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Exam 1 ReviewReview
Cathy Poliak, [email protected]: Fleming 11C
Department of MathematicsUniversity of Houston
Test 1 Review
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 1 / 20
Dates of Exam and Covered Chapters
Dates: June 22 & 24
Chapters: 1, 2, 3 and 4 and sections 9.1 & 9.2
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 2 / 20
Topics Covered
Random Variables and the different types
Probability Rules
Conditional Probability
Baye’s Rule
Descriptive Statistics: mean, median, sd, IQR, outliers
Graphs
Least-Squares Regression Line
Discrete Probability Distributions
Binomial, Hypergeometric and Poisson Distributions
Joint probability distributionCathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 3 / 20
Chapter 1
Sample - Simple random sample
Population versus sample
Parameter versus statistic.
Categorical variables
Quantitative variables; continuous and discrete
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 4 / 20
Chpater 2
Describing distributions with numbers and graphs
Center - mean, median, mode
Spread - range, interquartile range (IQR), standard deviation
Location - percentiles, quartiles (Q1 and Q3), z-scores, 1.5 × IQR
The five number summary
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 5 / 20
Chapter 2 Graphs
Describing distributions with graphs
Categorical variables - bar chart, pie chart, Pareto chart
Quantitative variables - histogram, stemplot, boxplot
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 6 / 20
Chapter 3 section 1: Sets and Venn Diagrams
Notation Descriptiona ∈ A The object a is an element of the set A.A ⊆ B Set A is a subset of set B.
That is every element in A is also in B.A ⊂ B Set A is a proper subset of set B.
That is every element that is is in A is also in set B andthere is at least one element in set B that is no in set A.
A ∪ B A set of all elements that are in A or B.A ∩ B A set of all elements that are in A and B.U Called the universal set, all elements we are interested in.AC The set of all elements that are in the universal set
but are not in set A.
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 7 / 20
Chapter 3 section 2: Counting Techniques
If an experiment can be described as a sequence of k steps withn1 possible outcomes on the first step, n2 possible outcomes onthe second step, and so on, then the total number of experimentaloutcomes is given by (n1)(n2) . . . (nk ).Permutations: allows one to compute the number of outcomeswhen r objects are to be selected from a set of n objects wherethe order of selection is important. The number of permutations isgiven by Pn
r = n!(n−r)!
When we allow repeated values, The number of orderings of nobjects taken r at a time, with repetition is nr .The number of permutations, P, of n objects taken n at a time withr objects alike, s of another kind alike, and t of another kind alikeis P = n!
r !s!t!
The number of circular permutations of n objects is (n − 1)!.
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 8 / 20
Combinations
Combinations counts the number of experimental outcomes when theexperiment involves selecting r objects from a (usually larger) set of nobjects. The number of combinations of n objects taken r unordered ata time is
Cnr =
(nr
)=
n!r !(n − r)!
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 9 / 20
Chapter 3 section 3 & 4: Basic Probability Models
For any event A, the probability of A is
P(A) =number of times A occurstotal number of outcomes
.
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 10 / 20
General Rules of Probability
1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.2. If S is the sample space in a probability model, then P(S) = 1.3. Complement rule: For any event A,
P(AC) = 1− P(A)
4. General rule for addition: For any two events A and B
P(A ∪ B) = P(A) + P(B)− P(A ∩ B)
5. General rule for multiplication: For any two events A and B
P(A ∩ B) = P(A)× P(B, given A)
orP(A ∩ B) = P(B)× P(A, given B)
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 11 / 20
Chapter 3 section 5: General Probability Models
Two events are disjoint if the occurrence of one prevents the otherfrom happening.
If two events A and B are disjoint then P(A and B) = P(A∩B) = 0.
Two events are independent if the occurrence of one does notchange the probability of the other.
If two events A and B are independent then
P(A and B) = P(A ∩ B) = P(A)× P(B).
Conditional Probability: For any two events A and B, theprobability of A given B is
P(A given B) = P(A|B) =P(A ∩ B)
P(B)
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 12 / 20
Chapter 4: Discrete Distributions
Define a discrete random variable.
Know how to calculate the expected values and variances from adiscrete probability distribution.
Know how to calculate the probability knowing that it is a binomial,hypergeometric and Poisson distribution. Also, be able todetermine the expected value and variances.
Know how to find expected values and covaraince from a jointprobability distribution.
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 13 / 20
Chapter 9: Least-Squares Regression
Scatterplot of two variables, in R plot(x,y)
Finding covariance, in R cov(x,y), correlation, in R cor(x,y) andLSLR, in R lm(y x).
Understand what each of these numbers mean.
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 14 / 20
What is on the Exam
There are 7 multiple choice questions that are worth 7 pointseach.
There are 4 free response questions.I Question 8 – worth 13 points +5 points bonusI Question 9 – worth 12 pointsI Question 10 – worth 13 pointsI Question 11 – worth 13 points
Do not forget to take the practice test. 5% of whatever you get onthat will be added to your score.
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 15 / 20
Possible Multiple Choice Questions
Decting outliers.
Looking at the graphs and determine the shape.
Know the difference between the types of variables.
Using the probability rules.
Know how to find probabilities from a discrete distribution tabl,binomial, hypergoemetric and Poisson distribution
Know how to find expected value from a discrete distribution table,binomial, hypergoementic and Poisson distribution and from alinear expression.
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 16 / 20
Possible Free Response Questions
Determining outliers, shape, center and spread from descriptivenumerical values.
Probability rules, including the Baye’s rule.
Know when two events are independent.
Know how to find probabilities from a discrete distribution table.
Know how to find expected value from a discrete distribution tableand for joint probabilities.
Creating a least-squares linear equation from the data,scatterplot, correlation, coefficient of determination, and residual.
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 17 / 20
Example 1
Suppose the probability that a company will be awarded a certaincontract is .25, the probability that it will be awarded a second contractis .21 and the probability that it will get both contracts is .13. What isthe probability that the company will win at least one of the twocontracts?
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 18 / 20
Example 2An appliance store is offering a special price on a complete set of kitchenappliances (refrigerator, oven, stove, dishwasher). A purchaser is offered achoice of manufacturer for each component:
Refrigerator: Kenmore, GE, LG, WhirlpoolOven: KitchenAid, Samsung, Frigidaire, KenmoreStove: Electrolux, Hotpoint, GEDishwasher: Bosch, Silhouette, Premier, WhirlpoolUse the product rules to answer the following questions:
a. In how many ways can one appliance of each type be selected?
b. In how many ways can appliances be selected if none is to be Kenmore?
c. If someone randomly chooses their appliances, what is the probability thatat least one Kenmore component is chosen?
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 19 / 20
Example 3
Inventory for a manufacturer are produced at three different plants,45% from plant 1, 30% from plant 2, and 25% from plant 3. In addition,each plant produces at different levels of quality. Plant 1 produces 2%defectives, plant 2 produces 5% defectives, and plant 3 produces 8%defectives.a. What is the probability that an item is defective?
b. If an item from the inventory is found to be defective, what is theprobability that is was produced in plant 2?
Cathy Poliak, Ph.D. [email protected] Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 20 / 20