Probability & Statistics Workbook: Classroom Edition

By Mel Friedman

REA’s Probability & Statistics Workbook is a Perfect Study Tool for Exit Exams, End-of-Course Exams, or Graduation Exams What makes REA’s workbooks different? For starters, students will actually like using them. Here’s why: * Math is explained in simple language, in an easy-to-follow style * The workbooks allow students to learn at their own pace and master the subject * More than 20 lessons break down the material into the basics * Each lesson is fully devoted to a key math concept and includes many step-by-step examples * Paced instruction with drills and quizzes reinforces learning * Innovative “Math Flash” feature offers helpful tips and strategies in each lesson—including advice on common mistakes to avoid * Skill scorecard measures the student’s progress and success * Every answer to every question, in every test, is explained in full detail * A final exam is included so students can test what they’ve learned Some of the math topics covered in the Ready, Set, Go! Probability & Statistics Workbook include: * Organizing Data into Tables and Charts * Measures of Central Tendency * Frequency Distributions * Classical and Empirical Probability * Compound Events * Counting Methods When students apply the skills they’ve mastered in our workbooks, they can do better in class, raise their grades, and score higher on the all-important end-of-course, graduation, and exit exams.

• Language: English
• Publisher: Research & Education Association
• Release date: Jan 1, 2013
• ISBN: 9780738669915

Book preview

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Lesson One

Organizing Data—Part I

In this lesson, we will explore the major categories of data, using statistics. Statistics is a science that deals with analyzing data. We will also explore a way in which data can be organized into picture form. Different types of data are present in almost every part of our lives. As examples, you can find data in (a) baseball batting averages, (b) the numbers of people insured by various insurance companies, (c) the most popular brand of designer jeans, and (d) the highest salaries for school principals.

Your Goal: When you have completed this lesson, you should be able to classify data into different categories and be able to present data in a popular pictorial form.

Organizing Data—Part 1

There are two major classifications of data: qualitative and quantitative. Qualitative data is data that can be identified as nonnumerical. Examples would be (a) gender, (b) subjects taught in high school, (c) leading causes of cancer, and (d) nationally recognized holidays.

Quantitative data is data that can be identified as numerical, which implies that it can be ranked in some way. Examples would be (a) grade point averages, (b) bowling scores, (c) rankings of major cities in crime prevention, and (d) prices of gasoline at different service stations.

Also, quantitative data can be split into two subcategories, namely, discrete and continuous. Discrete data can be assigned a specific value and can be counted. Examples would be (a) the number of children in a classroom; (b) the annual salary of a person, in whole dollars; (c) the number of months in a year; and (d) attendance at a concert.

Continuous data is data that is not discrete. This type of data includes (a) the temperature at different times of the day, (b) the weight of each player on a football team, (c) the number of gallons of water in different swimming pools, and (d) the heights of adults at a party. Continuous data is obtained by measuring, not by counting. For this reason, continuous data really is an estimation. Thus, if we claim that a person weighs 160 pounds, we recognize that our accuracy is only as good as the scale that is being used. That is, the 160-pound person might really weigh 159.85 pounds or possibly 160.003 pounds. We realize that the scale is only accurate to the nearest pound.

As another example, let’s suppose that you ran a 100-yard dash. Suppose that your time was clocked at 11.2 seconds. It is possible that your time was closer to 11.22 seconds or even 11.19 seconds. However, the watch that was used to record your time was only accurate to the nearest tenth of a second.

When continuous data is used, there are presumed boundaries associated with a given value. In the above examples, we saw that a weight of 160 pounds would have boundaries of 159.5 pounds and 160.5 pounds. We would include the lower boundary of 159.5 but not include the upper boundary of 160.5. The reason is that to the nearest pound, 159.5 rounds off to 160, but 160.5 would actually round off to 161. For the number 11.2, the lower boundary would be 11.15, and the upper boundary would be 11.25. As you can see, the number 11.15 would be included as a possible value, but 11.25 would not be included.

It is important to remember that boundaries are only used for continuous data.

Boundaries for continuous data are commonly written as intervals. An interval will indicate all allowable values. Instead of writing the lower boundary as 159.5 and the upper boundary as 160.5, the boundaries may be written as 159.5-160.5. This means that values such as 159.7 and 160.233 would be included.

Are you curious about the technique that is used to establish boundaries for continuous data?

If the given data x is an integer, the lower boundary is x − 0.5, and the upper boundary is x + 0.5.

If the given data x is written in tenths, the lower boundary is x − 0.05, and the upper boundary is x + 0.05.

If the given data x is written in hundredths, the lower boundary is x − 0.005, and the upper boundary is x + 0.005.

This process can be extended to any decimal number. Basically, we are subtracting and adding one-half unit to arrive at the boundaries.

Example:

What are the boundaries for 52.81?

Solution:

A unit for this number is 0.01, so we need to subtract and add one-half of 0.01, which is 0.005.

Thus, the boundaries are 52.805-52.815.

Remember that 52.805 is included in this interval, but 52.815 is not included.

Example:

What are the boundaries for 33.0?

Solution:

Be careful here! Even though 33.0 has the same numerical value as 33, the given measurement is shown to be accurate to the nearest tenth. We must subtract and add 0.05 to 33.0. The answer is 32.95–33.05.

Example:

The length of a stapler is measured to be 4.85 inches. How many of the following numbers would be included in the interval containing the associated boundaries? 4.848 inches, 4.855 inches, 4.8449 inches, 4.9 inches, 4.8452 inches

Solution:

The correct boundaries are given by subtracting and adding 0.005, so that the interval is 4.845–4.855. Thus, two of these five numbers, namely, 4.848 and 4.8452, are included in this interval. Be sure you understand that the upper boundary of 4.855 is not included.

Example:

Which one(s) of the following are representative of qualitative data? time needed to complete a project, number of houses in a city block, depth of an ocean, blood type

Solution:

Only blood type qualifies as qualitative data. Each of the other three represents a numerical value.

Quantities such as Social Security numbers, house addresses, and company badge numbers are also considered qualitative data. Even though they contain numbers, there is no ranking system involved. For example, we cannot say that one Social Security number has a greater value than another Social Security number.

One popular way to organize data is to represent it in a pie graph form. (Some books use the term pie chart or circle graph.) It is best to use this type of graph when we are looking at the percent contribution of the component parts of a particular category.

Example:

In the Growing Strong Hospital, a survey was taken of the marital status of each employee. The results showed that 20% are married with no children, 25% are married with at least one child, 15% are single, 30% are divorced, and the remaining 10% are widowed. Create an appropriate pie graph.

Solution:

Recall from your knowledge of geometry that there are 360° in a circle. In order to create the correct portions to represent each of these categories, we start in the center of the circle and construct an angle proportional to the percent assigned to each category. Here are the computations:

Married with No Children, (0.20)(360°) = 72°

Married with at Least One Child, (0.25)(360°) = 90°

Single, (0.15)(360°) = 54°

Divorced, (0.30)(360°) = 108°

Widowed, (0.10)(360°) = 36°

The pie graph should appear as follows:

GROWING STRONG HOSPITAL EMPLOYEES

Notice that the percents add up to 100%. This is a necessity for all pie graphs.

Example:

The mayor of the town of Peopleville has mailed a survey to each adult resident. The resident was asked to rate the quality of the mayor’s ability to govern the town. The five ratings given were (a) excellent, (b) very good, (c) average, (d) below average, and (e) poor. The results showed that

of the population ranked the mayor as excellent

ranked him as very good

ranked him as average

ranked him as below average

the remainder ranked him as poor

Create an appropriate pie graph.

Solution:

In order to determine the fraction of the population that ranked the mayor as poor, we calculate

Now, we must convert each fraction to the correct central angle of a circle. Again, remember that 360° represents a full circle.

The central angles corresponding to these five categories are as follows:

Excellent(360°) = 144°

Very Good, (360°) = 120°

Average, (360°) = 36°

Below Average(360°) = 30°

Poor(360°) = 30°

The pie graph should appear as follows:

RANKING OF THE MAYOR OF PEOPLEVILLE

Notice that these fractions must add up to 1, which is equivalent to 100%. Sometimes you will be asked to compute an actual number that is associated with a specific sector of the pie graph. (From geometry, you know that a sector is a portion of any circle that is bounded by two radii and an included arc.)

Example:

Return to Example 5. Suppose there are a total of 1500 employees in the Growing Strong Hospital. How many of them are either single or divorced?

Solution:

The corresponding percent of single or divorced employees is 15% + 30% = 45%. Then (1500)(0.45) = 675 employees.

Example:

Return to Example 6. Assuming that each adult completed the survey, if 1416 adults had ranked the mayor as excellent, how many adults live in Peopleville?

Solution:

Let x represent the number of adults in Peopleville.

(x) = 1416.

Thus, x = 3540.

Which one of the following represents qualitative data?

Number of golf balls in a golf bag

Days of the week

Amount of salt, in grams, in a slice of cake

Number of motorists in Pennsylvania

Which one of the following represents discrete data?

Street addresses

Weights of cats in an animal shelter

Number of quarts of milk in a large bottle

Number of dollar bills in a bank teller’s drawer

If a horse runs a race in 123.8 seconds, what is the lower boundary for this number when it is written in interval form?

123.75 seconds

123.85 seconds

123.95 seconds

124.0 seconds

A scientist is weighing a certain substance. She determines that its weight is 5.207 grams. What are the actual boundaries for this measurement?

5.2005–5.2015

5.2065–5.2075

5.206–5.208

5.15–5.25

A meteorologist states that the boundaries for the warmest temperature ever recorded in Antarctica are 59.035°–59.045° (Fahrenheit). Which one of the following represents the appropriate single measurement (Fahrenheit) for this interval?

59.0°

59.3°

59.04°

59.0355°

At Climb Higher High School, 32% of the students are in the freshman class, 25% are in the sophomore class, 16% are in the junior class, and the remaining students are in the senior class. In a pie graph, to the nearest degree, how many degrees would be contained in the central angle that represents the students in the senior class?

93°

95°

97°

99°

Refer back to question 6. If there are 64 students in the junior class, how many students are in the entire school?

1024

964

500

400

Refer back to question 6. Create an appropriate pie graph.

The Adams family has identified its major monthly expenses as follows:

is allocated for rentis allocated for foodis allocated for clothing, and the remaining amount is allocated for miscellaneous expenses. If the total of the monthly expenses is $4740, how much money is allocated for miscellaneous expenses?

$316

$632

$948

$1264

Refer back to question 9. Create an appropriate pie graph.

Lesson Two

Organizing Data—Part 2

In this lesson, we will explore two new ways to place data into picture form. These will be a type of bar graph known as a Pareto chart and a time series line graph. The Pareto chart will illustrate the comparison of several different components. The time series line graph will show the changes over time of just one particular component. We will also explore a system for compiling data into a more easily readable form. This system is called a stem-and-leaf plot.

Finally, we will also discuss the elements that lead to misleading graphs and mistaken conclusions.

Your Goal: When you have completed this lesson, you should be able to present data in two more picture forms, compile data into a stem-and-leaf plot form, and recognize the inherent errors in misleading graphs.

Organizing Data—Part 2

A Pareto chart is a bar graph in which rectangular bars are used to represent the data. The frequency of the data must be arranged in descending order. In this way, the corresponding bars are arranged from highest to lowest as you look at the graph from left to right. In almost every case, the Pareto chart is based on qualitative data.

Example:

In the city of Trailerville, there are four major food stores, namely, Apney, Bag-A-Way, Cost-Less, and Donacart. The total number of customers who visited each store last week was: Apney, 300; Bag-A-Way, 450; Cost-Less, 400; and Donacart, 650. Construct a Pareto chart.

Solution:

Each bar must have the same width, which will be displayed horizontally. The frequency of each bar will be shown vertically. Each of the tick marks on the vertical axis represents 50 units. This choice is really quite arbitrary; for that reason, each problem involving the Pareto chart must have the vertical scale drawn and appropriately marked. The categories being used are indicated on the horizontal axis.

Here is the finished product, with the food stores being shown in descending order of the number of customers.

Example:

A survey was recently conducted to determine the number of old style diners that were still operating in a select group of states. The results were as follows: Florida, 144; New Jersey, 208; Ohio, 80; Arizona, 48; California, 192; and Missouri, 160. Construct a Pareto chart.

Solution:

The vertical scale will be marked in units of 16 because it divides evenly into each of the given numbers. On the horizontal axis, we list New Jersey first and Arizona last. All rectangles have the same width.

Here is how the Pareto chart should appear.

Once a scale on the vertical axis is chosen, it must be used for each of the categories under discussion. Another possible scaling unit for Example 1 would be 100. In that case, a number such as 450 would lie halfway between 400 and 500.

Likewise, there is nothing magical about the number 16 that was just used in Example 2. Certainly, a number such as 8 or 10 could also be used. The difficulty would be in estimating the height of (for example) the Arizona rectangle if 10 were used. Notice that there is no space between successive bars.

Let’s now focus on the nature of a time series line graph. A time series line graph is best used to show trends of a single quantity over a period of time.

Example:

During the first six months of this year, Amanda kept track of how many different projects she worked on each month. Here are her results: January, 28; February, 22; March, 52; April, 8; May, 12; June, 40. Construct a time series graph.

Solution:

The months are placed on the horizontal axis, in order from January through June. Dots are placed that correspond to the respective values for each month. Finally, line segments are drawn to connect the monthly values. We will use a scaling unit of 4. Notice that all but one of the numbers divides evenly by 4. The number 22 will be measured by approximating one-half the distance between 20 and 24.

Here is how the time series graph should appear:

Example:

From the year 1980 through the year 1987, Ricky collected CDs of his favorite country music singers. Here are the results of the number of CDs that he bought per