Unit1complete PDF

Preview

Summary

Unit1complete...

Description

Page 1 of 11

Study Guide Unit 1 Descriptive Statistics Unit 1 introduces the fields of statistics; 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 his or her work by collecting a body of numerical facts, called raw data, through surveys, observation, or internal or external information sources. After gathering this data, the practitioner must organize it in an orderly fashion 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 she or he organizes it, presents it in tables and graphs, and computes 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 Mathematics 215 consists of the sections listed below. 1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9

Statistics, Population and Sample Types of Variables and the Nature of Statistical Data Organizing and Graphing Qualitative Data Organizing and Graphing Quantitative Data Measures of Central Tendency for Ungrouped Data Measures of Dispersion for Ungrouped Data Mean, Variance and Standard Deviation for Grouped Data Using Standard Deviation Measures of Position and Box-and-Whisker Plots

The unit also contains a self test (with solutions) and instructions for the first tutor-marked assignment.

Section 1-1 Statistics, Population and Sample Objective After completing the readings and exercises for this section, you should be able to define, and use in context, the key terms listed below.    

descriptive statistics and inferential statistics population, sample and census random sample and simple random sample sampling with replacement and sampling without replacement

Page 2 of 11

Reading This subsection directs you to the appropriate reading in the eText. Be prepared to read the material twice—first for a general overview of topics, and a second time to concentrate on the terms and examples presented. Note: In this and all subsequent units, all page numbers refer to the eText Mann, Prem S. Introductory Statistics, 8th ed. Hoboken, NJ: Wiley, 2013. This is a digital textbook (eTextbook). If you haven’t already done so, access or download it now through the link on the course home page. Read the introduction to Chapter 1, “Introduction,” and Sections 1.1 and 1.2, pages 1-8.

Exercises This subsection directs you to the exercises you must complete. Show your work as you develop your answers. Solutions are provided in the Student Solutions Manual in the left-hand navigation column of each chapter in your eText that you accessed from the Read, Study, & Practice link on your course home page. Remember, 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, do not hesitate to contact your tutor for assistance. Complete Exercises 1.3, 1.5, 1.7 on pages 7 and 8.

Section 1-2 Types of Variables and the Nature of Statistical Data Objectives After completing the readings and exercises for this section, you should be able to 1. define, and use in context, the key terms listed below.  elements (members), variables and data set  quantitative, discrete, continuous and qualitative (categorical) variables  cross section and time series data 2. compute the values for expressions that are presented in summation notation.

Page 3 of 11

Reading Read Sections 1.3, 1.4, 1.5, 1.6, and 1.7 on pages 8-18.

Exercises Complete each of the exercises listed below.   

Exercises 1.15 and 1.17 on page 12 Exercise 1.21 on page 15 Exercises 1.23, 1.25 and 1.29 on pages 17-18

Solutions are provided in the Student Solutions Manual in the left-hand navigation column of each chapter in your eText. Then, take the “Self-Review Test” on pages 20-21.

Optional Extra Practice Return to your course home page and follow the “Read, Study & Practice” link. When you are asked to complete exercises that have solutions provided in the Student Solutions Manual, you will find the solutions below the Reading Content in the left-hand navigation column of each chapter of your eText under the heading ‘Student Solutions Manual.’

Section 1-3 Organizing and Graphing Qualitative Data Objectives After completing the readings and exercises for this section, you should be able to 1. construct a frequency distribution that includes frequencies, relative frequencies and percentage frequencies, given raw data for a qualitative (categorical) variable. 2. construct a bar graph and a pie chart. 3. interpret frequencies, relative frequencies and percentage frequencies, given a frequency distribution or a graph relating to a frequency distribution.

Reading

Page 4 of 11

Read the introduction to Chapter 2, “Organizing and Graphing Data,” and Section 2.1, pages 28-34.

Exercises Complete Exercise 2.7 on page 35. Solutions are provided in the Student Solutions Manual.

Section 1-4 Organizing and Graphing Quantitative Data Objectives After completing the readings and exercises for this section, you should be able to 1. construct a frequency distribution that uses either a “less than” or not “less than” method for writing the classes, given raw data for a continuous variable. Note: This distribution will include class limits, class boundaries, midpoints, frequencies, relative frequencies, percentage frequencies, cumulative frequencies, cumulative relative frequencies and cumulative percentage frequencies. 2. construct the following graphs: histogram, relative frequency histogram, frequency polygon, relative or percentage frequency polygon, ogive, and relative or percentage ogive. 3. construct a frequency distribution using single-valued classes, given raw data. Note: This distribution will include frequencies, relative frequencies and percentage frequencies. 4. construct a bar graph for the distribution described in Objective 3, above. 5. interpret frequencies, relative and percentage frequencies, cumulative frequencies, cumulative relative frequencies and cumulative percentage frequencies, given a frequency distribution or a related graph. 6. interpret symmetric, skewed and uniform distributions for the frequency distribution or graph described in Objective 5, above. 7. construct stem-and-leaf displays and dotplots, and identify possible outliers, given raw data.

Reading Read Sections 2.2, 2.3, 2.4 and 2.5 on pages 36-64.

Page 5 of 11

Exercises Complete each of the exercises listed below.     

Exercises 2.15, 2.17 and 2.21 on pages 49-51 Exercise 2.29 on page 53 Exercise 2.35 on page 57 Exercises 2.53 and 2.55 on page 62 Exercise 2.63 on page 65

Solutions are provided in the Student Solutions Manual. Then, take the “Self-Review Test” on pages 77-78.

Optional Extra Practice Follow the “Read, Study & Practice” link on your course home page. The link will open in a new window. Choose the chapter of interest. Optional practice exercises with solutions may be found at the end of the Resources for each chapter.

Section 1-5 Measures of Central Tendency for Ungrouped Data Objectives After completing the readings and exercises for this section, you should be able to 1. compute the mean, median and mode, given ungrouped (raw) sample data or ungrouped population data. 2. compute the combined mean for two or more data sets. 3. compute the weighted mean for a data set. 4. identify the advantages and disadvantages of using the mean, weighted mean, combined mean, median and mode, as a measure of central tendency for different types of data sets. 5. determine how the skewness of a data set affects the relationship between the mean, median and mode.

Reading Read the introduction to Chapter 3 and Section 3.1 Measures of Central Tendency for Ungrouped Data, pages 86-94.

Page 6 of 11

Exercises Complete Exercises 3.5, 3.7, 3.9, 3.11, 3.17, 3.21, 3.25, 3.27, 3.29 and 3.35 on pages 95-99. Solutions are provided in the Student Solutions Manual.

Section 1-6 Measures of Dispersion for Ungrouped Data Objectives After completing the readings and exercises for this section, you should be able to 1. compute the range, variance and standard deviation, given ungrouped (raw) sample data or ungrouped population data. 2. identify the advantages and disadvantages of using the range and standard deviation as a measure of dispersion for different types of data sets. 3. distinguish between a “parameter” and a “statistic.”

Reading Read Section 3.2 on pages 99-104.

Exercises Complete Exercises 3.43, 3.49, 3.55 and 3.59 on pages 105-106. Solutions are provided in the Student Solutions Manual.

Section 1-7 Mean, Variance and Standard Deviation for Grouped Data Objective After completing the readings and exercises for this section, you should be able to compute the mean,

Page 7 of 11

variance and standard deviation, given grouped sample or grouped population data.

Reading Read Section 3.3 on pages 106-111.

Exercises Complete Exercises 3.61, 3.65 and 3.69 on pages 111-113. Solutions are provided in the Student Solutions Manual.

Section 1-8 Use of Standard Deviation Objectives After completing the readings and exercises for this section, you should be able to 1. use Chebyshev’s Theorem with any distribution to find the proportion or percentage of the total observations that fall within a given interval about the mean. 2. use the Empirical Rule with any bell-shaped distribution to find the proportion or percentage of the total observations that fall within a given interval about the mean.

Reading Read Section 3.4 on pages 113-117.

Exercises Complete Exercises 3.79, 3.81 and 3.83 on pages 117-118. Solutions are provided in the Student Solutions Manual.

Page 8 of 11

Section 1-9 Measures of Position; Box-and-Whisker Plots Objectives After completing the readings and exercises for this section, you should be able to 1. compute the three quartiles ( Q1, Q2, Q3), the interquartile range, percentiles and percentile ranks, given ungrouped (raw) sample data or ungrouped population data. 2. interpret the three quartiles ( Q1, Q2, Q3), the interquartile range, percentiles and percentile ranks in the context of a given problem. 3. construct a box-and-whisker plot, given ungrouped (raw) sample data or ungrouped population data. 4. determine the three quartiles, the lower and upper inner fences, the skewness, and outliers (if any), given a box-and-whisker plot.

Reading Read Sections 3.5 and 3.6 on pages 118-125. Omit the discussion of outer fences and mild versus extreme outliers on page 125.

Exercises Complete each of the exercises listed below.   

Exercises 3.91 and 3.95 on pages 122-123 Exercises 3.101 and 3.107 on pages 125-126 Supplementary Exercises 3.109, 3.111, 3.113, 3.115, 3.117, 3.119, 3.127 and 3.141 on pages 128133

Solutions are provided in the Student Solutions Manual. Then, take the “Self-Review Test” on pages 136-138. Omit #26.

Optional Extra Practice Follow the “Read, Study & Practice” link on your course home page. The link will open in a new window. Choose the chapter of interest. Optional practice exercises with solutions may be found at the end of the Resources for each chapter.

Page 9 of 11

Unit 1 Self Test 1. A midsized food store has just opened up in a new neighbourhood. The owners are trying to determine whether their customers view the store as a convenience store or as a store where they will do their major grocery shopping. If the store is typically used as a convenience store, then most of the sales transactions will be small dollar amounts. However, if most of the sale transactions are relatively large dollar amounts, customers tend to use the store for their major grocery shopping. A listing of 20 grocery sale transactions, in dollars, randomly selected from a much larger list containing all the sale transactions since the store opened is given below. $52.10 $168.12 $239.87 $124.62 $144.94 $4.95 $211.34 $217.25 $163.98 $233.16 $188.24 $94.67 $174.52 $196.77 $209.45 $132.42 $76.98 $204.11 $114.23 $249.80 a. Construct a frequency distribution for the sales transactions above, using a lower limit for the first class of $0.00 and a class width of $50.00. In your distribution, include the class limits, the class midpoints, the frequency, the relative frequency and the percentage cumulative frequency. Note: As this data contains fractional values, use the less than method for writing the classes.

b. Construct a percentage frequency polygon for the frequency distribution above. Use a ruler and the graph paper. c. Is the distribution of the sales transactions for the 20 customers described above skewed? If so, in which direction is the skew? Does the skewness suggest that the food store is being used mainly as a convenience store? Explain. d. Construct a cumulative frequency histogram for the frequency distribution above. Use a ruler and the graph paper. e. What percentage of the 20 sales transactions involved less than $50.00? f. What percentage of the 20 sales transactions involved $100.00 or more? 2. The table below describes the distribution of midterm marks for all 40 of the students in a class taking an introduction to statistics course. Midterm Mark (%) 51-60 61-70 71-80 81-90 91-100

Frequency (# of Students) 12 10 8 6 4

a. For the “81-90” class, determine the class boundaries, the class width, the midpoint, the percentage frequency and the cumulative relative frequency, b. Compute the mean, variance and standard deviation for the midterm marks displayed in the distribution above. Use the short-cut method when computing the variance and the standard deviation. Round off your final answer to two decimal places. c. Based on your observation of the skewness of the distribution of the midterm marks, would you say that the median midterm mark is smaller or lar ger than the mean midterm mark?

Page 10 of 11

Explain. 3. The four questions below were part of a recent health-care survey. 1. Please indicate your gender (Circle one below.) Female

Male

2. Please indicate your weight in pounds. 3. How many times did you visit a doctor last year? 4. Please indicate your height in inches. Assume that 100 people filled in responses to each of the four questions above. a. The survey question (questions) that generates (generate) observations for a continuous variable is (are) b. The survey question (questions) that generates (generate) observations for a qualitative variable is (are) c. The survey question (questions) that generates (generate) observations for a discrete variable is (are) d. The measure of central tendency that is appropriate to apply to the responses from Question 1 is the e. If the distribution of the 100 observations relating to Question 2 is significantly skewed to the left, the measure of central tendency that is more appropriate to apply to the responses from Question 2 is the ________________(mean or median). f. Suppose the distribution of the 100 observations relating to Question 4 is bell-shaped, with a mean of 68 inches and a standard deviation of 5 inches. Based on __________________ (the Empirical Rule or Chebyshev’s Theorem), we can conclude that _______________ % of the people surveyed have a height between 58 inches and 78 inches. g. Suppose the distribution of the 100 observations relating to Question 2 is skewed to the left, with a mean of 135 pounds and a standard deviation of 10 pounds. Based on _______________ (the Empirical Rule or Chebyshev’s Theorem), we can conclude that _________________% of the people surveyed weighed between 105 pounds and 165 pounds. 4. The fire chief in a large Canadian city randomly selected a sample of eight firefighters and observed the scores that each one achieved after completing a fitness test. The scores are shown below. The higher the score, the higher the level of fitness. 4

20

18

16

18

18

16

10

a. Calculate the median fitness score. b. What is the modal fitness score? c. Calculate the standard deviation for the fitness scores. Use the short-cut method. Keep your middle work to four decimal places and your final answer to two decimal places. d. Calculate the third quartile (Q 3). Interpret your answer. e. Calculate the interquartile range. Interpret your answer. f. Calculate the percentile rank of the firefighter who achieved a fitness score of 10. Interpret your answer. g. Sketch a box-and-whisker plot for the eight fitness scores. h. Based on your box-and-whisker plot, comment on the skewness of distribution of the fitness scores. What does the skewness suggest about most of the fitness scores? 5. The table below describes the dail y high temperatures experienced by a Canadian city during the

Page 11 of 11

month of June 2006. As an example, the table indicates that for two of the days in June, the city experienced a daily high temperature of 26 degrees Celsius, while for ten days in June the same city experienced a daily high of 22 degrees Celsius. Daily High Temperature ( C) 26 25 24 23 22

Number of Days 2 4 6 8 10

Compute the average daily high temperature for the entire month of June 2006, for the city described in the table above. Keep your final answer to two decimal places. 6. The ages of the 15 professors teaching in the Faculty of Nursing at a large university are shown in the following table.

52 27 53 54 47 41 43 58 45 56 32 34 38 55 29 a. Construct a stem-and-leaf display for the ages shown in the table above, using the last digit for each age as the leaf. Place the leaves in ascending order. b. Does the age data set have a mode? c. Calculate the three quartiles and the interquartile range for the 15 ages shown above. d. Calculate the (approximate) 60th percentile. Interpret your answer. e. Based on the skewness of the distribution of the 15 ages, would you conclude that the mean is less than or greater than the second quartile? Explain. 7. The mean annual income of all five of the lawyers employed in a small law firm is $240,000. The annual incomes of four of these five lawyers are $200,000, $260,000, $235,000 and $230,000. What is the annual income of the fifth lawyer? Access the solutions from your course home page....