Stat 321 – Review 1
Optional: Zoom review session, 6:30-8pm-sih, Tuesday
(see Chance Zoom Room link in Canvas)
Format: The exam will be closed books, closed notes except that you will be supplied with a list of formulas. Bring your calculator. The exam will cover topics from Chapter 1 and Chapter 2 (Sections 2.1-2.3). You should study from the text, lecture notes, homework exercises, labs, and quizzes. HW and Quiz solutions are posted in Canvas (let me know if you can’t find). In studying, I recommend reworking homework exercises and quiz problems (best guides of what exam questions will look like) without looking at the solutions, then checking your answers, then repeat. The questions will not be heavily computational, but you are expected to know how to carry out the calculations by hand and how to interpret Minitab output. You are also expected to explain your reasoning, show your steps, and interpret your results.
· The distinction between population and sample
· How to interpret a distribution in context (e.g., behavior of hot dog prices)
· How to anticipate the behavior of a distribution from context
· How to construct a stem and leaf display, how interpret Minitab’s (including repeated stems)
· How to construct/interpret histogram (equal class width, left-hand endpoint rule)
· How to construct/interpret boxplots
· How to describe shape, center, and spread of the graph (descriptively, numerically)
· How to calculate mean, median, mode, quartiles (small, quantitative data sets)
· How to interpret standard deviation as (roughly) average (or typical) deviation from mean across all observations (quantitative data)
· Which measures of center are resistant to outliers and expected relationship with skewed distributions (Figure 1.10)
· How to reason about measures of center, spread
· How to check for outliers with 1.5 IQR rule, e.g., Q1- 1.5 IQR (show calculations)
·
Remember to check both ends for outliers.
·
Conjecture possible explanations for an unusual
observation
· How to compare two or more distributions graphically and numerically (e.g., if given output)
·
Note: While the shape of the distribution gives
us a conjecture for the relationship between the mean and median, this
conjecture does not always hold. It is
particularly risky to compare the mean and median and use that relationship
alone to conjecture the shape of the distribution. Look at graphical displays to learn about the
shape.
From Chapter 2 (Sections 2.1 – 2.3) you
should know:
· How to define probability as long-run relative frequency
Definition of empirical probability (estimate using relative frequency), simulation
· How to list elements of sample space, S, and of particular events
· A probability has to be a number between (inclusive) 0 and 1, and P(S)=1
· How to represent events with set notation, e.g., A Ç B'
· Verbal translations for set notation (e.g., unions, intersections, complements)
· Set notation for verbal descriptions (e.g., neither, exactly one, at least one, either)
· How to use Venn Diagrams to represent events
· How and when to apply DeMorgan’s Laws
· How to find probability of events:
- when equally likely (# of outcomes in event/# of outcomes in sample space)
- Counting rules
Product Rule (n1n2…)
Number of different possibilities, with ni options at each “stage” (Tree Diagram)
Permutations – # of ordered arrangements of k items from n distinct objects
e.g., arrangements of books on a shelf, King and Queen of prom, spelling of words
Combinations – # of unordered combinations of k items from n distinct objects
e.g., subsets of 3 books from shelf with 10 books
Order doesn’t matter (two people): is M/F equivalent to F/M
- as sum of probabilities of elements in the event
- using probability rules (complement rule, addition rule, multiplication rule, law of total probability, Bayes’ theorem)
- consider using a two-way or probability table to set up the calculations
· How to decide whether two events are mutually exclusive/disjoint
and how the general addition rule simplifies in this case
· How to calculate and interpret conditional probability
Restriction of sample space
Direct calculation P(A|B) = P(A Ç B)/P(B)
· Definition of independence, e.g., P(A|B)=P(A) and interpretation
· How to decide whether two events are independent by checking whether P(A Ç B)=P(A)P(B) or whether P(A|B)=P(A) or whether P(B|A)=P(B)
and how the multiplication rule simplifies in this case
· How and when to use Law of Total Probability and Bayes’ Theorem
e.g., tree diagrams, probability tables
Law of Total Probability: given P(A|other events) and want P(A) and
BT: given P(B|A) and want to know P(A|B) (reverses the conditioning)
Some general strategies (meaning these may not always apply exactly)
· with “or” we add probabilities, but may need to subtract P(AÇB)
· with “and” we multiply probabilities, but may need to use conditional probability
check for independence before applying multiplication rule for independent
events
· with conditional probabilities, look for “of the …” or “if the…” as the event you want to condition on
· a question asking “how many” does not have to be a number between 0 and 1, but probabilities and proportions do have to be numbers between 0 and 1 (inclusive)
· If event of interest is complicated or large, try its complement
§
P(at
least one) = 1-P(none)
·
Conditional
probabilities
§
Multiplication
rule: looking for probability of intersection and know conditional probability
§
Law of
total probability: looking for unconditional probability and know conditional
ones
§
Bayes’
theorem: looking for “reverse” conditional probability
·
In
approaching a problem, clearly identify and define events, record given
information with appropriate symbols, identify what’s asked for with
appropriate symbols, decide (and specify) what rule(s) apply
o
Use good notation - make sure can “set up” event
notation even before calculation
Remember to read questions carefully and
think about the solution before you begin writing. Work efficiently, time could be an
issue. Be prepared to
think/explain/interpret (not just plug numbers in).
Suggested Review Problems
Problem solving hints:
· First, define the relevant events and put the probabilities given and requested into set notation
· Use Venn Diagrams or Tree Diagrams or Two-Way Tables to coordinate the information
· Always provide the name of any "theorem" or "law" you are using, e.g., "this is true because of DeMorgan’s law"
· Be careful with your notation
Note: This is not intended to represent the length of an exam, more trying to give a couple problems from each topic/section, and some previous exam questions. Even the questions from the book, you should try solving them without telling yourself which section they came from.
1) Suppose that Chris rolls a fair four-sided die (numbered 1-4) and Amy rolls a fair six-sided die (numbered 1-6). What is the probability that Amy rolls a larger number than Chris?
2) Suppose that a student is given 25 problems to solve for homework and that she correctly solves 20 of them. Then suppose that the instructor randomly picks 5 of the 25 problems to be graded.
(a) What is the probability that the student gets exactly four of the graded problems correct?
(b) What is the probability the student gets at least one of the graded problems correct?
3) If you roll two fair, six-sided dice and compute the sum of the two dice:
(a) Describe the sample space of this experiment.
(b) Are the events {sum is 3} and {sum is 6} equally likely? Explain.
4) p. 40-47: problems 8, 13, 16, Supplementary exercises: 1, 2, 6
Adding to problem 8
(b) Approximate the 90th percentile
5) I’ve heard that about 5 percent of men and 0.25 percent of women are color blind. And then suppose a color-blind person is chosen at random, and they identify as male or female.
(a) What is the probability of this person being male? Assume that there are an equal number of males and females in the population.
(b) How does your answer to (a) compare to P(male)? [P(male) is the unconditional probability of a member of the population being male.] Briefly explain why this makes sense.
6) Suppose A and B are two events with P(A)=.4, P(B)=.7. Find the minimum and maximum possible values of P(A Ç B) and the conditions under which each of these values is attained.
7) Two cards are randomly selected from an ordinary playing deck. What is the probability that they form a blackjack(one of the cards is an ace and the other is a ten, jack, queen, or king)?
8) p. 61-62: 17, 18; p. 68: 4; p. 88: 26, 34
9) Explain how a person can move from state A to state B and the mean IQ in both states can decrease as a result.
10) Create a hypothetical set of 10 exam scores such that 90% of them are greater than the mean.