Student Workbook for Research and Evaluation in Counseling, 2e
Student Workbook
to accompany
Research and Evaluation in Counseling
2ndedition
By Bradley T. Erford
Loyola University Maryland
Contents
Preface ………………………………………………………………………………… 3
Activities
A Quick Math Review…………………………………………………………. 4
Chapter 1 The Nature and Importance of Inquiry…………………………… 15
Chapter 2 Characteristics of a Research Study……………………………… 17
Chapter 3 Locating, Reviewing and Writing Research……………………… 20
Chapter 4 Outcomes Research in Counseling……………………………….. 22
Chapter 5 Qualitative Approaches to Research……………………………… 24
Chapter 6 Qualitative Research Design……………………………………… 27
Chapter 7 Quantitative Research Design in Counseling…………………….. 29
Chapter 8 Using Action Research and Single Subjects Research Design …... 35
Chapter 9 Needs Assessment………………………………………………… 39
Chapter 10 Program Evaluation and Accountability………………………….. 40
Chapter 11 Collecting Data……………………………………………………. 43
Chapter 12 Describing Data…………………………………………………… 47
Chapter 13 Deriving Standardized Scores…………………………………….. 50
Chapter 14 Statistical Hypothesis Testing…………………………………….. 53
Chapter 15 Using SPSS for Introductory Statistical Analysis………………….55
Chapter 16 Univariate Inferential Statistics…………………………………… 57
Chapter 17 Correlation and Regression……………………………………….. 59
Chapter 18 Nonparametric Tests of Statistical Inference……………………. 63
Chapter 19 MultivariateStatistical Analysis………………………………….. 68
Answer Keys
Chapter 1 The Nature and Importance of Inquiry…………………………… 74
Chapter 2 Characteristics of a Research Study……………………………… 74
Chapter 3 Locating, Reviewing and Writing Research……………………… 74
Chapter 4 Outcomes Research in Counseling……………………………….. 75
Chapter 5 Qualitative Approaches to Research……………………………… 75
Chapter 6 Qualitative Research Design……………………………………… 76
Chapter 7 Quantitative Research Design in Counseling…………………….. 77
Chapter 8 Using Action Research and Single Subjects Research Design …... 78
Chapter 9 Needs Assessment………………………………………………… 79
Chapter 10 Program Evaluation and Accountability………………………….. 79
Chapter 11 Collecting Data……………………………………………………. 80
Chapter 12 Describing Data…………………………………………………… 82
Chapter 13 Deriving Standardized Scores…………………………………….. 83
Chapter 14 Statistical Hypothesis Testing…………………………………….. 84
Chapter 15 Using SPSS for introductory Statistical Analysis…………………. 84
Chapter 16 Univariate Inferential Statistics…………………………………… 84
Chapter 17 Correlation and Regression……………………………………….. 85
Chapter 18 Nonparametric Tests of Statistical Inference……………………. 86
Chapter 19 Multiple Statistical Analysis……………………………………… 87
Preface
This student workbook was written to accompany Research and Evaluation in Counseling (2nd edition)by Bradley T. Erford. The workbook chapters correspond to the text chapters. Each chapter in this workbook includes activities to help students review and master the content of the text and help extend the learning of students beyond the textual readings through some experiential tasks that can be enacted both within and outside the classroom. Some activities may be appropriate for homework assignments or even additional course requirements. These activities are only meant as a start. Instructors are encouraged to alter and modify them to meet the individual and group needs of students. Answer keys for the activities are located at the end of the workbook.
A Quick Math Review
The Number World
Positive and Negative Numbers
Fractions, Percents, and Decimals
Exponents
Order of Operations
Algebra
I’m not very good in math.” This sentence is uttered by countless students in an infinite number of classrooms from elementary schools to high schools and, yes, even more so in colleges and universities. Math, often considered a four-letter word, has suffered through a long history of intolerance perpetuated by the belief that if your parents weren’t very good in math, then “genetically” neither are you! This section reviews some rules and skills that will help you survive aresearch course. Who knows—you may even end up liking math! What follows are the basic principles of mathematics that will serve as the basis for the statistical concepts later in this chapter.
The Number World
Within the number world, just like our world, there are families of numbers. Some numbers belong to more than one family, such as the whole numbers and the natural numbers.
Real Numbers are the set of numbers that consists of the positive numbers, negative numbers, and zero: –232.41, 0, 6, 9.3.
Rational Numbers are any real number that can be written as a fraction where the numerator and denominator are integers (the denominator cannot be zero). The resulting decimal either repeats itself in some form or terminates:–.
Irrational Numbers are any real number that cannot be written as a fraction using two integers. The resulting decimal neither repeats nor ends:= 1.41421356237 . . . , = 1.73205080756 . . . ,= 2.23606797749 . . . , π = 3.141592653589 . . . ,e = 2.718281828459045235 . . . .
Integers include the set {. . . , –2, –1, 0, 1, 2, . . .}. (The “. . .” means to infinity—in this case, in both positive and negative directions.)
Whole Numbers include the set {0, 1, 2, 3, . . .}.
Natural Numbers include the set {1, 2, 3, . . .}.
Positive and Negative Numbers
In an ideal world, there would be no debt and, therefore, no need for negative numbers. Thanks to the invention of credit cards and bank loans, however, we’re all seeinga little red! The sign in front of a number tells you where it is located in reference to zero. Negative numbers are to the left of zero, and positive numbers are to the right. For visual learners, it is helpful to look at a number line.
The farther you move to the left, the smaller the numbers become. This means that negative 5 (–5) is smaller than negative 2 (–2). In accounting terms, being “in the black” means one has made money (which is a positive!). The farther one is “in the red,” the more money one has lost (which is a negative!). The farther into the red one goes, the more money is owed (e.g., deeper in debt, deeper into the negative territory).
Concept 1. If a number or expression is written without a sign, it is positive.
Ex.5 and +5 represent the same value.5 = +5
Concept 2.When adding numbers with the same sign, add the numbers to get your answer.
Then take the sign of the original numbers as the sign of your answer.
Positive + Positive = PositiveNegative + Negative = Negative
Ex.4 + 7 = 11Ex.–3 + –6 = –9
Concept 3.When adding two numbers with opposite signs, subtract the two numbers (ignoring the signs) to get your answer.
To assign a prefix to your answer, look at the original two numbers. Use the sign of the number that is farthest from zero on the number line—in other words, the number that is the largest if they had no signs. It is clearer to see in these examples.
Ex.–4 + 8 = 4Ex.5 + –8 = –3
A concrete way of understanding these examples is to go a number line. To demonstrate the first example (–4 + 8 = 4), start at 0 and move 4 spaces tothe left. You should now be at –4. Now, move 8 spaces to the right (positive). You should now be at +4, the correct answer. Repeat this exercise for the second example (5 + –8 = –3).
Concept 4. Adding a negative number is the same as subtracting a positive number.
Ex.6 + (–4) = 2is the same as6 – (+4) = 2
Concept 5. Subtracting a negative number is the same as adding a positive number.
Two negatives (or subtraction signs) in a row equal a positive.
Ex.13 – (–5) = 18is the same as13 + (+5) = 18
Concept 6. When adding several numbers of mixed signs, follow these steps:
1.Add all of the positive numbers together. This answer is positive.
2.Add all of the negative numbers together, ignoring their signs for the moment. This answer is negative.
3.Then “add” your two answers by following Concept 3.
Ex.18 – 5 + 3 –7 + (–2) + 4
1.Add the positives: 18 + 3 + 4 = 25
2.Add the negatives: 5 = 7 + 2 = 14 or –14
3.Add these numbers: 25 + (–14) = 11
Concept 7.When multiplying or dividing numbers with the same sign, the answer is always positive.
Positive × Positive = PositiveNegative × Negative = Positive
Positive ÷ Positive = PositiveNegative ÷ Negative = Positive
Ex.4 × 7 = 28(-3)(-9) = 27-8 ·-4 = 32
Notice that there are three different ways of indicating multiplication: an “×” sign, a dot, or no sign if the numbers are in parentheses. Sometimes parentheses are used to enclose a single number or expression.
Ex.18 ÷ 6 = 3(–45) ÷ (–5) = 9
Concept 8.When multiplying or dividing numbers with opposite signs, the answer is always negative.
The order in which the signs appear does not matter.
Positive × Negative = NegativePositive ÷ Negative = Negative
Ex.(6) (–5) = –30-5 x 6 = -30
Concept 9. Multiplying any number by zero will ALWAYS equal zero.
Ex.0 × 8 = 0–4 · 0 = 0
Concept 10. Multiplying any number by one will give you back that number.
Ex.(1)(8) = 8(–3)(1) = –3
Concept 11. Dividing by zero is NOT possible in the real number system.
Try it. Enter 11 divided by 0 into your calculator. It should say you have an error. If within a problem you find you are dividing by 0 (0 is on the bottom of a fraction), go back and check your work. Mostly likely, there is a mathematical error.
Practice Set for Concepts 1–11:
1.–4 + 7 +5 –8 + 1 =2.(8)(–4) =
3.=4.5 + 7 – (–8) – 4 + 6 =
Fractions, Percents, and Decimals
When working with fractions, you will need to learn some basic vocabulary. The top number is referred to as the numerator, and the bottom number is called the denominator. To help you remember which is which, transpose the letters in denominator so it looks like “demon-inator,” and demons belong underneath! Fractions, percentages, and decimals are just three ways of representing the same value.
Concept 12.Comparing decimals.
Each place in a decimal number represents a value. As with money, the more you have to the left of the decimal point, the greater the value. The numbers to the right of the decimal point, however, are fractions of numbers.
The farther out to the right you go, the smaller the decimal value.
Ex. 0.01= and0.0.001=(one hundredth and one thousandth, respectively). So 0.001 is smaller than 0.01
Ex. 0.0236 = and0.154 = (236 ten thousandths and 154 thousandths, respectively). So 0.0236 is smaller than 0.154
Concept 13.Rounding decimals.
When rounding decimals, first decide how many places to which you plan to round. Typically, rounding to two decimal places is acceptable in most statistical studies, but the rules apply to rounding for any place value. Look one place past where you plan to round, and apply the phrase “Five or above, give it a shove. Four or below, leave it alone.” This phrase tells you what to do with the number you are rounding.
Ex.Round to two decimal places:3.14159
Look at the third-place number (thousandths). This is a 1. Since 1 is “four or below,” leave the second decimal place alone. Cut off the rest of the numbers. Your answer is 3.14.
Ex.Round to two decimal places:6.32815
Look at the third-place number (thousandths). This is an 8. Since 8 is “five or above,” give the second decimal place number a shove to the next number. Your answer is 6.33.
Concept 14.Simplifying fractions.
Simplifying a fraction means to find a number that divides evenly into both the numerator and the denominator.
Ex.Ex.
In the preceding example, 4 is the largest number that divides into both 12 and 8. When you divide 12 by 4, you get 3, and when you divide 8 by 4, you get 2. Therefore, your answer is , which is the same as the mixed number 1. If you don’t come up with the largest divisor right away, then continue to find numbers that go into both the numerator and denominator, and divide until you are out of numbers that go into both. The same process is used for the second example, . In this example, the number can be divided by 6, thus simplifying toand, finally to 2, which is the mixed number equivalent written in lowest terms.
Concept 15. Adding, subtracting, multiplying, and dividing decimals.
Plain and simple: Use a calculator! In this century, there is no reason to be without one.
Concept 16. Adding and subtracting fractions.
Before you can add or subtract fractions, their denominators must be the same number. If they are already the same, then add or subtract just the numerators, and leave the denominators alone.6
Ex.Ex.
If the denominators are not the same, then make them the same without changing the value of the fraction. Do this by multiplying both the numerators and denominators by a number that will make the denominators equal.
Ex.
Notice how multiplying byordoes not change the value of the fraction—just the look!
Ex.
Concept 17.Multiplying fractions.
When multiplying fractions, follow these steps:
1.Multiply the numerators together.
2.Multiply the denominators together.
3.Simplify if necessary.3
Ex.Ex.
Note: You can make any whole number a fraction by placing the number one underneath it.
Concept 18.Dividing fractions.
Dividing fractions is the same as multiplying by the reciprocal. In other words, leave the first fraction alone, flip the second fraction, and multiply using Concept 6.
Ex.Ex.
Concept 19.Rewriting fractions, percents, and decimals.
a.Fractions to Decimals: Fractions are just another way of representing division. Using your calculator, take the numerator (top number), and divide it by the denominator (bottom number). To changeto a decimal, enter 5 divided by 8 (5 ÷ 8) into your calculator. This gives you 0.625.
b.Fractions to Percents: First change your fraction to a decimal, and then multiply the decimal by 100% to get your percent.
Ex.= 0.8, (0.8)(100%) = 80%Ex.= 0.625, (0.625)(100%) = 62.5%
c.Percents to Decimals: Remove the percent sign, and then use your calculator to divide the number by 100. This usually results in the same number, except the decimal place has moved to the right two places.
Ex. 43.5% = Ex.
d.Percents to Fractions: Remove the percent. Place the number over 100, and simplify if necessary.
Ex. 22% = Ex. 250% =
e.Decimals to Percents: Multiply by 100!
Ex..26 × 100% = 26%Ex.3.91 × 100% = 391%
f.Decimals to Fractions: Leave decimals as decimals. It is rare they will ever need to be fractions!
Practice Set for Concepts 12–19:
Round the following numbers to two decimal places:
A.8.9315B.104.39842
Simplify the following fractions:
A.B.
Perform the indicated operations on the following fractions:
A.B.
C.D.
Complete the table by rewriting the number given as a fraction, decimal, and percent.
FRACTION / DECIMAL / PERCENT0.7
35%
Exponents
Exponents are used to represent repeated multiplication. It is shorthand for multiplying several of the same numbers or letters in a row.
Ex.3 · 3 · 3 · 3 · 3 · 3 = 36 = 729
Ex.x·x·x = x3
The general form of an exponent looks like an, where ais called the base, and n is called the exponent. The entire term is called a power.
arepresents the number tobe multiplied ←an→ n tells you how many times tomultiply the number by itself
Concept 20. If there is no exponent number, it is the same as a 1 there to hold the spot.
Ex.m = m1
Concept 21.When multiplying powers with the same base, add the exponents.
Ex.23· 22 = 25 = 32Ex.x2·x4·y2·y3·x = x7y5
(3 + 2) =5(x : 2 + 4 +1 = 7, y : 2 + 3 = 5)
(Note: If it is possible to multiply out the base (because it is a number and not a letter), then by all means, multiply it out.)
Concept 22. When an exponent is raised to another exponent, multiply the two exponents.
The exponent on the outside of the parentheses is distributed to each base on the inside.
Ex.(x3)2 = x6Ex.(3y3)2 = (3)2(y3)2 = 9y6
Concept 23.When dividing powers that have the same base, subtract the exponents.
Ex.*Sincethe first four 6’s on the top andbottom cancel out to equal 1;there are only two 6’s left on top.
Ex.Ex.
Concept 24. A nonzero number raised to the zero power is equal to one.
In other words, any number (or letter) raised to the power of zero cancels out to become the value of 1.
Ex.50 = 1Ex.6a4b0c3 = 6a4(1)c3 = 6a4c3
Practice Set for Concepts 20–24:
Simplify the following:
1.a2·a4·a =2.(4x2y3)2 =
3.4.(3b4c0d2)3 =
Concept 25.Radicals.
Radicals are the inverse operation of exponents. For example, addition is the inverse operation of subtraction because one operation undoes or cancels out the other. Therefore, to cancel out the operation of raising a number to the second power, one would take the square root of that number. It is easier to see with numbers:
Ex.52 = 25and
The general form of a radical (sometimes called a root) is , where m represents the root (or “shelf” number), and a is called the radicand. More often than not, m is 2 and left off of the shelf. So, if there is no number on the shelf, assume it is 2. This is asking you to think of what number times itself will give you the radicand. For example,is asking for a number that when multiplied by itself will give you 36. The answer is 6 because 62is 36. A common mistake is to take half of the radicand. If one did that with , one would have 18 and 182 is not 36!
Order of Operations
Concept 26.PEMDAS.
An order must be followed when performing operations (addition, subtraction,multiplication, and division) on numbers so that everyone comes out with the same answer. Here’s what could happen if there were NO order:
Ex.4 + 3 · 2 =or4 + 3 · 2 =
7 · 2 = 144 + 6 =10
1410
In the first expression, 4 and 3 were added before multiplying by 2. In the second expression, 3 and 2 were multiplied before adding to 4. Which way is correct?
To determine what order to perform operations, follow this simple phrase, “Please Excuse My Dear Aunt Sally.” In PEMDAS, each letter represents an operation and the order of the sentence explains what comes first!
Ex.6(4 + 1)2 – 7 · 2ParenthesesEx.3 – 4(2 · 5) + 52
6(5)2 – 7 · 2Exponents3 – 4(10) + 52
6(25) – 7 · 2Multiply/Divide3 – 4(10) + 25
150 – 14Add/Subtract3 – 40 + 25
136–12
If an operation is not included, then skip it and go onto the next! Just as with reading, work left to right when you are multiplying and dividing, or when you are adding and subtracting.
Ex.9 – 12 ÷ 4 + 6 · 4There are no parentheses or exponents; therefore, multiply/divide from left to right.
9 – 3 + 24Now add/subtract from left to right.
6 + 24
30
Practice Set for Concept 26:
Solve.
1.14 ÷ 7 + 32 =2.2 · 32 – 9 =
3.5 (4 + 2) – 12 =4.42 ÷ 3 + 32· (12 – 8) + 3 =
Algebra
Algebra is an often misunderstood math subject. Most students believe that there is a concrete explanation and practical use for algebra when, in fact, it is highly unlikely that you’ll be asked to solve an equation or graph a line unless you are in the math field! Algebra is more accurately described as a “workout for your brain.” You lift weights to strengthen your muscles, and you exercise to strengthen your heart and lungs. Doing these activities can help you be healthy and live longer. Well, algebra is strengthening your thinking skills and your problem-solving abilities. Doing algebra will help the brain think quicker and more effectively. This applies to all thinking activities done throughout the day! Now, here’s the one concept that throws off all of the math we’ve ever learned: