# 11.13: Exercises - Mathematics

## Skills

1. A political scientist surveys 28 of the current 106 representatives in a state’s congress. Of them, 14 said they were supporting a new education bill, 12 said there were not supporting the bill, and 2 were undecided.
1. What is the population of this survey?
2. What is the size of the population?
3. What is the size of the sample?
4. Give the sample statistic for the proportion of voters surveyed who said they were supporting the education bill.
5. Based on this sample, we might expect how many of the representatives to support the education bill?
2. The city of Raleigh has 9500 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 350 randomly selected registered voters was conducted. 112 said they’d vote for Brown, 207 said they’d vote for Feliz, and 31 were undecided.
1. What is the population of this survey?
2. What is the size of the population?
3. What is the size of the sample?
4. Give the sample statistic for the proportion of voters surveyed who said they’d vote for Brown.
5. Based on this sample, we might expect how many of the 9500 voters to vote for Brown?
3. Identify the most relevant source of bias in this situation: A survey asks the following: Should the mall prohibit loud and annoying rock music in clothing stores catering to teenagers?
4. Identify the most relevant source of bias in this situation: To determine opinions on voter support for a downtown renovation project, a surveyor randomly questions people working in downtown businesses.
5. Identify the most relevant source of bias in this situation: A survey asks people to report their actual income and the income they reported on their IRS tax form.
6. Identify the most relevant source of bias in this situation: A survey randomly calls people from the phone book and asks them to answer a long series of questions.
7. Identify the most relevant source of bias in this situation: A survey asks the following: Should the death penalty be permitted if innocent people might die?
8. Identify the most relevant source of bias in this situation: A study seeks to investigate whether a new pain medication is safe to market to the public. They test by randomly selecting 300 men from a set of volunteers.
9. In a study, you ask the subjects their age in years. Is this data qualitative or quantitative?
10. In a study, you ask the subjects their gender. Is this data qualitative or quantitative?
11. Does this describe an observational study or an experiment: The temperature on randomly selected days throughout the year was measured.
12. Does this describe an observational study or an experiment? A group of students are told to listen to music while taking a test and their results are compared to a group not listening to music.
13. In a study, the sample is chosen by separating all cars by size, and selecting 10 of each size grouping. What is the sampling method?
14. In a study, the sample is chosen by writing everyone’s name on a playing card, shuffling the deck, then choosing the top 20 cards. What is the sampling method?
15. A team of researchers is testing the effectiveness of a new HPV vaccine. They randomly divide the subjects into two groups. Group 1 receives new HPV vaccine, and Group 2 receives the existing HPV vaccine. The patients in the study do not know which group they are in.
1. Which is the treatment group?
2. Which is the control group (if there is one)?
3. Is this study blind, double-blind, or neither?
4. Is this best described as an experiment, a controlled experiment, or a placebo controlled experiment?
16. For the clinical trials of a weight loss drug containing Garcinia cambogia the subjects were randomly divided into two groups. The first received an inert pill along with an exercise and diet plan, while the second received the test medicine along with the same exercise and diet plan. The patients do not know which group they are in, nor do the fitness and nutrition advisors.
1. Which is the treatment group?
2. Which is the control group (if there is one)?
3. Is this study blind, double-blind, or neither?
4. Is this best described as an experiment, a controlled experiment, or a placebo controlled experiment?

## Concepts

1. A teacher wishes to know whether the males in his/her class have more conservative attitudes than the females. A questionnaire is distributed assessing attitudes.
1. Is this a sampling or a census?
2. Is this an observational study or an experiment?
3. Are there any possible sources of bias in this study?
2. A study is conducted to determine whether people learn better with spaced or massed practice. Subjects volunteer from an introductory psychology class. At the beginning of the semester 12 subjects volunteer and are assigned to the massed-practice group. At the end of the semester 12 subjects volunteer and are assigned to the spaced-practice condition.
1. Is this a sampling or a census?
2. Is this an observational study or an experiment?
3. This study involves two kinds of non-random sampling: (1) Subjects are not randomly sampled from some specified population and (2) Subjects are not randomly assigned to groups. Which problem is more serious? What affect on the results does each have?
3. A farmer believes that playing Barry Manilow songs to his peas will increase their yield. Describe a controlled experiment the farmer could use to test his theory.
4. A sports psychologist believes that people are more likely to be extroverted as adults if they played team sports as children. Describe two possible studies to test this theory. Design one as an observational study and the other as an experiment. Which is more practical?

## Exploration

1. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average length of time in months patients live once starting the treatment. Two researchers each follow a different set of 50 AIDS patients from the start of treatment until their deaths.
1. What is the population of this study?
2. List two reasons why the data may differ.
3. Can you tell if one researcher is correct and the other one is incorrect? Why?
4. Would you expect the data to be identical? Why or why not?
5. If the first researcher collected her data by randomly selecting 40 states, then selecting 1 person from each of those states. What sampling method is that?
6. If the second researcher collected his data by choosing 40 patients he knew. What sampling method would that researcher have used? What concerns would you have about this data set, based upon the data collection method?
2. Find a newspaper or magazine article, or the online equivalent, describing the results of a recent study (the results of a poll are not sufficient). Give a summary of the study’s findings, then analyze whether the article provided enough information to determine the validity of the conclusions. If not, produce a list of things that are missing from the article that would help you determine the validity of the study. Look for the things discussed in the text: population, sample, randomness, blind, control, placebos, etc.

Step 1 - Remove all the parentheses and distribute subtraction sign. This will change the sign of the terms inside the set of parenthesis that follows the subtraction sign.

Step 2 - Arrange each polynomial with the term with the highest degree first then in decreasing order of degree.

Step 3 - Group like terms then combine them by either adding or subtracting them. Remember that like terms are terms that has the same variable and exponent.

## MATH 280 (Spring 2013): Applied Differential Equation

Midterm Exam 1: (date to be annouced) in class.
Midterm Exam 2: (date to be annouced) in class.
Midterm Exam 3: (date to be annouced) in class.

### Lecture times and locations

Mondays & Wednesdays 2:00 pm - 3:15 pm in CR (Chaparral Hall) 5117

### Course text

A First Course in Differential Equations with modeling Applications (10th edition) by Dennis G. Zill.

### Announcements

1. QUIZ 1 will be in class on Monday (Jan 28, 2013) . Content on integration methods
2. The date of Midterm 1 will be announced in class.
3. Extra office hours: 10:00am - 11:45am (Friday, Mar 1, 2013)
4. Midterm 2 will be in class on Wednesday (April 3, 2013). There will be extra office hours (3pm - 5pm) on Tuesday April 2, 2013.
5. Midterm 3 will be in class on Monday (April 29, 2013). There will be extra office hours (3pm - 5pm) on Friday April 26, 2013 and (11am - 12pm) on Monday April 29, 2013.

### Course syllabus and tentative timetable

I will post all assigments, solutions and additional material in this space. You should therefore consult this spot frequently.

## Class 11 - Maths

Get NCERT solutions for Class 11 Maths Free with videos. All exercise questions, supplementary questions, examples and miscellaneous are solved with important questions marked.

Most of the chapters we will study in Class 11 forms a base of what we will study in Class 12. Forming a good base in Class 11 is important for good marks Class 12 Boards.

In each chapter, we have divided it into two parts - Serial Order Wise and Concept Wise.

Serial Order Wise is studying the chapter from the NCERT Book. This is useful when you want to look for a particular question or example.

Concept Wise is the Teachoo (टीचू) way of doing the chapter. First a topic is explained, and then their questions of that topic - from easy to difficult.

We suggest you do all the chapters from Concept Wise, so that your concepts are cleared. Which is important in competitive exams like JEE, GRE, GMAT as well as in Class 12.

In this class, the chapters and their topics include

Chapter 1 Sets &ndash What are sets, Roster & Set-builder form, Types of sets - Empty Set, Equal set, Finite & Infinite sets, Subsets, Universal Set, Power Set, Intervals, Venn Diagrams, Operation of sets - Intersection, Union, Complement, Difference

1. Direct Method
2. Contrapositive method
4. Using counter example

## MATH 117

Functions Modeling Change: A Preparation for Calculus, 5th Edition, Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason packaged with WileyPlus.

Chapter 1: Linear Functions and Change

1.1 Functions and Function Notation
1.2 Rates of Change
1.3 Linear Functions
1.4 Formulas for Linear Functions
1.5 Modeling with Linear Functions
1.6 Fitting linear functions to data

Chapter 2: Functions

2.1 Input and Output
2.2 Domain and Range
2.3 Piecewise-defined functions
2.4 Preview of Transformations: Shifts
2.5 Preview of Composite and Inverse Functions
2.6 Concavity

3.1 Introduction to the Family of Quadratic Functions
3.2 The vertex of a Parabola

Chapter 6: Transformations and their graphs

6.1 Shifts, reflections, and symmetry
6.2 Vertical stretches and compressions
6.3 Horizontal stretches and combinations of transformations

Chapter 11: Polynomial and Rational Functions

11.1 Power functions and proportionality
11.2 Polynomial functions
11.3 The short-run behavior of polynomials
11.4 Rational functions
11.5 The short-run behavior of rational functions

Abbreviations:

SR=Skills Review, E&P=Exercises and Problems, GT=Go Tutorial, AQ=Additional Questions, R=Review Exercises, IT=Intelligent Tutoring

Chapter 1: Linear Functions and Change

1.1 Functions and Function Notation
SR: 5, 8
E&P: 1, 5, 7, 9, 10, 13, 14, 17, 21, 22, 23, 24, 25, 26, 29, 32, 33, 35, 42, 46b

1.2 Rates of Change
SR: 4, 9
E&P: 8, 13, 14, 18, 26, 29a, 31ab

1.3 Linear Functions
SR: 1, 3, 6, 8
E&P: 2, 6, 8, 10, 13, 17, 18, 28, R14, R15, R35cd, R36ab, IT27

1.4 Formulas for Linear Functions
SR: 4, 9
E&P: 3, 6, 8, 10, 14, GT18, 27, 30, 36, GT41, 42, 45, 46ab, 49, 52ab, 53, 59

1.5 Modeling with Linear Functions
SR: 5, 7
E&P: 1, 3, 5, 8, 15, 16, 18, 23, 28acde, 30, 32, IT19

1.6 Fitting linear functions to data (optional)

Chapter 2: Functions

2.1 Input and Output

SR: 4, 8
E&P: 2, 4, 9, 10, 11, 14, 18, 20, 23, 26, 30, 44

2.2 Domain and Range

SR: 2, 7
E&P: 1, 6, 7, IT11, 13, 15, 24, 26, 29, 36, 38

2.3 Piecewise-defined functions

SR: 5, 8
E&P: 3, 4, 6, 9, 13, 15, 20, 22, 25abc, 26, 28, 31

2.4 Preview of Transformations: Shifts

E&P: IT1, 1, 7, 8, 9, 11, 13, 14, 16, 26

2.5 Preview of Composite and Inverse Functions

SR: 6, 10
E&P: 1, 4, 7, 8, 12, 15, 17, 24, IT26, R30, R31, 32, 34, 39, 40, 43, 47, 48, 55, GT56, 58

2.6 Concavity

E&P: 1, 3, 4, 7, 8, 10, 12, 13, 14, 16, 18, 19, 22, IT24, 24, 25

3.1 Introduction to the Family of Quadratic Functions

SR: 1, 6, 9
E&P: 3, 5, 9, 12, 13, 14, 24, 26, 29, 30, 32, 35, 37, 42ab

3.2 The vertex of a Parabola

SR: 3, 8
E&P: IT3, 3, 9, 10, 13, 28, 29, 31, 39, 40

Chapter 6: Transformations and their graphs

Skip any problems in this chapter using exponential or logarithmic functions.

6.1 Shifts, reflections, and symmetry

SR: 8
E&P: 3, AQ5, IT7, 8, 12, 14, 15, 16, 21, 22, 23, 24, 25, 27, 32, 38a, 44

6.2 Vertical stretches and compressions

SR: 1, 3
E&P: 1, 2, 4, GT6, 7acf, 13, 16, 19, 22, 25, 27, 28, 29, 35, 36, 37, 39

6.3 Horizontal stretches and combinations of transformations

SR: 4, 14
E&P: 1, 2, AQ7, 8, 10, GT12, 14, 15ab, IT15c, 17, 20, 25, 29, 30, 31, 38, 41, 44, 48, 51, 52

Chapter 11: Polynomial and Rational Functions

11.1 Power functions and proportionality

SR: 3, 5, 10
E&P: 1, 5, 8, 9, 10, 11, 12, 14, 15, 18, 22, 23, 26, 32, 35, 39, 43, 46

11.2 Polynomial functions

E&P: IT3, 6, 7, 11, 18, R18, 20, R21, GT26, 33, 34, 35, 36, 38, 40, 41, GT43, 44

11.3 The short-run behavior of polynomials

E&P: AQ1, 3, 6, 13, IT14, 19, 21, 26, 29, 31, R40, 42c, R68

11.4 Rational functions

SR: 2, 9
E&P: AQ1, 3, 9, 11, 13, 14, 15, 16, 19, R29, 23, 24, 25, 26, IT19, 31a, 33

11.5 The short-run behavior of rational functions

E&P: 1, 3, IT1, 5, 8, 11, 13, 14, 17, 20, 23, 24, 26, 34, 36, 39, 40, 41

### Center for Tutoring and Academic Excellence

The Center for Tutoring & Academic Excellence offers free collaborative learning opportunities that include small group tutoring and tutor-led study halls to Loyola students. To learn more or request tutoring services, visit the Center for Tutoring & Academic Excellence online at http://www.luc.edu/tutoring.

### Loyola Math Club Tutoring

The Loyola Math Club offers free tutoring to students in 100-level MATH courses (and others).

## 11.13: Exercises - Mathematics

Instructor: Steven G. Krantz

Office: Room 103, Cupples I

Office Hour: to be announced

Course Web Page: http://www.math.wustl.edu/

Department Office: Room 100, Cupples I

Department Phone : (314) 935-6760

Textbook: Multivariable Mathematics by Theodore Shifrin

This is a course in multivariable calculus, linear algebra, and introductory mathematical

analysis . We plan to cover at least the first six chapters of the text. More material may

be treated as time permits.

Two Midterms, each 20 % of the grade.

Homework, worth 20% of the grade.

A Final Exam, worth 40A% of the grade.

Be familiar with this Web page. This is where homework assignments will be posted,

due dates posted, and exams and other course events announced.

First Homework Assignment

Exercises 1.1: 1bdf, 2,5, 9, 12bdf

Exercises 1.2: 1bdf, 2, 5, 7, 12, 15

Exercises 1.3: 1bdf, 3, 6, 9, 12

Due Friday, January 25, 2013 in class.

Second Homework Assignment

Exercises 1.4: 1bdfh, 2a, 3, 7, 11, 13, 18, 25

Exercises 1.5: 4a, 5, 6bd, 7bd, 9, 14

Due Monday, February 4, 2013 in class.

Third Homework Assignment>

Exercises 2.1: 1bd, 2a, 9bd, 11a, 12ab

Exercises 2.2: 1bdf, 4, 7, 9, 14

Due Wednesday, February 13, 2013 in class.

The first Midterm Exam will be on Wednesday, February 20. It will be in class. It will cover Chapters 1, 2.

#### Solutions to Second Homework Assignment

Fourth Homework Assignment

Exercises 2.3: 1, 2, 7, 8abce, 13, 15

Exercises 3.1: 1ace, 2bd, 3b, 7, 9 10a

Due Friday, March 1 in class.

#### Solutions to First Midterm Exam

Fifth Homework Assignment

Exercises 3.2: 1bd, 2bd, 3bde, 7, 10, 15

Exercises 3.3: 1, 2, 3, 8, 11, 14

Due Monday, March 25, 2013 in class.

#### Solutions to Fourth Homework Assignment

Sixth Homework Assignment

Exercises 3.4: 1b, 2bd, 4, 6, 10

Exercises 3.6: 1, 2bd, 3b, 6, 7, 10a

Due Wednesday, April 3, 2013 in class.

#### Solutions to Fifth Homework Assignment

The Second Midterm Exam will be on Wednesday, April 10, 2013 in class. There will be no calculators or computers, but you can bring a 3 x 5 card with notes tothe exam.

Seventh Homework Assignment

Exercises 4.1: 3bdf, 4b, 6b, 9b, 12b

Exercises 4.2: 2bd, 3bd, 4ab, 7a

Due Monday, April 8, 2013 in class.

#### Solutions to Second Midterm Exam

Eighth Homework Assignment

Exercises 4.3: 1, 2bd, 3, 5, 12bd, 14bd

Exercises 5.1: 1bdf, 2, 4b, 8, 11

Due Monday, April 22, 2013 in class.

Ninth Homework Assignment (This is the last homework assignment for this class.)

Due Friday, April 26 in class.

The Final Exam for this class is Tuesday, May 7 at 10:30am in our usual meeting room (Seigle Hall Room L002).

## Study Guide :: Unit 4

In Unit 3, you learned about the probability distribution of a random variable and how to compute the associated mean, variance, and probabilities. Numerical quantities that describe probability distributions are called parameters. In practice, because this information is not available it must be estimated using statistical techniques.

The most accurate way of obtaining information about a population parameter would be to collect the relevant data from every member of that population. Such a procedure is impractical in most cases. For example, a tire manufacturer who wanted to know the average lifespan of their tires could not stay in business if they tested every tire until it wore out. Similarly, if we wanted to know the average lifespan of Canadians, we could not wait until all members of the Canadian population had died.

Hence, the best way to gather information about a population is to collect data from a representative sample of the population and make inferences about the population. The numerical descriptive measures (such as sample mean, sample proportion, sample standard deviation, etc.) are called statistics. However, statistics vary from sample to sample. Stated simply, if we consider all possible samples from a given population, we will find variability in the sample statistic that is, each sample statistic will have its own distribution. If all possible values of a sample statistic that might occur are organized into a probability distribution, the resulting distribution is called the sampling distribution.

This unit begins by discussing sampling distribution of means and proportions, and how the mean and standard deviation are related to the mean and standard deviation of the parent population. Central limit theorem, the foundation for the inferential branch of statistics, is introduced in this context. Once you understand the concept of sampling distributions and central limit theorem, you are ready to begin the study of inferential statistics, which is concerned with estimating a population parameter (characteristic) based on results observed from a sample.

In the rest of this unit, we discuss the two categories of inferential statistics: estimation and hypothesis testing. Estimation is the process of obtaining a single numerical value (point estimate) or a set of values (interval estimate or confidence interval) intended as a &ldquobest guess&rdquo of the unknown population parameter. In hypothesis testing, we test claims regarding a characteristic of one or more populations. The claims that we test concern the population mean and the population proportion.

### Sampling Distributions and the Central Limit Theorem

##### Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

1. Explain the term &ldquosampling distribution&rdquo and verify its properties.
2. Calculate the mean and standard deviation of the sampling distribution of sample means.
3. Describe and interpret central limit theorem.
4. Finding Probabilities for the Sample Mean.

Important Note: For help accessing the e-text resources referred to below, see the navigation notes under eText on the course home page.

Elementary Statistics, Chapter 5, Section 5.4 Sampling Distributions and the Central Limit Theorem

##### Try It Yourself Examples

Work through each Try It Yourself example in this section of the e-textbook. Check your work against the solutions provided.

Do the following exercises in your e-textbook:

Chapter 5, Section 5.4 Exercises 1, 5, 7, 9, 11, 13, 15. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

##### Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 5 Section 5.4 are available on the textbook publisher&rsquos MyStatLab website.

### Chapter 5.4 Review ( Extra Online Practice )

For more practice working with the topics in this chapter of the e-textbook, work through this review. Or, if you feel you have mastered this material, you may skip to the computer lab section of this unit.

##### Review Learning Objectives

Before proceeding to the online exercises, briefly review the Learning Objectives for the topic (below) presented in the previous section of this study guide:

##### Optional Practice in Study Plan at MyStatLab

For more practice on the topics/sections of this chapter of your e-textbook, visit MyStatLab, and work interactively through the exercises in the Study Plan. For help accessing this resource, see MyStatLab navigation hints on the course home page.

### Confidence Intervals for the Mean (&sigma Known)

##### Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

1. Explain the meaning of the key terms:
• point estimate interval estimate
• confidence interval level of confidence
• margin of error
2. Compute point estimate and margin of error for the population mean.
3. Construct and interpret intervals for the population mean.
4. Determine the minimum sample size required when estimating &mu.

Important Note: For help accessing the e-text resources referred to below, see the navigation notes under eText on the course home page.

Elementary Statistics, Chapter 6, Section 6.1 Confidence Intervals for the Mean (&sigma Known)

##### Try It Yourself Examples

Work through each Try It Yourself example in this section of the e-textbook. Check your work against the solutions provided.

Do the following exercises in your e-textbook:

Chapter 6, Section 6.1 Exercises 3, 35, 37, 41, 49, 55. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

##### Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 6, Section 6.1 are available on the textbook publisher&rsquos MyStatLab website.

### Confidence Intervals for the Mean (&sigma Unknown)

##### Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

1. Interpret t distribution and use a t-distribution table.
2. Know the properties of students&rsquo t-distribution.
3. Construct confidence intervals when n < 30, the population is normally distributed, and &sigma is known.

Important Note: For help accessing the e-text resources referred to below, see the navigation notes under eText on the course home page.

Elementary Statistics, Chapter 6, Section 6.2 Confidence Intervals for the Mean (&sigma Unknown)

##### Try It Yourself Examples

Work through each Try It Yourself example in this section of the e-textbook. Check your work against the solutions provided.

Do the following exercises in your e-textbook:

Chapter 6, Section 6.2 Exercises 1, 3, 5, 7, 9, 13, 17, 21. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

##### Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 6 Section 6.2 are available on the textbook publisher&rsquos MyStatLab website.

### Confidence Intervals for Population Proportions

##### Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

1. Obtain a point estimate for the population proportion.
2. Construct and interpret confidence intervals for the population proportion.
3. Determine the minimum sample size required for estimating a population proportion within a specified margin of error.

Important Note: For help accessing the e-text resources referred to below, see the navigation notes under eText on the course home page.

Elementary Statistics, Chapter 6, Section 6.3 Confidence Intervals for Population Proportions

##### Try It Yourself Examples

Work through each Try It Yourself example in this section of the e-textbook. Check your work against the solutions provided.

Do the following exercises in your e-textbook:

Chapter 6, Section 6.3 Exercises 1, 3, 7, 13, 19, 21, 23, 25. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

##### Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 6 Section 6.3 are available on the textbook publisher&rsquos MyStatLab website.

### Chapter 6 Review ( Extra Online Practice )

For more practice working with the topics in this chapter of the e-textbook, work through this review. Or, if you feel you have mastered this material, you may skip to Computer Lab 4A.

##### Review Learning Objectives

Before proceeding to the online exercises, briefly review the Learning Objectives for each of the following topics (listed below), which are presented in previous sections of this study guide.

• Confidence Interval for Mean (&sigma Known)
• Confidence Interval for Mean (&sigma Unknown)
• Confidence Interval for Population Proportion
##### Optional Practice in Study Plan at MyStatLab

For more practice on the topics/sections of Chapter 6, visit MyStatLab, and work interactively through the exercises in the Study Plan. For help accessing this resource, see MyStatLab navigation hints on the course home page.

### Computer Lab 4A

In Computer Lab 4A, you will learn to use StatCrunch to develop solutions to exercises related to topics in the e-textbook&rsquos Chapter 5 and 6.

##### Computer Lab 4A Detailed Instructions

Your Computer Lab activities and the detailed step-by-step instructions (Guided Solutions) that will guide you in using StatCrunch to complete these are in the Computer Lab 4A file.

##### Computer Lab 4A Quick Reviews

The Quick Reviews (QRs) summarize a few key steps (but not all steps) needed to complete each Activity in Computer Lab 4A. These QRs will be useful when you are preparing for the computer components of the assignments, midterm exam, and final exam. To access, the QRs, click Computer Lab 4A QRs.

### Introduction to Hypothesis Testing with One Sample

##### Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

1. Determine the null and alternative hypotheses from a claim.
2. Distinguish between type I and type II errors.
3. Interpret the level of significance.
4. Determine whether to use a one-tailed or two-tailed statistical test.
5. Compute and interpret P-value.
6. Make and interpret a decision based on the results of a hypothesis test.

Important Note: For help accessing the e-text resources referred to below, see the navigation notes under eText on the course home page.

Elementary Statistics, Chapter 7, Section 7.1 Introduction to Hypothesis Testing

##### Try It Yourself Examples

Work through each Try It Yourself example in this section of the e-textbook. Check your work against the solutions provided.

Do the following exercises in your e-textbook:

Chapter 7, Section 7.1 Exercises 1, 11, 13, 15, 21, 29, 31, 33, 35, 37, 41, 43, 45, 51. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

##### Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 7 Section 7.1 are available on the MyStatLab website.

### Hypothesis Testing for the Mean (&sigma Known)

##### Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

1. Use P-values to make decisions.
2. Use P-values in a z-test.
3. Construct critical (rejection) regions and critical values.
4. Use rejection regions in a z-test.

Important Note: For help accessing the e-text resources referred to below, see the navigation notes under eText on the course home page.

Elementary Statistics, Chapter 7, Section 7.2 Hypothesis Testing for the Mean (&sigma Known)

##### Try It Yourself Examples

Work through each Try It Yourself example in this section of the e-textbook. Check your work against the solutions provided.

Do the following exercises in your e-textbook:

Chapter 7, Section 7.2 Exercises 1, 3, 9, 15, 19, 25, 33, 37, 39. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

##### Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 7, Section 7.2 are available on the MyStatLab website.

### Hypothesis Testing for the Mean (&sigma Unknown)

##### Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

1. Find critical values in a t distribution.
2. Apply the t-test to test a mean &mu, using the critical values/rejection region approach.

Important Note: For help accessing the e-text resources referred to below, see the navigation notes under eText on the course home page.

Elementary Statistics, Chapter 7, Section 7.3 Hypothesis Testing for the Mean (&sigma Unknown)

##### Try It Yourself Examples

Work through each Try It Yourself example in this section of the e-textbook. Check your work against the solutions provided.

Do the following exercises in your e-textbook:

Chapter 7, Section 7.3 Exercises 1, 3, 9, 11, 13, 19, 21, 25, 27. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

##### Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 7 Section 7.3 are available on the MyStatLab website.

Note 1: Unless otherwise stated, always use the critical values/rejection region approach when using your calculator to work through hypotheses test exercises and problems in the Exercises sections of your textbook, in the unit Self-Test Theory Components, and Assignment Theory Components for the remainder of this course.

Note 2: Unless otherwise stated, always use the P-value approach when using your computer, with StatCrunch, to work through hypotheses test exercises and problems in the Computer Labs, Unit Self-test Computer Components, and Assignment Computer Components for the remainder of this course.

### Hypothesis Testing for Proportions

##### Learning Objective

After completing the readings and exercises assigned for this topic, you should be able to:

1. Apply z in hypotheses tests involving a population proportion, using the critical values/rejection region approach.

Important Note: For help accessing the e-text resources referred to below, see the navigation notes under eText on the course home page.

Elementary Statistics, Chapter 7, Section 7.4 Hypothesis Testing for Proportions

##### Try It Yourself Examples

Work through each Try It Yourself example in this section of the e-textbook. Check your work against the solutions provided.

Do the following exercises in your e-textbook:

Chapter 7, Section 7.4 Exercises 1,3, 5, 7, 9, 11, 13. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

##### Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 7 Section 7.4 are available on the MyStatLab website.

Note: Use the critical values/rejections region approach to conduct test of hypotheses exercises, using your calculator.

### Chapter 7 Review ( Extra Online Practice )

For more practice working with the topics in this chapter of the e-textbook, work through this review. Or, if you feel you have mastered this material, you may skip to Computer Lab 4B.

##### Review Learning Objectives

Before proceeding to the online exercises, briefly review the Learning Objectives for each of the following topics (listed below), which are presented in previous sections of this study guide.

• Introduction to Hypothesis Testing with One Sample
• Hypothesis Testing for Mean (&sigma Known)
• Hypothesis Testing for Mean (&sigma Unknown)
• Hypothesis Testing for Proportions
##### Optional Practice in Study Plan at MyStatLab

If you would like more practice on the various topics/sections of Chapter 7, you may wish to visit the website that accompanies your textbook, and work interactively through online exercises located in the Study Plan. For help accessing this resource, see MyStatLab navigation hints on the course home page.

### Computer Lab 4B

In Computer Lab 4B, you will learn to use StatCrunch to develop solutions to exercises related to topics in the e-textook&rsquos Chapter 7.

##### Computer Lab 4B Detailed Instructions

Your Computer Lab activities and the detailed step-by-step instructions (Guided Solutions) that will guide you in using StatCrunch to complete these are in the Computer Lab 4B file.

##### Computer Lab 4B Quick Reviews

The Quick Reviews (QRs) summarize a few key steps (but not all steps) needed to complete each Activity in Computer Lab 4B. These QRs will be useful when you are preparing for the computer components of the assignments, midterm exam, and final exam. To access, the QRs, click Computer Lab 4B QRs.

### Self-test 4

To access Self-Test 4, click MATH 216 Self-Test 4.

It is important that you work through all the exercises in the unit self-tests and the e-text chapter quizzes. No grades are assigned to the self-tests. They are designed to, along with the unit assignments, help you master the content presented in each unit.

Each unit self-test has two parts: one on theory (A) and one on computer work (B). Working through these will help you review key exercises in the unit, which will help you prepare for assignments and exams.

### Assignment 4

After completing Self-Test 4, complete Assignment 4, which you will find on the course home page. Submit your solutions to this assignment to your tutor for marking.

## 11.13: Exercises - Mathematics

### (v) A = , B = Φ

(i) X = <1, 3, 5>Y = <1, 2, 3>

So, the union of the pairs of set can be written as

X ∪ Y= <1, 2, 3, 5>

(ii) A = B =

B = = <7, 8, 9>

So, the union of the pairs of set can be written as

A∪ B = <2, 3, 4, 5, 6, 7, 8, 9>

Hence, A∪ B =

(v) A = <1, 2, 3>, B = Φ

So, the union of the pairs of set can be written as

A∪ B =

### Question 2. Let A = , B = . Is A ⊂ B? What is A ∪ B?

It is given that

A = and B =

Yes, A ⊂ B

So, the union of the pairs of set can be written as

A∪ B = = B

### Question 3. If A and B are two sets such that A ⊂ B, then what is A ∪ B?

If A and B are two sets such that A ⊂ B, then A ∪ B = B.

### Question 4. If A = , B = , C = and D = find

(vii) B ∪ C ∪ D

It is given that

A = <1, 2, 3, 4], B = <3, 4, 5, 6>, C = <5, 6, 7, 8>and D = <7, 8, 9, 10>

(i) A ∪ B = <1, 2, 3, 4, 5, 6>

(ii) A ∪ C = <1, 2, 3, 4, 5, 6, 7, 8>

(iii) B ∪ C = <3, 4, 5, 6, 7, 8>

(iv) B ∪ D = <3, 4, 5, 6, 7, 8, 9, 10>

(v) A ∪ B ∪ C = <1, 2, 3, 4, 5, 6, 7, 8>

(vi) A ∪ B ∪ D = <1, 2, 3, 4, 5, 6, 7, 8, 9, 10>

(vii) B ∪ C ∪ D =

### (i) X = Y = (ii) A = B = (iii) A = (v) A = , B = Φ

(i) X = <1, 3, 5>, Y = <1, 2, 3>

So, the intersection of the given set can be written as

X ∩ Y = <1, 3>

(ii) A = , B =

So, the intersection of the given set can be written as

A ∩ B =

(iii) A = = (3, 6, 9 …>

B = = <1, 2, 3, 4, 5>

So, the intersection of the given set can be written as

A ∩ B = <3>

(iv) A = = <2, 3, 4, 5, 6>

B = = <7, 8, 9>

So, the intersection of the given set can be written as

A ∩ B = Φ

(v) A = <1, 2, 3>, B = Φ

So, the intersection of the given set can be written as

A ∩ B = Φ

### Question 6. If A = , B = , C = and D = find

(iii) A ∩ C ∩ D

(viii) A ∩ (B ∪ D)

(ix) (A ∩ B) ∩ (B ∪ C)

(x) (A ∪ D) ∩ (B ∪ C)

(i) A ∩ B = <7, 9, 11>

(ii) B ∩ C = <11, 13>

(iii) A ∩ C ∩ D = ∩ D

= <11>∩ <15, 17>

= Φ

(iv) A ∩ C = <11>

(v) B ∩ D = Φ

(vi) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

= <7, 9, 11>∪ <11>

= <7, 9, 11>

(vii) A ∩ D = Φ

(viii) A ∩ (B ∪ D) = (A ∩ B) ∪ (A ∩ D)

= <7, 9, 11>∪ Φ

= <7, 9, 11>

(ix) (A ∩ B) ∩ (B ∪ C) = <7, 9, 11>∩ <7, 9, 11, 13, 15>

= <7, 9, 11>

(x) (A ∪ D) ∩ (B ∪ C) = <3, 5, 7, 9, 11, 15, 17) ∩ <7, 9, 11, 13, 15>

=

### (iii) and

(i) <1, 2, 3, 4>

= <4, 5, 6>

So, we get

<1, 2, 3, 4>∩ <4, 5, 6>= <4>

Hence, this pair of sets is not disjoint.

(ii) ∩ (c, d, e, f> =

Hence, and (c, d, e, f> are not disjoint.

(iii) = Φ

Hence, this pair of sets is disjoint.

### Question 9. If A = , B = , C = , D = find

(i) A – B = <3, 6, 9, 15, 18, 21>

(ii) A – C = <3, 9, 15, 18, 21>

(iii) A – D = <3, 6, 9, 12, 18, 21>

(iv) B – A = <4, 8, 16, 20>

(v) C – A = <2, 4, 8, 10, 14, 16>

(vi) D – A = <5, 10, 20>

(vii) B – C = <20>

(viii) B – D = <4, 8, 12, 16>

(ix) C – B = <2, 6, 10, 14>

(x) D – B = <5, 10, 15>

(xi) C – D = <2, 4, 6, 8, 12, 14, 16>

(xii) D – C =

(i) X – Y =

(ii) Y – X =

(iii) X ∩ Y =

### Question 11. If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?

We know that

R – Set of real numbers

Q – Set of rational numbers

Hence, R – Q is a set of irrational numbers.

### Question 12. State whether each of the following statement is true or false. Justify your answer.

(i) <2, 3, 4, 5>and <3, 6>are disjoint sets.

(ii) and are disjoint sets.

(iii) <2, 6, 10, 14>and <3, 7, 11, 15>are disjoint sets.

## NCERT Solutions class-11 Maths Exercise 13.1

CBSE, NCERT, JEE Main, NEET-UG, NDA, Exam Papers, Question Bank, NCERT Solutions, Exemplars, Revision Notes, Free Videos, MCQ Tests & more.

Exercise 13.1

Evaluate the following limits in Exercises 1 to 22.

Ans. 3 + 3 = 6

Dividing numerator and denominator by

23. Find and where

24. Find where

25. Evaluate where

Therefore, this limit does not exist at

26. Find where

Therefore, this limit does not exist at

27. Find where

Therefore, this limit exists at and

28. Suppose and if what are possible values of and ?

Ans. Given: and

Putting values from eq. (ii) and (iii) in eq. (i), we get

On solving these equation, we get and

29. Let be fixed real numbers and define a function What is ? For some compute

By an arithmetic progression of $m$ terms, we mean a finite sequence of the form

$a, a + d, a + 2d, a + 3d, . . . , a + ( m - 1)d.$

The real number $a$ is called the first term of the arithmetic progression, and the real number $d$ is called the difference of the arithmetic progression.

Consider the sequence of numbers

$1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23$

The property of this sequence is that the difference between successive terms is constant and equal to 2.

Here we have: $a = 1$ $d = 2$.

Consider the sequence of numbers:

$2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32$

The property of this sequence is that the difference between successive terms is constant and equal to 3.

Here we have: $a = 2$ $d = 3$.

### General term of arithmetic progression:

The general term of an arithmetic progression with first term $a_1$ and common difference $d$ is:

Example 3: Find the general term for the arithmetic sequence $-1, 3, 7, 11, . . .$ and then find $a_<12>$.

Here $a_1 = 1$. To find $d$ subtract any two adjacent terms: $d = 7 - 3 = 4$. The general term is:

Example 4: If $a_3 = 8$ and $a_6 = 17$, find $a_<14>$.

Use the formula for $a_k$ with the given terms

$egin a_3 &= a_1 + (3 - 1) cdot d 8 &= a_1 + 2d a_6 &= a_1 + (6 - 1) cdot d 17 &= a_1 + 5d end$

This gives us a system of two equations with two variables. By solving them, we can find that $a_1 = 2$ and $d=3$.

Use the formula for $a_k$ to find $a_<14>$

$egin a_k &= a_1 + (k - 1) cdot d a_ <14>&= 2 + (14 - 1) cdot 3 color> &color <=>color <41>end$