exam 1 review - review - math.uh.educathy/math3339/lecture/exam 1 review 3339c.pdf · know how to...

Exam 1 ReviewReview

Cathy Poliak, Ph.D.cathy@math.uh.eduOffice: Fleming 11C

Department of MathematicsUniversity of Houston

Test 1 Review

Cathy Poliak, Ph.D. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu 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. cathy@math.uh.edu Office: Fleming 11C (Department of Mathematics University of Houston )Review Test 1 Review 20 / 20