikki will be server?
c. Express the probability in part b above as a percentage.
Q 15 In a class of 24 students, 7 are left-handed, and the rest right-handed. If 8 people
are selected at random from this group, what is the probability that
a. 3 are left-handed and 5 are right-handed?
b. all are right handed?
c. all are left-handed?
Q16 A box contains 14 identical balls of which 6 are white, 5 red and 3 black. $ balls
are drawn. What is the probability that
a.
two are red
b.
one is white
Q17 A bag contains 6 red &4 white balls. Two balls are drawn at random from the
bag. Find the probability that (i) both are white and (ii) one is red and one is white
Q18 A bag contains 4 white and 3 black balls. Another bag contains 5 black and 3
white balls. A ball is drawn from one of the bags selected at random. What is the
probability that it is white?
Q19. A can hit a target 4 times in 5 shots, B can hit 3 times in 4 shots and C twice in 3
shots. All three sleep in a dorm. Early one morning a little birdie starts cheeping really
loudly. All three fire a volley. What is the probability that at least 2 shots hit (and
hopefully kill or permanently disable the cute little bird)?
By using the complementary rule with the multiplication rule, one can find the
probability of at least one event being what we want. At least one means the same as
one or more. The complement of one or more is none. So the multiplication rule is
used to find P(none) and then take the complement of it. P(at least one) = 1 P(none)!!!
Now attempt the following question.
Q20 A person deals you a new five cards hand. What is the probability of having at
least one heart?