Mathematics 215: Introduction to Statistics
Study Guide
Unit 1: Descriptive Statistics
Unit 1 introduces the field of statistics and the areas within it, presents many of the terms used throughout this course, and examines common methods employed to organize, display, and summarize data.
Typically, the statistics practitioner, faced with a specific problem, research objective, or decision, begins their work by collecting a body of numerical facts, called raw data, through surveys, through observation, or from internal or external information sources. After gathering this data, the practitioner must organize it and present the results in such a way that coherent, relevant information about the problem, objective, or decision emerges.
The set of methods used to organize, display, and describe data is called descriptive statistics, and is the subject of this unit.
We will now examine what a statistics practitioner does with the vast quantity of numbers that form the raw data: how they organize it, present it in tables and graphs, and compute various summary measures of location, variability, and position.
As the seemingly unrelated raw data takes on a meaningful form, we can appreciate how these numbers say something about our lives, our society, and our universe.
Unit 1 of MATH 215 consists of the following sections:
1-1 Statistics and Basic Terms 1-2 Types of Variables and the Nature of Statistical Data 1-3 Population, Sampling, Design of Experiments, and Summation Notation 1-4 Organizing and Graphing Qualitative Data 1-5 Organizing and Graphing Quantitative Data 1-6 Measures of Central Tendency for Ungrouped Data 1-7 Measures of Dispersion for Ungrouped Data 1-8 Mean, Variance, and Standard Deviation for Grouped Data 1-9 Using Standard Deviation 1-10 Measures of Position and Box-and-Whisker Plots
The unit also contains a self-test. When you have completed the material for this unit, including the self-test, complete Assignment 1.
Section 1-1: Statistics and Basic Terms
Outcomes
After completing the readings and exercises for this section, you should be able to define, and use in context, the following key terms:
- descriptive statistics and inferential statistics
- element, variable, observation and data set
Reading
Read the following sections in Chapter 1 of the textbook:
- Chapter 1 Introduction
- Section 1.1
- Section 1.2
Note: In this and all subsequent units, all readings are in the textbook Introductory Statistics, 9th ed, by Prem S. Mann. Any page numbers given here refer to the downloadable digital version of the textbook (eText) made available by VitalSource. If you haven’t already done so, access or download it now through the link on the course home page. You may also complete the readings in the interactive textbook in WileyPLUS, but the page numbers given for the exercises do not apply to that version.
Be prepared to read the material in Chapter 1 twice—the first time for a general overview of topics, and the second time to concentrate on the terms and examples presented. Return to these sections when you need to review these topics.
Supplementary Video Resources
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Chapter 1 of the textbook.
- Introduction to Statistics and Graphical Displays of Data (Jeremy Haselhorst)
- Introduction to Descriptive Statistics – an Overview (Teresa Johnson)
- Sampling: Simple Random, Convenience, Systematic, Cluster, Stratified (Dr Nic’s Maths and Stats)
- Note: You can also check Appendix A for more information on sampling techniques.
- Types of Data: Nominal, Ordinal, Interval/Ratio – an Overview (Dr Nic’s Maths and Stats)
- Appropriate Data Displays – an Overview (Rob Oliver)
Exercises
Complete the following exercises from Chapter 1 of the textbook (page numbers are for the downloadable eText):
- Exercise 1.3 on page 5
- Exercise 1.5 on page 6
Show your work as you develop your answers.
Solutions to these exercises are provided in the Student Solutions Manual for Chapter 1 in the left-hand navigation column in the interactive textbook (accessible from the Read, Study & Practice link on the course home page) and on page AN1 in the Answers to Selected Odd-Numbered Exercises section in the downloadable eText.
It is very important that you make a concerted effort to answer each question independently before you refer to the solutions. If your answers differ from those provided and you cannot understand why, contact your tutor for assistance.
Note: At the end of each chapter of the textbook, there are instructions for how to complete the statistical calculations, graphs, and processes for that chapter using a TI-84 calculator, Microsoft Excel, and Minitab. You are not required to use a TI-84 calculator or to learn these statistical software programs for MATH 215. However, if you happen to have access to this calculator or these applications, you may use them to double-check your work.
You are also not permitted to use a TI-84 calculator, Microsoft Excel, or Minitab on the midterm or the final exam for this course. The only calculator you are allowed to bring into the exam room is the Texas Instruments TI-30Xa Scientific Calculator. You should familiarize yourself with its functionality now so that you can complete the calculations as required on the assignments and exams.
See the Calculators section of the Course Orientation for more information.
Section 1-2: Types of Variables and the Nature of Statistical Data
Outcomes
After completing the readings and exercises for this section, you should be able to do the following:
- define, and use in context, the following key terms:
- quantitative, discrete, continuous, and qualitative (categorical) variables
- cross-section and time-series data
- compute the values for expressions that are presented in summation notation.
Reading
Read the following sections in Chapter 1 of the textbook:
- Section 1.3
- Section 1.4
Exercises
Complete the following exercises from Chapter 1 of the textbook (page numbers are for the downloadable eText):
- Exercises 1.7 and 1.9 on page 8
- Exercise 1.11 on page 10
Solutions are provided in the Student Solutions Manual for Chapter 1 in the interactive textbook. Please note that the solutions to exercises 1.7 and 1.11 are not included in the Answers to Selected Odd-Numbered Exercises section of the downloadable eText.
Section 1-3: Population, Sampling, Design of Experiments, and Summation Notation
Outcomes
After completing the readings and exercises for this section, you should be able to do the following:
- contrast the following:
- population vs. sample
- census vs. sample survey
- sampling with replacement vs. sampling without replacement
- random vs. non-random samples
- sampling vs. non-sampling errors
- identify types of non-random samples
- identify random sampling techniques
- define the following terms:
- treatment
- randomization
- designed experiment
- contrast the following:
- designed experiment vs. observational study
- treatment vs. control group
Reading
Read the following sections in Chapter 1 of the textbook:
- Section 1.5
- Section 1.6
- Section 1.7
Exercises
- Complete the following exercises from 1 of the textbook (page numbers are for the downloadable eText):
- Exercises 1.13, 1.15, and 1.19 on page 17
- Exercises 1.21, 1.23, 1.25, and 1.27 on page 18
- Exercises 1.31, 1.33, and 1.35 on page 22
- Exercises 1.37 and 1.39 on page 24
Complete the Self-Review Test for Chapter 1 (pages 28–29 of the downloadable eText).
Solutions are provided in the Student Solutions Manual for Chapter 1 (interactive textbook) and on pages AN1 to AN2 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).
Note: At the end of each chapter of the textbook, there are instructions for how to complete the statistical calculations, graphs, and processes for that chapter using a TI-84 calculator, Microsoft Excel, and Minitab. You are not required to use a TI-84 calculator or to learn these statistical software programs for MATH 215. However, if you happen to have access to this calculator or these applications, you may use them to double-check your work.
You are also not permitted to use a TI-84 calculator, Microsoft Excel, or Minitab on the midterm or the final exam for this course. The only calculator you are allowed to bring into the exam room is the Texas Instruments TI-30Xa Scientific Calculator. You should familiarize yourself with its functionality now so that you can complete the calculations as required on the assignments and exams.
See the Calculators section of the Course Orientation for more information.
Optional Extra Practice
At the end of Chapter 1 (pages 26–28 of the downloadable eText), there are both Supplementary Exercises and Advanced Exercises, with solutions provided for the odd-numbered questions. You can work through these exercises for additional practice with these concepts and techniques.
Section 1-4: Organizing and Graphing Qualitative Data
Outcomes
After completing the readings and exercises for this section, you should be able to do the following:
- construct a frequency distribution that includes frequencies, relative frequencies, and percentage frequencies, given raw data for a qualitative (categorical) variable.
- construct a bar graph and a pie chart.
- interpret frequencies, relative frequencies, and percentage frequencies, given a frequency distribution or a graph relating to a frequency distribution.
Reading
Read the following sections in Chapter 2 of the textbook:
- Chapter 2 Introduction
- Section 2.1
Be prepared to read the material in Chapter 2 at least twice—the first time for a general overview of topics, and the second time to concentrate on the terms and examples presented. Return to these sections when you need to review these topics.
Supplementary Video Resources
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 2.1 of the textbook.
Videos Related to Section 2.1
- Categorical Frequency Distributions (mattemath)
- Art of Problem Solving: Bar Charts and Pie Charts (Art of Problem Solving)
- Calculating values for a pie chart (Mr Potts Math)
- Using a pie graph to find the amount (adumas2009)
Exercises
Complete Exercise 2.5 (page 43 of the downloadable eText).
Solutions are provided in the Student Solutions Manual for Chapter 2 (interactive textbook) and on page AN2 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).
Section 1-5: Organizing and Graphing Quantitative Data
Outcomes
After completing the readings and exercises for this section, you should be able to do the following:
-
construct a frequency distribution table that uses either a “less than” or “not less than” method for writing the classes, given raw data for a continuous variable.
Note: This type of distribution can include class limits, class boundaries, midpoints, raw data frequencies, relative frequencies, percentage frequencies, cumulative frequencies, cumulative relative frequencies, and cumulative percentage frequencies.
- construct the following graphs: histogram, relative frequency histogram, frequency polygon, relative or percentage frequency polygon, ogive, and relative or percentage ogive.
-
construct a frequency distribution table using single-valued classes, given raw data.
Note: This type of distribution can include frequencies, relative frequencies, and percentage frequencies.
- construct a bar graph for the distribution described in Outcome 3, above.
- interpret frequencies, relative and percentage frequencies, cumulative frequencies, cumulative relative frequencies, and cumulative percentage frequencies, given a frequency distribution or a related graph.
- interpret symmetric, skewed and uniform distributions for the frequency distribution or graph described in Outcome 5, above.
- construct stem-and-leaf displays and dotplots, and identify possible outliers, given raw data.
Reading
- Read the following sections in Chapter 2 of the textbook:
- Section 2.2
- Section 2.3
- Section 2.4
- Read Additional Topics 1A and 1B in this Study Guide, below.
Supplementary Video Resources
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned textbook readings.
Videos Related to Section 2.2
- Dancing Statistics: explaining the statistical concept of ‘frequency distributions’ through dance (BPSOfficial)
- Statistics – Displaying Data (DrCraigMcBridePhD)
- How to Construct a Grouped Frequency Distribution (Joshua French)
- Constructing a Grouped Frequency Distribution Part 1 (Math and Stats)
- Constructing a Grouped Frequency Distribution Part 2 (Math and Stats)
- How to Create a Frequency Polygon (mattemath)
- Relative frequency histogram, polygon and ogive graphs (mrandersonmath)
- How to Draw a Histogram by Hand (MathBootcamps)
- Reading and Analyzing a Histogram (Dan Ozimek)
- Analyzing Histograms Reminder (Michael Branson)
- The Different Shapes of Frequency Distributions (mattemath)
- How to Construct a Cumulative Frequency Graph or Ogive (mattemath)
Videos Related to Section 2.3
- Statistics – How to Make a Stem-and-Leaf Plot (MySecretMathTutor)
- Stem-and-Leaf Plots (Anywhere Math)
- Stem-and-Leaf Plot – Mean, Median and Mode (Robert Boulet)
- Practice Exercises: Graphs and Plots (lbowen11235)
Videos Related to Section 2.4
- Dot Plots and Frequency Tables (Nicole Pellegrino)
- Dot Plots and Frequency Tables (Greg Wood)
Exercises
Complete the following exercises from Chapter 2 of the textbook (page numbers are for the downloadable eText):
- Exercises 2.11 and 2.13 on page 58
- Exercises 2.15 on page 59
- Exercise 2.21 on page 60
- Exercise 2.29 on page 64
- Exercise 2.31on page 66
Solutions are provided in the Student Solutions Manual for Chapter 2 (interactive textbook) and on pages AN2 and AN3 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).
- Complete Exercise for Additional Topics 1A and 1B in this Study Guide, below.
- Complete the Self-Review Test for Chapter 2 (pages 70–71 of the downloadable eText).
Note: At the end of each chapter of the textbook, there are instructions for how to complete the statistical calculations, graphs, and processes for that chapter using a TI-84 calculator, Microsoft Excel, and Minitab. You are not required to use a TI-84 calculator or to learn these statistical software programs for MATH 215. However, if you happen to have access to this calculator or these applications, you may use them to double-check your work.
You are also not permitted to use a TI-84 calculator, Microsoft Excel, or Minitab on the midterm or the final exam for this course. The only calculator you are allowed to bring into the exam room is the Texas Instruments TI-30Xa Scientific Calculator. You should familiarize yourself with its functionality now so that you can complete the calculations as required on the assignments and exams.
See the Calculators section of the Course Orientation for more information.
Optional Extra Practice
For extra practice with the material presented in this section, you can complete the following questions and exercises, for which the solutions are provided in the textbook:
- Any odd-numbered chapter-section practice questions that are not assigned above
- The odd-numbered Supplementary Exercises and Advanced Exercises found at the end of Chapter 2 (pages 68–70 of the downloadable eText)
Required Reading: Additional Topic 1A: Class Boundaries
The following notes on class boundaries are taken from the previous edition of the textbook. You can expect to have questions involving class boundaries in the assignments and exams for this course.
Definition
Class Boundary A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class.
| Class Limits | Class Boundaries | Class Width | Class Midpoint |
|---|---|---|---|
| 801 to 1000 | 800.5 to less than 1000.5 | 200 | 900.5 |
| 1001 to 1200 | 1000.5 to less than 1200.5 | 200 | 1100.5 |
| 1201 to 1400 | 1200.5 to less than 1400.5 | 200 | 1300.5 |
| 1401 to 1600 | 1400.5 to less than 1600.5 | 200 | 1500.5 |
| 1601 to 1800 | 1600.5 to less than 1800.5 | 200 | 1700.5 |
| 1801 to 2000 | 1800.5 to less than 2000.5 | 200 | 1900.5 |
Table 2.8: Class Boundaries, Class Widths, and Class Midpoints for Table 2.7
Note that in Table 2.8, when we write classes using class boundaries, we write “to less than” to ensure that each value belongs to one and only one class. As we can see, the upper boundary of the preceding class and the lower boundary of the succeeding class are the same.
[Source: Prem S. Mann, Introductory Statistics, 8th ed. (Wiley, 2012) [VitalSource], 37–38. This material is reproduced with the permission of John Wiley & Sons Canada, Ltd.]
Required Reading: Additional Topic 1B: Ogives
The following notes on ogives are taken from the previous edition of your textbook. You can expect to have questions involving ogives in the assignments and exams for this course.
Definition
Ogive An ogive is a curve drawn for the cumulative frequency distribution by joining with straight lines the dots marked above the upper boundaries of classes at heights equal to the cumulative frequencies of respective classes.
When plotted on a diagram, the cumulative frequencies give a curve that is called an ogive (pronounced o-jive). Figure 2.12 gives an ogive for the cumulative frequency distribution of Table 2.14 which has been constructed from the following frequency distribution table (include the table on page 54 in Example 2–7 in the Mann v.8e version).
Note that the variable is the number of iPods sold per day by a company over a period of 30 days. The frequencies represent the number of days on which the number of iPods indicated by each class were sold. [As an example, for the second class in the Cumulative Frequency Distribution table below, 14 or fewer iPods were sold in 9 days. According to the third class, 19 or fewer iPods were sold in 17 days.]
To draw the ogive in Figure 2.12, the variable, which is the total number of iPods sold by a company in each of 30 days [emphasis added], is marked on the horizontal axis and the cumulative frequencies on the vertical axis. Then the dots are marked above the upper boundaries of various classes at the heights equal to the corresponding cumulative frequencies. The ogive is obtained by joining consecutive points with straight lines. Note that the ogive starts at the lower boundary of the first class and ends at the upper boundary of the last class.
| Class Limits | Class Boundaries | Cumulative Frequency |
|---|---|---|
| 5–9 | 4.5 to less than 9.5 | |
| 5–14 | 4.5 to less than 14.5 | |
| 5–19 | 4.5 to less than 19.5 | |
| 5–24 | 4.5 to less than 24.5 | |
| 5–29 | 4.5 to less than 29.5 |
Table 2.14: Cumulative Frequency Distribution of iPods Sold
Figure 2.12: Ogive for the cumulative frequency distribution of Table 2.14
One advantage of an ogive is that it can be used to approximate the cumulative frequency for any interval. For example, we can use Figure 2.12 to estimate the number of days for which 17 or fewer iPods were sold. First, draw a vertical line from 17 on the horizontal axis up to the ogive. Then draw a horizontal line from the point where this line intersects the ogive to the vertical axis. This point gives the estimated cumulative frequency of the class 5 to 17. In Figure 2.12, this cumulative frequency is (approximately) 13, as shown by the dashed line. Therefore, 17 or fewer iPods were sold on (approximately) 13 days.
We can draw an ogive for cumulative relative frequency and cumulative percentage distributions the same way as we did for the cumulative frequency distribution.
[Source: Prem S. Mann, Introductory Statistics, 8th ed. (Wiley, 2012) [VitalSource], 54–56. This material is reproduced with the permission of John Wiley & Sons Canada, Ltd.]
Exercise for Additional Topics 1A and 1B
The following exercise is reproduced from Exercise 2.35 in the previous edition of the textbook. The solution is provided.
[The table below] gives the frequency distribution of the number of days to expiry date for all containers of yogurt in stock at a local grocery store. Containers that had already expired but were still on the shelves were given a value of 0 for number of days to expiry date.
| Number of Days | Number of Containers |
|---|---|
| 0 to 5 | 32 |
| 6 to 11 | 67 |
| 12 to 17 | 44 |
| 18 to 23 | 20 |
| 24 to 29 | 11 |
[Source: Prem S. Mann, Introductory Statistics, 8th ed. (Wiley, 2012) [VitalSource], 57. This material is reproduced with the permission of John Wiley & Sons Canada, Ltd.]
Prepare a cumulative frequency distribution table that also displays cumulative relative frequencies and cumulative percentage frequencies.
Solution:
Number of Days Cumulative Frequency Cumulative Relative Frequency Cumulative Percentage 0 to 5 0.184 18.4 0 to 11 0.569 56.9 0 to 17 0.822 82.2 0 to 23 0.937 93.7 0 to 29 1.000 100.0 Sketch an ogive for the cumulative percentage distribution in part a, using class boundaries on the horizontal X-axis.
Solution:
Using the ogive, estimate the percentage of containers that will expire in fewer than 20 days.
Solution: Approximately 85% of the containers will expire in fewer than 20 days, as indicated on the ogive in part b.
Section 1-6: Measures of Central Tendency for Ungrouped Data
Outcomes
After completing the readings and exercises for this section, you should be able to do the following:
- compute the mean, median, and mode, given ungrouped (raw) sample data or ungrouped population data.
- compute the weighted mean for a data set.
- identify the advantages and disadvantages of using the mean, weighted mean, median, and mode as a measure of central tendency for different types of data sets.
- determine how the skewness of a data set affects the relationship between the mean, median, and mode.
Reading
Read the following sections in Chapter 3 of the textbook:
- Chapter 3 Introduction
- Section 3.1
- Note: Omit Section 3.1.4: Trimmed Mean.
Be prepared to read the material in Chapter 3 at least twice—the first time for a general overview of topics, and the second time to concentrate on the terms and examples presented. Return to these sections when you need to review these topics.
Supplementary Video Resources
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned textbook readings.
Videos Related to Chapter 3
- Numerical Summaries (Jeremy Haselhorst)
- Numerical Summaries for a Quantitative Variable (DWR447)
- Descriptive Statistics, Part 1 (The Doctoral Journey)
Videos Related to Section 3.1
- Statistics – Mean, Median and Mode (Math Meeting)
- Measures of Central Tendency (jbstatistics)
Exercises
Complete the following exercises from Chapter 3 of the textbook (page numbers are for the downloadable eText):
- Exercises 3.5, 3.7, 3.9, 3.11, and 3.13 (parts a and b only) on page 88
- Exercises 3.19, 3.21, 3.23, and 3.25 on page 89
Solutions are provided in the Student Solutions Manual for Chapter 3 (interactive textbook) and on page AN4 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).
Section 1-7: Measures of Dispersion for Ungrouped Data
Outcomes
After completing the readings and exercises for this section, you should be able to do the following:
- compute the range, variance, standard deviation, and coefficient of variation, given ungrouped (raw) sample data or ungrouped population data.
- identify the advantages and disadvantages of using the range, standard deviation, and coefficient of variation as a measure of dispersion for different types of data sets.
- distinguish between a parameter and a statistic.
Reading
Read Section 3.2 in Chapter 3 of the textbook.
Supplementary Video Resources
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 3.2 of the textbook.
Videos Related to Section 3.2
- Dancing Statistics: explaining the statistical concept of ‘variance’ through dance (BPSOfficial)
- Measuring Spread: the Standard Deviation (Jeremy Haselhorst)
- Statistics – Standard Deviation (Math Meeting)
- Standard Deviation and Variance (statisticsfun)
- Why are degrees of freedom () used in Standard Deviation (statisticsfun)
- Measures of Variability (jbstatistics)
- The Sample Variance: Why divide by ? (jbstatistics)
Exercises
Complete the following exercises from Chapter 3 of the textbook (page numbers are for the downloadable eText):
- Exercises 3.33, 3.35, and 3.39 on page 96
- Exercise 3.43 on page 97
Solutions are provided in the Student Solutions Manual for Chapter 3 (interactive textbook) and on page AN4 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).
Section 1-8: Mean, Variance, and Standard Deviation for Grouped Data
Outcomes
After completing the readings and exercises for this section, you should be able to do the following: Compute the mean, variance, and standard deviation, given grouped sample or grouped population data.
Reading
Read Section 3.3 in Chapter 3 of the textbook.
Supplementary Video Resources
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 3.3 of the textbook.
Videos Related to Section 3.3
- Statistics (Mean and Standard Deviation) for Grouped Data (lbowen11235)
- Mean, Median and Mode from a Frequency Distribution (ChattState Math)
- Variance and Standard Deviation for Grouped Data (searching4math)
Exercises
Complete Exercises 3.47 and 3.49 from Chapter 3 of the textbook (page 102 of the downloadable eText).
Solutions are provided in the Student Solutions Manual for Chapter 3 (interactive textbook) and on page AN4 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).
Section 1-9: Using Standard Deviation
Outcomes
After completing the readings and exercises for this section, you should be able to do the following:
- use Chebyshev’s theorem with any distribution to find the proportion or percentage of the total observations that falls within a given interval about the mean.
- use the empirical rule with any bell-shaped distribution to find the proportion or percentage of the total observations that falls within a given interval about the mean.
Reading
Read Section 3.4 in Chapter 3 of the textbook.
Supplementary Video Resources
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 3.4 of the textbook.
Videos Related to Section 3.4
- The Normal Distribution and the 68-95-99.7 Rule (patrickJMT)
- Chebychev’s Theorem EXPLAINED (Don Davis)
- In this video, be sure to compare the excellent graphical representations of Chebyshev’s theorem and the empirical rule, and pay attention to the helpful table showing Chebyshev’s theorem computations.
Exercises
Complete Exercises 3.61 and 3.63 from Chapter 3 of the textbook (page 107 of the downloadable eText).
Solutions are provided in the Student Solutions Manual for Chapter 3 (interactive textbook) and on page AN5 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).
Section 1-10 Measures of Position and Box-and-Whisker Plots
Outcomes
After completing the readings and exercises for this section, you should be able to do the following:
- compute the three quartiles (Q 1, Q 2, Q 3), the interquartile range, percentiles, and percentile ranks, given ungrouped (raw) sample data or ungrouped population data.
- interpret the three quartiles (Q 1, Q 2, Q 3), the interquartile range, percentiles, and percentile ranks in the context of a given problem.
- construct a box-and-whisker plot, given ungrouped (raw) sample data or ungrouped population data.
- determine the three quartiles, the lower and upper inner fences, the skewness, and the outliers (if any), given a box-and-whisker plot.
Reading
Read the following sections in Chapter 3 of the textbook:
- Section 3.5
- Section 3.6
- Note: Omit Appendix 3.1.
Supplementary Video Resources
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned textbook readings.
Videos Related to Section 3.5
- Quartiles & Interquartile Range (Colette Tropp)
Videos Related to Section 3.6
- Intro to Box and Whisker Plots (Mashup Math)
- How to Make Box & Whisker Plots (The Organic Chemistry Tutor)
- How to Read a Box Plot (SmithMathAcademy)
- Comparing Box Plots (Mark Dolan)
- Understanding & Comparing Boxplots (Box & Whisker Plots) (MATHRoberg)
- Box Plot & Skew (Mona Schraer)
- The Five Number Summary, Boxplots, and Outliers (Simple Learning Pro)
Exercises
Complete the following exercises from Chapter 3 of the textbook (page numbers are for the downloadable eText):
- Exercises 3.69 and 3.73 on page 112
- Exercises 3.77 and 3.79 on page 115
- Supplementary Exercise 3.81 on page 117
- Supplementary Exercises 3.83, 3.85, 3.87, 3.89, and 3.91 on page 118
Solutions are provided in the Student Solutions Manual for Chapter 3 (interactive textbook) and on pages AN5 and AN6 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).
- Complete the Self-Review Test for Chapter 3 (pages 122–124 of the downloadable eText). Omit questions 25 and 27.
- Complete the Unit 1 Self-Test below.
- Complete Assignment 1.
Note: At the end of each chapter of the textbook, there are instructions for how to complete the statistical calculations, graphs, and processes for that chapter using a TI-84 calculator, Microsoft Excel, and Minitab. You are not required to use a TI-84 calculator or to learn these statistical software programs for MATH 215. However, if you happen to have access to this calculator or these applications, you may use them to double-check your work.
You are also not permitted to use a TI-84 calculator, Microsoft Excel, or Minitab on the midterm or the final exam for this course. The only calculator you are allowed to bring into the exam room is the Texas Instruments TI-30Xa Scientific Calculator. You should familiarize yourself with its functionality now so that you can complete the calculations as required on the assignments and exams.
See the Calculators section of the Course Orientation for more information.
Optional Extra Practice
For extra practice with the material presented in this section, you can complete the following questions and exercises, for which the solutions are provided in the textbook:
- Any odd-numbered chapter-section practice questions and Supplementary Exercises that are not assigned above
- The odd-numbered Advanced Exercises found at the end of Chapter 3 (pages 119–120 of the downloadable eText)
Assignment 1
Once you have completed the Unit 1 Self-Test below, complete Assignment 1. You can access the assignment in the Assessment section of the course home page. Once you have completed the assignment, submit it to your tutor for marking using the drop box on the page for Assignment 2.
Unit 1 Self-Test
The self-test questions are shown here for your information. Download the Unit 1 Self-Test document and write out your answers. Show all your work and keep your calculations to four decimal places. You can access the solutions to this self-test on the course home page.
-
The following short survey was completed by nine randomly selected regular, paying customers who shop at Savemore, a large supermarket located on the outskirts of a large city.
1. Please indicate your area of residence: _____ Rural _____Urban 2. Please indicate your annual family income before taxes (nearest $1,000): $__________ 3. Please estimate the number of visits you make to Savemore in a typical month: ______. The survey responses of the nine randomly selected, regular customers are summarized in the table below.
Customer last name Residence Annual income (000’s) Visits Jackson rural 85 10 Wong urban 65 4 Manderson urban 60 22 Miles rural 66 8 Selenas urban 78 7 Goldman urban 75 5 Jang urban 72 9 Jones rural 64 24 Khan urban 68 4 - Referring to the Savemore survey, fill in the blanks for each question below.
- The name “Jackson” would be referred to as an _____________ (observation or element).
- The first income value of “85” would be referred to as an _____________ (observation or element).
- How many variables does the Savemore short survey focus on? _____
- The responses to the “residence” question would be classed as _____________ (qualitative or quantitative).
- The responses to the “income” question would be classed as _____________ (qualitative or quantitative).
- The responses to the “visits” question would be classed as _____________ (discrete or continuous).
- The responses to the “income” question would be classed as _____________ (discrete or continuous).
- Data collected from the Savemore survey would be called _____________ (cross-section data or time-series data).
- The measure of central tendency most appropriate for summarizing the “residence” question responses would be _____________ (mean or median or mode).
- If Savemore plans to use the nine customer survey responses to make decisions regarding ALL Savemore customers, this would be an example of _____________ (descriptive statistics or inferential statistics).
- Would mean or median be the better measure of central tendency for the responses to the “visits” data? Explain.
- Suppose the population distribution of incomes related to survey question 2 (above) has a mean of $70,000 and a standard deviation of $20,000. If the shape of the population distribution is unknown, ___________% of the population earns between $30,000 and $110,000.
- Suppose the population distribution of incomes related to survey question 2 (above) has a mean of $70,000 and a standard deviation of $20,000. If the shape of the population distribution is bell-shaped ___________% of the population earns between $30,000 and $110,000.
- Provide two possible reasons why Savemore did not use a census survey to collect information from its customers:
________________________________________
________________________________________
- If Savemore selected the sample of customers in a way that ensured each customer had an equal chance of being selected, this would be called a _____________________ (random sample or simple random sample).
- Suppose that Savemore plans to estimate the true average annual income (before taxes) for ALL its regular paying customers, based on computing the average annual income of the sample of nine regular paying customers surveyed. The fact that the mean income calculation of the sample will differ from the population mean income, because the sample is a subset of the population, leads to a _____________________ (sampling error or non-sampling error).
- If Savemore selected the sample of customers in a way that ensured that three customers were randomly selected from rural residences and six customers were selected from urban areas, this would be an example of a _____________________ (systematic random sample or stratified random sample)
- Referring to the Savemore survey responses, determine the median customer monthly visits.
- Referring to the Savemore survey responses, determine the mean customer monthly visits. By comparing the mean with the median customer monthly visits, determine if the distribution of the visits is positively or negatively skewed. Explain. In computing the mean, keep your work to four decimals.
- Calculate the standard deviation for the customer monthly visits. Use the short-cut method. Keep your work to four decimals.
- Referring to the Savemore survey responses, which variable—income or visits—exhibits greater relative variability? Show the appropriate calculations. Hint: The standard deviation of the nine monthly incomes equals 7.8899.
- Compute the first quartile (Q1) and third quartile (Q3) for customer monthly visits. Interpret your answers.
- Calculate the interquartile range for monthly customer visits. Interpret your answer.
- Calculate the percentile rank of the customer who typically makes 10 visits per month. Interpret your answer.
- Construct a stem-and-leaf display for the annual customer income data. Display the leaves in ascending order.
- Sketch a box-and-whisker plot for the annual customer family income data. In your sketch, indicate all the quartiles and the minimum and maximum values. Using the appropriate computations, determine if there are mild outliers.
- Referring to the Savemore survey, fill in the blanks for each question below.
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A government healthcare clinic that serves older adults is trying to decide whether to open up a clinic location in a new subdivision. The clinic surveyed a random sample of 20 homeowners and recorded their ages as follows:
81 77 56 44 86 65 84 74 72 71 79 63 83 82 58 52 41 68 67 88 - Construct a frequency distribution for the ages above, using a lower limit for the first class of 40 and a class width of 10. In your distribution, include the class limits, the class boundaries, the class midpoints, the frequency, the relative frequency and the cumulative frequency.
Construct a percentage histogram for the frequency distribution above. Use a ruler and the grid provided below.
- Is the distribution of the ages for the 20 homeowners described above skewed? If so, in which direction is the skew? Does the skewness of the age data support the decision to open a new clinic location in this subdivision? Explain.
Construct an ogive for the frequency distribution above. Use a ruler and the grid provided below. Hint: An ogive is a curve that describes the cumulative frequencies. It does this by joining with lines the dots marked above the upper boundaries of classes at heights that are equal to the cumulative frequencies of the respective classes.
- What percentage of the twenty ages are below 60?
- What percentage of the twenty ages are 70 or above?
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The table below describes the commute times for all 30 employees working at a car dealership.
Commute time
(in minutes)Frequency
(# of employees)1–15 10 16–30 9 31–45 8 46–60 2 61–75 1 - Would you classify the 30 commute times as sample or population data? Explain.
- Compute the mean, standard deviation and variance for the commute times displayed in the distribution above. Use the short-cut method when computing the variance and the standard deviation.
- Based on your observation of the skewness of the distribution of the commute times, is the median commute time smaller or larger than the mean commute time? Explain.
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The Golf Depot sold all 100 KPOW-SUPRA golf sets at different prices during the 2018 golf season, as follows.
Month in 2018 Number of golf sets Price per golf set May 35 $1,200 June 25 $1,100 July 20 $1,000 August 15 $900 September 5 $800 Compute the overall average price (per golf set) for the 100 KPOW-SUPRA golf sets sold during the 2018 golf season.
- The mean annual income of all five of the doctors employed in a small medical center is $260,000. The annual incomes of three of these five doctors are $210,000, $250,000 and $275,000. Find the annual income of the fourth and fifth doctors, assuming these two doctors make the same annual income. Hint: In getting started with your solution, let the annual income of the fourth doctor be $X.
References
Mann, Prem S. Introductory Statistics, 8th ed. Wiley, 2012. [VitalSource].