Test 1 Review - University of Houstoncathy/Math2311/Lectures/Spring 18...Test 1 Scheduled February...

Test 1 ReviewReview

Cathy Poliak, Ph.D.cathy@math.uh.eduOffice in Fleming 11c

Department of MathematicsUniversity of Houston

Exam 1 Review

Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department of Mathematics University of Houston )Reveiw Exam 1 Review 1 / 28

Outline

1 Test 1 Reveiw

2 Chapter 1

3 Chapter 2

4 Chapter 3

Test 1

Scheduled February 15 - 17 & 19

Covers chapters 1, 2, and 3

What to Expect on the Exam

The test has three parts1. 24% of the grade is based on true/false questions. Three

questions.2. 32% of the grade is based on multiple choice questions. Four

questions.3. 44% of the grade is based on free response questions. Three free

response questions with multiple parts.Each T/F and mutliple choice question is worth 8 points, and one freeresponse is worth 16 points, the other two are worth 14 points each.

Examples of True/False Questions

1. A sample is the set of all possible data values for a given subjectunder consideration.

a. True b. False

2. X is the number of days it rained last month where you lived. X isan example of a discrete random variable.

a. True b. False

3. X is the amount of rainfall in your state last month. X is anexample of a continuous random variable.

a. True b. False

Possible Free Response Questions

From a list of observations find the mean, median, quartiles,variance and standard deviation.From a discrete probability distribution determine the mean,variance and standard deviation. Also, know the rules of meansand variances.From a two-way table determine the probability of the events.Using the probability general rules to find probability.Given sets determine the unions and intersections for each casedescribed.Finding probability of a binomial random variable.

Example 1

Among 13 electrical components exactly 3 are known not to functionproperly. If 7 components are randomly selected, find the followingprobabilities:a) The probability that all selected components function properly.

b) The probability that exactly 2 are defective.

c) The probability that at least 1 component is defective.

Example 2

Let A = {2,7}, B = {7,16,22}, D = {34} and S = sample space =A ∪ B ∪ D.a) Identify (Ac ∩ Bc)c .

b) Identify Ac ∩ B.

Example 3

a) A researcher randomly selects 4 fish from among 8 fish in a tankand puts each of the 4 selected fish into different containers. Howmany ways can this be done?

b) A person eating at a cafeteria must choose 4 of the 16 vegetableson offer. Calculate the number of elements in the sample space forthis experiment.

c) How many license plates can be made using 2 digits and 4 letters ifrepeated digits and letters are allowed?

Example 4

Suppose P(E) = 0.74, P(F) = 0.33, and P(E ∩ F ) = 0.19. Find each ofthe following:

a) P(E ∪ F )

b) P(E ∩ F c)

c) P(Ec ∩ F )

d) P(E ∪ F )c

e) P(E |F )

f) Are events E and Findependent?

Example 5

The following is a probability distribution:

X 1 2 3 4P(X) 0.15 0.2 0.1 ?

a) Find P(X = 4).

b) Find P(1 < X ≤ 3).

c) Find the mean of X.

d) Find the variance of X.

e) Find the standard deviation of X.

f) Define a new random variable Y = 2X - 1. Determine the mean andstandard deviation of Y.

Example 6

According to government data, 20% of employed women have neverbeen married. If 10 employed women are selected at random, what isthe probabilitya) That exactly 2 have never been married?

b) That at most 2 have never been married?

c) That at least 8 have been married?

d) What is the expected number of employed women that have neverbeen married out of the 10?

Example 7

The probability that a randomly selected person has high bloodpressure (the event H) is P(H) = 0.4 and the probability that a randomlyselected person is a runner (the event R) is P(R) = 0.4. The probabilitythat a randomly selected person has high blood pressure and is arunner is 0.1. Find the probability that a randomly selected personeither has high blood pressure or is a runner or both.

What You Need an What is Provided

ProvidedI Formula sheet; see CASA calendar for the formula sheet provided.I Online calculator; it will be a link you see in the exam.I R studio; it will be a link you see in the exam.

Can bringI Calculator; if it is memory based CASA will remove the memory.I Pencil; you will need something to write with for the free response

questions.I Your Cougar Card.

Chapter 1 Section 1

Sample - Simple random sample

Population versus sample

Parameter versus statistic.

Categorical variables

Quantitative variables; continuous and discrete

Chapter 1 sections 2, 3 and 4

Describing distributions with numbers

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

Chapter 1 section 5

Describing distributions with graphs

Categorical variables - bar chart, pie chart

Quantitative variables - histogram, stemplot, boxplot, dotplot

Chapter 2 section 1: 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

n!(n − r)!

Combinations counts the number of experimental outcomeswhen the experiment involves selecting r objects from a (usuallylarger) set of n objects. The number of combinations of n objectstaken r unordered at a time is

n!r !(n − r)!

Chapter 2 section 2: 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.

Chapter 2 section 3: Basic Probability Models

For any event A, the probability of A is

P(A) =number of times A occurstotal number of outcomes

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)

Chapter 2 section 4: 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)

Chapter 3 section 1: Random Variables

Discrete random variables has either a finite number of valuesor a countable number of values, where countable refers to thefact that there might be infinitely many values, but they result froma counting process.

Continuous random variables are random variables that canassume values corresponding to any of the points contained inone or more intervals.

Discrete Random Variable Probability Distribution

Suppose that X is a discrete random variable whose distribution is

Values of X x1 x2 x3 · · · xkProbability p1 p2 p3 · · · pk

To find the mean of the random variable X , multiply each possiblevalue by its probability, then add all the products:

µX = x1p1 + x2p2 + x3p3 + · · ·+ xkpk

xipi .

The variance of a discrete random variable X is

σ2X = (x1 − µX )

2p1 + (x2 − µX )2p2 + · · ·+ (xk − µX )

(xi − µX )2pi

The standard deviation of X is the square root of the variance

σX =√σ2

X .Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department of Mathematics University of Houston )Reveiw Exam 1 Review 25 / 28

Rules for Means and Variances

If X is a random variable and a and b be are fixed numbers, such thatwe add a to X and multiply b to X , a + bX , then

The mean of a + bX changes by what is added and multiplied:

µ(a+bX) = a + bµX

The variance of a + bX changes by the square of the multipliedvalue:

σ2(a+bX) = b2σ2

The standard deviation of a + bX is the square root of thevariance:

σ(a+bX) =√

b2σ2X = bσX

Chapter 3 section 2: Binomial Distribution

The distribution of the count X of successes in the Binomialsetting has a Binomial probability distribution.

Where the parameters for a binomial probability distribution is:I n the number of observationsI p is the probability of a success on any one observation

The possible values of X are the whole numbers from 0 to n.

The probability of selecting k successes out of n observationsuses the formula:

P(X = k) =n Ck (p)k (1− p)n−k

Popper Set Up

Fill in all of the proper bubbles.

Make sure your ID number is correct.

Make sure the filled in circles are very dark.

This is popper number 04.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department of Mathematics University of Houston )Section 4.1 Lecture 8 - 2311 3 / 29

Popper 04 Questions

Newsweek in 1989 reported that 60% of young children have bloodlead levels that could impair their neurological development. Assuminga random sample from the population of all school children at risk.Determine the type of distribution we have.

1. The probability that at least 5 children out of 10 in a sample takenfrom a school may have a blood lead level that may impairdevelopment.

a) Geometric b) Binomial c) None of these

2. The probability you will need to test 10 children before finding achild with a blood lead level that may impair development.

a) Geometric b) Binomial c) None of these

3. The probability you will need to test no more than 10 childrenbefore finding a child with a blood lead level that may impairdevelopment.

a) Geometric b) Binomial c) None of theseCathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department of Mathematics University of Houston )Section 4.1 Lecture 8 - 2311 4 / 29

Popper 04 Questions

For each random variable, determine if it is:

a. Discrete b. Continuous

4. The number of cars passing a busy intersection between 4:30 PMand 6:30 PM.

5. The weight of a fire fighter.

6. The amount of soda in a can of Pepsi.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department of Mathematics University of Houston )Section 4.1 Lecture 8 - 2311 5 / 29

Test 1 Review - University of Houstoncathy/Math2311/Lectures/Spring 18...Test 1 Scheduled February...

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