Understanding Statistics in the Behavioral Sciences 10th Edition by Robert R. Pagano Test Bank

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Understanding Statistics in the Behavioral Sciences 10th Edition by Robert R. Pagano Test Bank

Table of content

1. Statistics and Scientific Method.
2. Basic Mathematical and Measurement Concepts.
3. Frequency Distributions.
4. Measures of Central Tendency and Variability.
5. The Normal Curve and Standard Scores.
6. Correlation.
7. Linear Regression.
8. Random Sampling and Probability.
9. Binomial Distribution.
10. Introduction to Hypothesis Testing Using the Sign Test.
11. Power.
12. Sampling Distributions, Sampling Distribution of the Mean, the Normal Deviate (z) Test.
13. Student’s t-Test for Single Samples.
14. Student’s t-Test for Correlated and Independent Groups.
15. Introduction to the Analysis of Variance.
16. Introduction to Two-Way Analysis of Variance.
17. Chi-Square and Other Nonparametric Tests.
18. Review of Inferential Statistics. Appendices.

CHAPTER 2

Basic Mathematical and  Measurement Concepts LEARNING OBJECTIVES

After completing Chapter 2, students should be able to:

1. Assign subscripts using the X variable to a set of numbers.

2 Do the operations called for by the summation sign for various values of i and N.

3. Specify the differences in mathematical operations between (ΣX)² and ΣX² and compute each.

4. Define and recognize the four measurement scales, give an example of each, and state the mathematical operations that are permissible with each scale.

5. Define continuous and discrete variables, and give an example of each.

6. Define the real limits of a continuous variable; and determine the real limits of values obtained when measuring a continuous variable.

7. Round numbers with decimal remainders. 

8. Understand the illustrative examples, do the practice problems and understand the solutions.

DETAILED CHAPTER SUMMARY

I. Study Hints for the Student

AReview basic algebra but don’t be afraid that the mathematics will be too hard.

B.Become very familiar with the notations in the book. 

C.Don’t fall behind.  The material in the book is cumulative and getting behind is a bad idea.

D.Work problems!

II Mathematical Notation

A.Symbols.  The symbols X (capital letter X) and sometimes Y will be used as symbols to represent variables measured in the study.

1.For example, X could stand for age, or height, or IQ in any given study.

2.To indicate a specific observation a subscript on X will be used; e.g., X2 would mean the second observation of the X variable.

B.Summation sign.  The summation sign (Σ) is used to indicate the fact that the scores following the summation sign are to be added up.  The notations above and below the Σ sign are used to indicate the first and last scores to be summed.

C.Summation rules.

1.The sum of the values of a variable plus a constant is equal to the sum of the values of the variable plus N times the constant.  In equation form 

2.  The sum of the values of a variable minus a constant is equal to the sum of the variable minus N times the constant.  In equation form

 

3.The sum of a constant times the values of a variable is equal to the constant times the sum of the values of the variable.  In equation form

 

4. The sum of a constant divided into the values of a variable is equal to the constant divided into the sum of the values of the variable.  In equation form

 

III. Measurement Scales.

A. Attributes.  All measurement scales have one or more of the following three attributes.

1.Magnitude.

2.Equal intervals between adjacent units.

3.Absolute zero point.

B.Nominal scales.  The nominal scale is the lowest level of measurement.  It is more qualitative than quantitative.  Nominal scales are comprised of elements that have been classified as belonging to a certain category.  For example, whether someone’s sex is male or female.  Can only determine whether A = B or A ≠ B.

C.Ordinal scales.  Ordinal scales possess a relatively low level of the property of magnitude.  The rank order of people according to height is an example of an ordinal scale.  One does not know how much taller the first rank person is over the second rank person.  Can determine whether A > B, A = B or A < B.

D.Interval scales.  This scale possesses equal intervals, magnitude, but no absolute zero point.  An example is temperature measured in degrees Celsius.  What is called zero is actually the freezing point of water, not absolute zero.  Can do same determinations as ordinal scale, plus can determine if A – B = C − D, A − B > C – D, or A − B < C − D.

E.Ratio scales.  These scales have the most useful characteristics since they possess attributes of magnitude, equal intervals, and an absolute zero point.  All mathematical operations can be performed on ratio scales.  Examples include height measured in centimeters, reaction time measured in milliseconds.

IV. Additional Points Concerning Variables

A.Continuous variables.  This type can be identified by the fact that they can theoretically take on an infinite number of values between adjacent units on the scale.  Examples include length, time and weight.  For example, there are an infinite number of possible values between 1.0 and 1.1 centimeters.

B.Discrete variables.  In this case there are no possible values between adjacent units on the measuring scale.  For example, the number of people in a room has to be measured in discrete units.  One cannot reasonably have 6 1/2 people in a room.

C.Continuous variables.  All measurements on a continuous variable are approximate.  They are limited by the accuracy of the measurement instrument.  When a measurement is taken, one is actually specifying a range of values and calling it a specific value.  The real limits of a continuous variable are those values that are above and below the recorded value by 1/2 of the smallest measuring unit of the scale (e.g., the real limits of 100∞C are 99.5∞ C and 100.5∞ C, when using a thermometer with accuracy to the nearest degree).

D.Significant figures.  The number of decimal places in statistics is established by tradition.  The advent of calculators has made carrying out laborious calculations much less cumbersome.  Because solutions to problems often involve a large number of intermediate steps, small rounding inaccuracies can become large errors.  Therefore, the more decimals carried in intermediate calculations, the more accurate is the final answer.  It is standard practice to carry to one or more decimal places in intermediate calculations than you report in the final answer.

E.Rounding.  If the remainder beyond the last digit is greater than 1/2 add one to the last digit.  If the remainder is less than 1/2 leave the last digit the same.  If the remainder is equal to 1/2 add one to the last digit if it is an odd number, but if it is even, leave it as it is.

TEACHING SUGGESTIONS AND COMMENTS

This is also a relatively easy chapter.  The chapter flows well and I suggest that you lecture following the text.  Some specific comments follow:  

1.Subscripting and summation.  If you want to use new examples, an easy opportunity to do so, without confusing the student is to use your own examples to illustrate subscripting and summation.  It is very important that you go over the difference between the operations called for by  and  .  These terms appear often throughout the textbook, particularly in conjunction with computing standard deviation and variance.  If students are not clear on the distinction and don’t learn how to compute each now, it can cause them a lot of trouble down the road.  They also get some practice in Chapter 4.  I suggest that you use your own numbers to illustrate the difference.  It adds a little variety without causing confusion.    Regarding summation, I usually go over in detail, explaining the use of the terms beneath and above the summation sign, as is done in the textbook.  However, I don’t require that students learn the summation rules contained in note 2.1, p. 44. 

2.Measurement scales.  The material on measurement scales is rather straight forward with the following exceptions.  

a.Regarding nominal scales, students often confuse the concepts that there is no quantitative relationship between the units of a nominal scale and that it is proper to use a ratio scale to count items within each unit (category).  Be sure to discuss this.  Going through an example usually clears up this confusion.

b.Students sometimes have a problem understanding the mathematical operations that are allowed by each measuring scale, except of course, the mathematical operations allowed with a ratio scale, since all are allowed.  A few examples usually helps.  Again, I recommend using your own numbers with these examples 

3.Real limits of a continuous variable.  This topic can be a little confusing to some students.  However, a few examples explained in conjunction with the definition on p. 35 seems to work well in dispelling this confusion.

4.Rounding.  This is an easy section with the exception of rounding when the decimal remainder is ½.  To help correct this, I suggest you go over several examples.  I recommend you make up your own examples since it is easy to do so and adds some variety.  Students sometime wonder why such a complicated rule is used and ask, “Why not just round up.”  The answer is that if you did this systematically over many such roundings, it would introduce a systematic upwards bias.