# Statistics for Undergraduate Studies

University Preparation

ePrep Course

##### Statistics for Undergraduate Studies is one of the ten ePrep courses, specially developed by NTU, in collaboration with Cengage, the publishers of the popular probability and Statistics textbook by Mendenhall, Beaver and Beaver. This ePrep course is meant to help NSFs, NSMen, and others, to better prepare for their university studies, whether in Singapore or overseas.

##### In addition to providing this textbook at no additional cost, this Statistics for Undergraduate Studies ePrep course also comes with excellent eLearning materials provided by the book publishers, Cengage.

##### There are also lots of materials on other subjects such as various branches of mathematics (including algebra, geometry, calculus and statistics), physics, mechanics, biotechnology, life science, business finance, corporate finance, engineering economy, economics, business and engineering ethics, psychology, Python programming, discrete mathematics, etc., so that the students not only get to build up a strong foundation on probability and statistics, they also get to strengthen their knowledge on many other subjects as well. Samples of materials provided can be found below. Most of these materials can be downloaded for later studies.

##### In this course, there is also a retired NTU professor acting as a personalized tutor. He can be reached via email or WhatsApp messaging. He will be happy to answer any queries or questions the students may have, not only during the course but until the students enter the universities and have their own tutors.

**Audio**： Fuller Intro to Stats for Undergrad studies

**Statistics for Undergraduate Studies**

**Learning Contents**

**I. Compulsory Chapters**

**Ch 1 Describing Data with Graphs**

1.1 Variables and Data

1.2 Types of Variables

1.3 Graphs for Categorical Data 11 Exercises

1.4 Graphs for Quantitative Data

Pie Charts and Bar Charts

Line Charts

Dotplots

Stem and Leaf Plots

Interpreting Graphs with a Critical Eye

1.5 Relative Frequency Histograms

**Ch 2 Describing Data with Numerical Measures**

2.1 Describing a Set of Data with Numerical Measures

2.2 Measures of Center

2.3 Measures of Variability

2.4 On the Practical Significance of the Standard Deviation

2.5 A Check on the Calculation of s

2.6 Measures of Relative Standing

2.7 The Five-Number Summary and the Box Plot

**Ch 3 Describing Bivariate Data**

3.1 Bivariate Data

3.2 Graphs for Categorical Variables

3.3 Scatterplots for Two Quantitative Variables

3.4 Numerical Measures for Quantitative Bivariate Data

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**Ch 4 Probability and Probability Distributions**

4.1 The Role of Probability in Statistics

4.2 Events and the Sample Space

4.3 Calculating Probabilities Using Simple Events

4.4 Useful Counting Rules (Optional)

4.5 Event Relations and Probability Rules

Calculating Probabilities for Unions and Complements

4.6 Independence, Conditional Probability, and the Multiplication Rule

4.7 Bayes’ Rule (Optional)

4.8 Discrete Random Variables and Their Probability Distributions

Random Variables

Probability Distributions

The Mean and Standard Deviation for a Discrete Random Variable

** **

**Ch 5 Several Useful Discrete Distributions**

5.1 Introduction

5.2 The Binomial Probability Distribution

5.3 The Poisson Probability Distribution

5.4 The Hypergeometric Probability Distribution

**Ch 6 The Normal Probability Distribution**

6.1 Probability Distributions for Continuous Random Variables

6.2 The Normal Probability Distribution

6.3 Tabulated Areas of the Normal Probability Distribution

The Standard Normal Random Variable

Calculating Probabilities for a General Normal Random Variable

6.4 The Normal Approximation to the Binomial Probability Distribution (Optional)

** **

**Ch 7 Sampling Distributions**

7.1 Introduction

7.2 Sampling Plans and Experimental Designs

7.3 Statistics and Sampling Distributions

7.4 The Central Limit Theorem

7.5 The Sampling Distribution of the Sample Mean

Standard Error

7.6 The Sampling Distribution of the Sample Proportion

7.7 A Sampling Application: Statistical Process Control (Optional)

A Control Chart for the Process Mean: The x Chart

A Control Chart for the Proportion Defective: The p Chart

**Ch 8 Large-Sample Estimation**

8.1 Where We’ve

8.2 Where We’re Going—Statistical Inference

8.3 Types of Estimators

8.4 Point Estimation

8.5 Interval Estimation

Constructing a Confidence Interval

Large-Sample Confidence Interval for a Population Mean mu

Interpreting the Confidence Interval

Large-Sample Confidence Interval for a Population Proportion p

8.6 Estimating the Difference between Two Population Means

8.7 Estimating the Difference between Two Binomial Proportions

8.8 One-Sided Confidence Bounds

8.9 Choosing the Sample Size

**II. Optional Chapters**

- Large-Sample Tests of Hypotheses
- Inference from Small Samples
- The Analysis of Variance
- Linear Regression and Correlation
- Multiple Regression Analysis
- Analysis of Categorical Data
- Non-Parametric Statistics

For complete listing of detailed topics, see here

**Statistics for Undergraduate Studies **

**What You Get in this University Preparation ePrep Course**

**I. Free Textbook**

##### “Introduction to Probability & Statistics” is a very popular introductory probability and statistics textbook, authored by Mendendall, Beaver and Beaver, 14th Ed.

**II. Free Consultation**

##### A retired NTU professor is acting as the tutor. You can consult him via email or WhatsApp.

**III. Materials Online**

##### 1 Notes, video lessons and PowerPoint files.

##### 2 Answers/solutions to all questions/problems in the textbook.

##### 3 Online exercises.

##### 4 Problems and solutions in files.

##### 5 Bonus learning materials in other branches of mathematics, including calculus, algebra, geometry, and trigonometry, as well as on other subjects such as business finance, corporate finance, engineering economy, economics, physics, ethics, life science, biotechnology, Python programming, and psychology.

**IV. Digital Certificate**

##### A digital certificate will be issued if you have successfully completed the course and passing all the tests at the end of each of the ten compulsory chapters.

**Audio**: What do you get in this Statistics for Undergraduate Studies?

**Statistics for Undergraduate Studies**

**Samples of**** Course Materials**

**1. Video Lesson **(Z-Scores)

This short video lesson discusses z-score which is the number of standard deviations from the mean and provides an illustrative example of its usefulness.

**2. Video Lesson **(Estimation of Standard Deviation)

##### This short video lesson discusses z-score which is the number of standard deviations from the mean and provides an illustrative example of its usefulness.

**3. Question and Answer **(Random Sampling)

**Question**: A psychology professor has received a grant to examine the association of words occurring within a given distance of each other in everyday prose writing. One way of attracting students to participate in this studywould be to advertise for volunteers with an offer to pay $10 an hour for participants. Would you consider students selected using this mechanism to comprise a random sample? Why or why not?

**Answer**: Although the mechanism used by the professor does not constitute a true random sample, the students selectedusing this method will probably behave like a random sample. The fact that they are being paid to participate should not cause their responses to the experiment to be substantially different from the responses of a randomly selected group of students.

**4. Multiple Choice Question Exercise** (Normal Distribution)

**Question**: Which of the following correctly describes the normal probability distribution?

(a) It is single-peaked above the random variable’s mean, median, and mode, all of which are equal to one another.

(b) It is perfectly symmetric about this peaked central value and, thus, said to be bell-shaped.

(c) It features tails extending indefinitely in both directions from the center, approaching (but never touching) the horizontal axis, which implies a positive probability for finding values of the random variable anywhere between minus infinity and plus infinity.

(d) All of the above.

(e) None of the above

**Answer**: (d)

**5. Problem with Solution** (Point Estimation)

**Question**: If a mental health agency would like to estimate the percentage of local clinic patients that are referred to their counseling center to within 5 percentage points with 90% accuracy, how many patient records should be sampled?

**Solution**: Maximum variation occurs when p = .5 Since 90% accuracy is involved, solve

1.645 √(*pq*/*n*)=.05,

1.645√((.5)(.5)/*n*)= .05,

√*n* = 1.645(.5) /.05 =16.45,

*n*= 270.6

271 patient records should be sampled.

**Samples of Bonus Materials beyond Statistics**

**1. Video on Mathematics** (Basic Rules of Differentiation)

##### This short video lesson discusses the sum and difference rules for derivatives and how to apply them

**2. Question and Answer on Corporate Finance **(Callable Bond and Convertible Bond)

**Question**: Explain who benefits from the option to call a bond, and who benefits from the option to convert a bond into shares.**Answer**: The call option benefits the issuer because it allows them to repurchase bonds at a fixed price. Issuers are likely to exercise this option when interest rates have fallen. Issuers repurchase the bonds and then issue new ones at a lower interest rate. The option to convert bonds into common stock benefits bondholders. Once the stock price rises high enough, the value of the bonds starts to behave like the value of the stock. So convertible bonds offer investors some minimal level of return plus a lot of upside potential

**3. Cross Word Puzzle on Biotechnology**

**4. Video Lesson on Physics** (Electro-Magnetic Field)

##### This short video lesson discusses the deflection of an electron beam under the influence of an electric field and a magnetic field.

**5. Video Lesson on Business Finance** (Free Cash Flow)

##### This short video explains how free cash flow is derived, what it measures, and why it is important.

**6. Video on Life Science** (Metastasis and Cancer)

##### This short video lesson discusses the metastasis (breaking loose and invading other parts of the body) of the cancer cells and the spread of cancer.

**7. Worked Example on Engineering Economy** (Cost Estimation)

**Question**: A car rental agency is considering a modification in its oil change procedure. Currently, it uses a Type X filter, which costs $5 and must be changed every 7,000 miles along with the oil (5 quarts). Between each oil change, one quart of oil must be added after each 1,000 miles. The proposed filter (Type Y) has to be replaced every 5,000 miles (along with 5 quarts of oil) but does not require any additional oil between filter changes. If the oil costs $1.08 per quart, what is the maximum acceptable price for the Type Y filter?

**Answer**. Type X filter: cost = $5, changed every 7,000 miles along with 5 quarts oil between each oil change 1 quart of oil must be added after each 1,000 miles

Type Y filter: cost = ?, changed every 5,000 miles along with 5 quarts of oil no additional oil between filter changes

oil = $1.08 / quart

Common multiple = 35,000 miles

For filter X = 5 oil changes: 5($5 + 5($1.08) + 6($1.08)) = (5)$16.88 = $84.40

For filter Y = 7 oil changes: 7C_{Y} + 7(5)($1.08) = 7X + $37.8

Equating the cost,

$84.40 = 7C_{Y} + $37.8

$46.60 = 7C_{Y}

CY = $6.66

**8. Glossary on Psychology** (Learning and Conditioning)

avoidance conditioning | The process of learning particular responses that avoid an aversive stimulus. |

classical conditioning | A procedure in which a neutral stimulus is paired with a stimulus that triggers a reflexive response until the neutral stimulus alone comes to trigger a similar response. |

cognitive map | A mental representation of the environment. |

conditioned response (CR) | The response triggered by the conditioned stimulus. |

conditioned stimulus (CS) | An originally neutral stimulus that now triggers a conditioned response. |

discriminative conditioned stimuli | Stimuli that signal whether reinforcement is available if a certain response is made. |

escape conditioning | The process of learning responses that stop an aversive stimulus. |

extinction | The gradual disappearance of a conditioned response. |

habituation | Reduced responsiveness to a repeated stimulus. |

higher order conditioning | A process through which a conditioned stimulus comes to signal another conditioned stimulus that is already associated with an unconditioned stimulus. |

insight | A sudden understanding of what is required to solve a problem. |

latent learning | Learning that is not demonstrated at the time it occurs. |

**9. Objective Question Exercise on Physics** (Current and Resistance)

**Question**: Three wires are made of copper having circular cross-sections. Wire 1 has a length *L* and radius *r*. Wire 2 has a length *L* and radius 2*r*. Wire 3 has a length 2*L* and radius 3*r*. Which wire has the smallest resistance?

1. wire 1

2. wire 2

3. wire 3

4. All have the same resistance.

5. Not enough information is given to answer the question.

**Answer**: (3). The resistances are:

*R1* = *ρ**L */ *A *= *ρ**L */ *π**r ^{2}*, :

*R2* = *ρ**L */ *A *= *ρ**L */ *π*(2*r*)^{2} = (¼) *ρ**L */ *π**r ^{2}*,

*R3* = *ρ* (2*L*) / *π*(3*r*)^{2} = (2/9) *ρ**L */ *π**r**2**.*

**10. Python Programming **(Insertion Sort)

**Code:**

def insertion_sort(array):

for i in range(1, len(array)):

temp = array[i]

j = i – 1

while (j >= 0 and temp < array[j]):

array[j + 1] = array[j]

j = j – 1

array[j + 1] = temp

print(‘temp array: ‘, array)

return array

oArray = [2, 8, 9, 4, 26, 82, 56, 43]

print(‘Orig Array: ‘, oArray)

nArray = insertion_sort(oArray)

print(‘Sorted Array’, nArray)

**Output:**

Orig Array: [2, 8, 9, 4, 26, 82, 56, 43]

temp array: [2, 8, 9, 4, 26, 82, 56, 43]

temp array: [2, 8, 9, 4, 26, 82, 56, 43]

temp array: [2, 4, 8, 9, 26, 82, 56, 43]

temp array: [2, 4, 8, 9, 26, 82, 56, 43]

temp array: [2, 4, 8, 9, 26, 82, 56, 43]

temp array: [2, 4, 8, 9, 26, 56, 82, 43]

temp array: [2, 4, 8, 9, 26, 43, 56, 82]

Sorted Array [2, 4, 8, 9, 26, 43, 56, 82]

**11. Economics** (Evaluating the Market Equilibrium)

- Buyers who value the product more than the equilibrium price will purchase the product; those who do not will not purchase the product.
- In other words, the free market allocates the supply of a good to the buyers who value it most highly, as measured by their willingness to pay.
- Sellers whose costs are less than the equilibrium price will produce the product; those whose costs are higher will not produce the product.
- In other words, the free market allocates the demand for goods to the sellers who can produce it at the lowest cost.

- At any quantity of output smaller than the equilibrium quantity, the value of the product to buyers is greater than the cost to sellers so total surplus would rise if output increases.
- At any quantity of output greater than the equilibrium quantity, the value of the product to buyers is less than the cost to sellers so total surplus would rise if output decreases.

3. Note that this is one of the reasons that economists believe the principle: Markets are usually a good way to organize economic activity.

**12. Discrete Mathematics** (Logic Circuit)

**Question**:

**Solution**:

**Remarks**

### Shown above are some samples of the bonus materials on other subjects and they illustrate how comprehensive and broad-base this Statistics for Undergraduate Studies e_Prep course is for preparing students for their university studies or for their careers. While not all the bonus course materials may be of interest to the students who take up this ePrep course on Statistics, they can choose which of these bonus course materials are of interest to them and disregard the rest.

Remember not to short-change yourself – do not go for any of those low-grade courses prepared by any “Tom-Dick-And-Harry” who self-claims to be an industry expert or subject expert, especially if you are preparing for further academic studies or career advancement! You do not need thousands of such courses. As you can see, with this single Statistics course, you have a good collection of materials on many other subjects as well. You also get a hard copy Introduction to Probability and Statistics textbook.

Go only for a high-quality specially-designed academic course such as this Statistics for Undergraduate Studies e_prep course by NTU for getting you a head start in university, or in your career.

**Who should take this Statistics University Preparation Course?**

### It is a very important course for students doing data science, computer science, psychology, sociology, economics, health care, politics, and engineering degrees, bearing in mind that statistics is about the most difficult discipline to comprehend and many students do face difficulties with it.

Even for those not going to any university due to various reasons, this is an opportunity to prove that they are capable of completing a university-level course on statistics.

Everyone is welcome and there is no pre-requisite.

**How to Sign Up for this Statistics University Preparation Course**

**1 NSF/NSMen**

Please sign up via **NS Portal** in order to obtain the NS e-PREP subsidy under the NS e-PREP Scheme. Please select “Acess ePREP”, and select “NTU” as the course provider. Then select “Statistics for Undergraduate Studies” among the NTU e-PREP courses listed.

There are reports by some NSFs that the course may be temporarily not listed in NS Portal due to unknown reasons. When that happens, please be patient and report the matter to eprep_enquiry@defence.gov.sg

Please note that since the course fee is $342.39, but the maximum NS e-PREP Credits is $350, there is no need for any payment from you. In case you encounter any registration issue, please contact the MINDEF e-Prep administrator at Tel No: (+65) 63731221, or eMail: EPrep_Enquiry@defence.gov.sg

**2 Non-NS Folks**

Please sign up with **NTU PaCE** directly. The course fees is S$342.40. Please note that if you reside outside Singapore, you may have to pay for the postage of the textbook.

**Course Duration**

### The official duration of the course is three months but may be extended upon request. Unofficially, however, the support by the tutor extends beyond the official course period. Also, most of the course materials can be downloaded for later study.

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**email: PaCE@NTU.edu.sg**

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