UNDERSTANDING STATISTICS IN THE BEHAVIORAL SCIENCES 10TH EDITION BY PAGANO TEST BANK

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UNDERSTANDING STATISTICS IN THE BEHAVIORAL SCIENCES 10TH EDITION BY PAGANO TEST BANK

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WITH ANSWERS
UNDERSTANDING STATISTICS IN THE BEHAVIORAL SCIENCES 10TH EDITION BY PAGANO TEST BANK

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.

 

  1. Specify the differences in mathematical operations between (X)2 and X2 and compute each.

 

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

 

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

 

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

 

  1. Round numbers with decimal remainders.

 

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

 

 

DETAILED CHAPTER SUMMARY

 

 

  1. Study Hints for the Student

 

A   Review basic algebra but dont be afraid that the mathematics will be too hard.

 

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

 

 

  1. Dont fall behind.  The material in the book is cumulative and getting behind is a bad idea.

 

  1. Work problems!

 

II      Mathematical Notation

 

  1. 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.

 

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

 

  1. 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

 

 

  1. 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

 

 

 

 

  1. 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

 

 

 

 

  1. 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.

 

  1. 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.

 

  1. 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 someones sex is male or female.  Can only determine whether A = B or A B.

 

  1. 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.

 

  1. 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.

 

  1. 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.

 

  1. Additional Points Concerning Variables

 

  1. 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.

 

  1. 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.

 

  1. 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 100C are 99.5 C and 100.5 C, when using a thermometer with accuracy to the nearest degree).

 

  1. 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.

 

  1. 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 dont 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 dont require that students learn the summation rules contained in note 2.1, p. 44.

 

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

 

  1.    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.
  2.    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

 

  1. 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.

 

  1. Rounding.  This is an easy section with the exception of rounding when the decimal remainder is .  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.

 

 

DISCUSSION QUESTIONS

 

  1. Are the mathematical operations called for by the same as those called for by?  Use an example to illustrate your answer.

 

  1. The Psychology Department faculty is considering four candidates for a faculty position. Each of the current twenty faculty members rank orders the four candidates, giving each a rank of 1, 2, 3, or 4, with a rank of 1 being the highest choice and a rank of 4 being the lowest.  The twenty rankings given for each candidate are then averaged and the candidate with the value closest to 1 is offered the job.  Is this a legitimate procedure?  Discuss.

 

  1. The procedure for rounding when the decimal remainder is seems a bit cumbersome. Why do you think it is used?  Discuss.

 

  1. Does it make sense to talk about the real limits of a discrete variable? Discuss.

 

 

TEST QUESTIONS

 

Multiple Choice

 

  1. Given the following subjects and scores, which symbol would be used to represent the score of 3?

 

Subject 1 2 3 4 5
Score 12 21 8 3 30

 

  1. X8
  2. X4
  3. X3
  4. X2

 

 

 

  1. We have collected the following data:

 

X1 = 6, X2 = 2, X3 = 4, X4 = 1, X5 = 3

 

For these data,  is equal to _________.

  1. 16
  2. 10
  3. 7
  4. 13

 

 

 

  1. Reaction time in seconds is an example of a(n) _________ scale.
  2. ratio
  3. ordinal
  4. interval
  5. nominal

 

 

 

 

  1. After performing several clever calculations on your calculator, the display shows the answer 53.655001. What is the appropriate value rounded to two decimal places?
  2. 53.65
  3. 53.66
  4. 53.64
  5. 53.60

 

 

 

  1. Consider the following points on a scale:

 

 

If the scale upon which A, B, C, and D are arranged is a nominal scale, we can say _________.

  1. B = 2A
  2. B A = D C
  3. both a and b
  4. neither a nor b

 

 

 

  1. When rounded to two decimal places, the number 3.175000 becomes _________.
  2. 3.17
  3. 3.20
  4. 3.18
  5. 3.10

 

 

 

  1. Given the data X1= 1, X2 = 4, X3 = 5, X4 = 8, X5 = 10, evaluate S X.
  2. 1
  3. 18
  4. 27
  5. 28

 

 

 

 

  1. Given the data X1 = 1, X2 = 4, X3 = 5, X4 = 8, X5 = 10, evaluate S X2.
  2. 56
  3. 784
  4. 206
  5. 28

 

 

 

  1. Given the data X1 = 1, X2 = 4, X3 = 5, X4 = 8, X5 = 10, evaluate (S X)2.
  2. 56
  3. 784
  4. 206
  5. 28

 

 

 

  1. Given the data X1= 1, X2 = 4, X3 = 5, X4 = 8, X5 = 10, evaluate .
  2. 17
  3. 27
  4. 28
  5. 23

 

 

 

  1. Given the data X1= 1, X2 = 4, X3 = 5, X4 = 8, X5 = 10, evaluate .
  2. 53
  3. 47
  4. 48
  5. 32

 

 

 

  1. Given the data X1= 1, X2 = 4, X3 = 5, X4 = 8, X5 = 10, evaluate .
  2. 47
  3. 53
  4. 48
  5. 32

 

 

 

 

  1. A discrete scale of measurement _________.
  2. is the same as a continuous scale
  3. provides exact measurements
  4. necessarily uses whole numbers
  5. b and c

 

 

 

  1. Consider the following points on a scale:

 

 

If the scale upon which A, B, C, and D are arranged is an interval scale, we can say _________.

  1. B = 2A
  2. B A = D C
  3. both a and b
  4. neither a nor b

 

 

 

  1. The number 83.476499 rounded to three decimal places is _________.
  2. 83.477
  3. 83.480
  4. 83.476
  5. 83.470

 

 

 

  1. The number 9.44650 rounded to two decimal places is _________.
  2. a. 45
  3. 99.46
  4. 99.44
  5. 99.40

 

 

 

  1. Brand of soft drink is measured on a(n) _________.
  2. nominal scale
  3. ordinal scale
  4. interval scale
  5. ratio scale

 

 

 

  1. At the annual sailing regatta, prizes are awarded for 1st, 2nd, 3rd, 4th, and 5th place. These places comprise a(n) _________.
  2. nominal scale
  3. ordinal scale
  4. interval scale
  5. ratio scale

 

 

 

  1. Which of the following numbers is rounded incorrectly to two decimal places?
  2. 10.4763410.48
  3. 15.3648515.36
  4. 21.47500 21.47
  5. 8.24501 8.25
  6. 6.66500 6.66

 

 

 

  1. Consider the following points on a scale:

 

 

If the scale upon which points A, B, C, and D are shown is an ordinal scale, we can meaningfully say _________.

  1. B A < D C
  2. B < C/2
  3. B = 2A
  4. C>B

 

 

 

  1. A continuous scale of measurement is different than a discrete scale in that a continuous scale _________.
  2. is an interval scale, not a ratio scale
  3. never provides exact measurements
  4. can take an infinite number of intermediate possible values
  5. never uses decimal numbers
  6. b and c

 

 

 

 

  1. Sex of children is an example of a(n) _________ scale.
  2. ratio
  3. nominal
  4. ordinal
  5. interval

 

 

 

  1. Which of the following variables has been labeled with an incorrect measuring scale?
  2. the number of students in a psychology class ratio
  3. ranking in a beauty contest ordinal
  4. finishing order in a poetry contest ordinal
  5. self-rating of anxiety level by students in a statistics class ratio

 

 

 

  1. A nutritionist uses a scale that measures weight to the nearest 0.01 grams. A slice of cheese weighs 0.35 grams on the scale.  The variable being measured is a _________.
  2. discrete variable
  3. constant
  4. continuous variable
  5. random variable

 

 

 

  1. A nutritionist uses a scale that measures weight to the nearest 0.01 grams. A slice of cheese weighs 0.35 grams on the scale.  The true weight of the cheese _________.
  2. is 0.35 grams
  3. may be anywhere in the range 0.345-0.355 grams
  4. may be anywhere in the range 0.34-0.35 grams
  5. may be anywhere in the range 0.34-0.36 grams

 

 

 

  1. In a 10-mile cross-country race, all runners are randomly assigned an identification number. These numbers represent a(n) _________.
  2. nominal scale
  3. ratio scale
  4. interval scale
  5. ordinal scale

 

 

 

  1. In the race mentioned in question 26, a comparison of each runners finishing time would represent a(n) _________.
  2. nominal scale
  3. ratio scale
  4. interval scale
  5. ordinal scale

 

 

 

  1. The sum of a distribution of 40 scores is 150. If we add a constant of 5 to each score, the resulting sum will be _________.
  2. 158
  3. 350
  4. 150
  5. 195

 

 

 

  1. Given the following set of numbers, X1= 2, X2 = 4, X3 = 6, X4 = 10, what is the value for S X?
  2. 12
  3. 156
  4. 480
  5. 22

 

 

 

  1. Given the following set of numbers, X1= 2, X2 = 4, X3 = 6, X4 = 10, what is the value of S X2?
  2. 156
  3. 22
  4. 480
  5. 37

 

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  1. Given the following set of numbers, X1= 2, X2 = 4, X3 = 6, X4 = 10, what is the value of X42?
  2. 4
  3. 6
  4. 100
  5. 10

 

 

 

  1. Given the following set of numbers, X1= 2, X2 = 4, X3 = 6, X4 = 10, what is the value of (S X)2?
  2. 480
  3. 484
  4. 156
  5. 44

 

 

 

  1. Given the following set of numbers, X1= 2, X2 = 4, X3 = 6, X4 = 10, what is the value of N?
  2. 2
  3. 4
  4. 6
  5. 10

 

 

 

  1. Given the following set of numbers, X1= 2, X2 = 4, X3 = 6, X4 = 10, what is the value of (S X)/N?
  2. 5
  3. 4
  4. 6
  5. 5.5

 

 

 

  1. Classifying subjects on the basis of sex is an example of using what kind of scale?
  2. nominal
  3. ordinal
  4. interval
  5. ratio
  6. bathroom

 

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  1. Number of bar presses is an example of a(n) _______ variable.
  2. discrete
  3. continuous
  4. nominal
  5. ordinal

 

 

 

  1. Using an ordinal scale to assess leadership, which of the following statements is appropriate?
  2. A has twice as much leadership ability as B
  3. X has no leadership ability
  4. Y has the most leadership ability
  5. all of the above

 

 

 

  1. The number of legs on a centipede is an example of a(an) _______ scale.
  2. nominal
  3. ordinal
  4. ratio
  5. continuous

 

 

 

  1. What are the real limits of the observation of 6.1 seconds (measured to the nearest second)?
  2. 6.05-6.15
  3. 5.5-6.5
  4. 6.0-6.2
  5. 6.00-6.20

 

 

 

  1. What is 17.295 rounded to one decimal place?
  2. 17.1
  3. 17.0
  4. 17.2
  5. 17.3

 

 

 

  1. What is the value of 0.05 rounded to one decimal place?
  2. 0.0
  3. 0.1
  4. 0.2
  5. 0.5

 

 

 

 

  1. The symbol S means:
  2. add the scores
  3. summarize the data
  4. square the value
  5. multiply the scores

 

 

 

  1. A therapist measures the difference between two clients. If the therapist can say that Rebeccas score is higher than Sarahs, but cant specify how much higher, the measuring scale used must have been a(an)_______ scale.
  2. nominal
  3. ordinal
  4. interval
  5. ratio

 

 

 

  1. An individual is measuring various objects. If the measurements made are to determine into which of six categories each object belongs, the measuring scale used must have been a(an)_______ scale.
  2. nominal
  3. ordinal
  4. interval
  5. ratio

 

 

 

  1. If an investigator determines that Carlos score is five times as large as the score of Juan, the measuring scale used must have been a(an)_______ scale.
  2. nominal
  3. ordinal
  4. interval
  5. ratio

 

 

 

The following questions test basic algebra

 

  1. Where 3X = 9, what is the value of X?
  2. 3
  3. 6
  4. 9
  5. 12

 

 

 

  1. For X + Y = Z, X equals _______.
  2. Y + Z
  3. Z Y
  4. Z/Y
  5. Y/Z

 

 

 

  1. 1/X + 2/X equals _______.
  2. 2/X
  3. 3/2X
  4. 3/X
  5. 2/X2

 

 

 

  1. What is (4 2)(34)/(6/3)?
  2. 24
  3. 1.3
  4. 12
  5. 6

 

 

 

  1. 6 + 43 1 simplified is _______.
  2. 29
  3. 48
  4. 71
  5. 17

 

 

 

  1. X = Y/Z can be expressed as _______.
  2. Y = (Z)(X)
  3. X = Z/Y
  4. Y = X/Z
  5. Z = X + Y

 

 

 

 

  1. 24equals _______.
  2. 4
  3. 32
  4. 8
  5. 16

 

 

 

  1. equals _______.
  2. 3
  3. 81
  4. 9
  5. 27

 

 

 

  1. X(Z + Y) equals _______.
  2. XZ + Y
  3. ZX + YX
  4. (X)(Y)(Z)
  5. (Z + Y)/X

 

 

 

  1. 1/2 + 1/4 equals _______.
  2. 1/6
  3. 1/8
  4. 2/8
  5. 3/4

 

 

 

  1. X6/X2equals _______.
  2. X8
  3. X4
  4. X2
  5. X3

 

 

 

 


True/False

 

  1. When doing summation, the number above the summation sign indicates the term ending the summation and the number below indicates the beginning term.

 

 

 

  1. S X2 and (S X)2 generally yield the same answer.

 

 

 

  1. With nominal scales there is a numerical relationship between the units of the scale.

 

 

 

  1. If IQ was measured on a ratio scale, and John had an IQ of 40 and Fred an IQ of 80, it would be correct to say that Fred was twice as intelligent as John.

 

 

 

  1. An ordinal scale possesses the attributes of magnitude and equal interval.

 

 

 

  1. Most scales used for measuring psychological variables are either ratio or interval.

 

 

 

  1. Measurement is always approximate with a continuous variable.

 

 

 

  1. It is standard practice to carry all intermediate calculations to four more decimal places than will be reported in the final answer.

 

 

 

  1. In 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 as it is.

 

 

  1. It is legitimate to do ratios with interval scaling.

 

 

 

  1. The number of students in a class is an example of a continuous variable.

 

 

 

  1. The real limits of a discrete variable are those values that are above and below the recorded value by one half of the smallest measuring unit of the scale.

 

 

 

  1. When rounding, if the decimal remainder is equal to and the last digit of the answer is even, add 1 to the last digit of the answer.

 

 

 

  1. A fundamental property of a nominal scale is equivalence.

 

 

 

  1. An interval scale is like a ratio scale, except that the interval scale doesnt possess an absolute zero point.

 

 

 

  1. A discrete variable requires nominal or interval scaling.

 

 

 

  1. Classifying students into whether they are good, fair, or poor speakers is an example of ordinal scaling.

 

 

 

  1. Determining the number of students in each section of introductory psychology involves the use of a ratio scale.

 

 

 

  1. In a race, Sam came in first and Fred second. Determining the difference in time to complete the race between Sam and Fred involves an ordinal scale

 

 

 

  1. If the remainder of a number = , we always round the last digit up.

 

 

 

  1. All scales possess magnitude, equal intervals between adjacent units, and an absolute zero point.

 

 

 

  1. Nominal scales can be used either qualitatively or quantitatively.

 

 

 

  1. With an ordinal scale one cannot be certain that the magnitude of the distance between any two adjacent points is the same.

 

 

 

  1. With the exception of division, one can perform all mathematical operations on a ratio scale.

 

 

 

  1. The average number of children in a classroom is an example of a discrete variable.

 

 

 

  1. When a weight is measured to 1/1000th of a gram, that measure is absolutely accurate.

 

 

 

  1. If the quantity S X = 400.3 for N observations, then the quantity S X will equal 40.03 if each of the original observations is multiplied by 0.1.

 

 

 

  1. One generally has to specify the real limits for discrete variables since they cannot be measured accurately.

 

 

 

  1. The symbol S means square the following numbers and sum them.

 

 

 

  1. Rounding 55.55 to the nearest whole number gives 55.

 

 

Short Answer

 

  1. Define continuous variable.

 

 

 

  1. Define discrete variable.

 

  1. Define interval scale.

 

  1. Define nominal scale.

 

  1. Define ratio scale.

 

 

 

  1. Define real limits of a continuous variable.

 

  1. How does an interval scale differ from an ordinal scale?

 

  1. Give two differences between continuous and discrete scales.

 

  1. What are the four types of scales and what mathematical operations can be done with each?
  2. Prove algebraically that .
  3. What is a discrete variable? Give an example.

 

  1. Student A claims that because his IQ is twice that of Student B, he is twice as smart as Student B. Is student A correct?  Explain.

 

  1. What is meant by the real limits of a continuous variable.

 

 

 

  1. The faculty of a psychology department are trying to decide between three candidates for a single faculty position. The department chairperson suggests that to decide, each faculty person should rank order the candidates from 1 to 3, and the ranks would then be averaged.  The candidate with the highest average would be offered the position.  Mathematically, what is wrong with that proposal.

 

 

 

 

  1. Consider the following sample scores for the variable weight:

 

X1 = 145, X2 = 160, X3 = 110, X4 = 130, X5 = 137, X6 = 172, and X7 = 150

 

  1. What is the value for ?
  2. What is the value for  ?
  3. What is the value for ?
  4. What is the value for ?
  5. What is the value for ?
  6. What is the value for ?
  7. What is the value for ?

 

 

 

  1. Round the following values to one decimal place.

 

  1. 25.15
  2. 25.25
  3. 25.25001
  4. 25.14999
  5. 25.26

 

 

 

  1. State the real limits for the following values of a continuous variable.

 

  1. 100 (smallest unit of measurement is 1)
  2. 1.35 (smallest unit of measurement is 0.01)
  3. 29.1 (smallest unit of measurement is 0.1)

 

 

 

  1. Indicate whether the following variables are discrete or continuous.

 

  1. The age of an experimental subject.
  2. The number of ducks on a pond.
  3. The reaction time of a subject on a driving task.
  4. A rating of leadership on a 3-point scale.

 

 

 

 

  1. Identify which type of measurement scale is involved for the following:

 

  1. The sex of a child.
  2. The religion of an individual
  3. The rank of a student in an academic class.
  4. The attitude score of a subject on a prejudice inventory.
  5. The time required to complete a task.
  6. The rating of a task as either easy, mildly difficult, or difficult.

 

 

 

  1. In an experiment measuring the number of aggressive acts of six children, the following scores were obtained.

 

 

Subject

Number of

Aggressive Acts

1

2

3

4

5

6

            15

25

5

18

14

22

 

  1. If X represents the variable of Number of Aggressive Acts, assign each of the scores its appropriate X symbol.
  2. Compute S X for these data.

 

 

 

  1. Given the following sample scores for the variable length (cm):

 

X1 = 22, X2 = 35, X3 = 32, X4 = 43, X5 = 28

 

  1. What is the value for ?
  2. What is the value for ?
  3. What is the value for ?

 

 

 

 

  1. Using the scores in Problem 21:

 

  1. What is the value for ?
  2. What is the value for ?

 

 

 

  1. Using the scores in Problem 21:

 

  1. What is the value for ?
  2. What is the value for ?

 

 

 

  1. Round the following to two decimal place accuracy.

 

  1. 75.0338
  2. 75.0372
  3. 75.0350
  4. 75.0450
  5. 75.045000001

 

 

 

CHAPTER 8

 

Random Sampling and Probability

 

 

 

 

 

LEARNING OBJECTIVES

 

After completing Chapter 8, students should be able to:

 

  1. Define a random sample; specify why the sample used in a study should be a random sample; and explain two methods of obtaining a random sample.

 

  1. Define sampling with replacement, sampling without replacement, a priori and a posteriori probability.

 

  1. List three basic points concerning probability values.

 

  1. Define the addition and multiplication rules, and solve problems involving their use.

 

  1. Define independent, mutually exclusive, and mutually exhaustive events.

 

  1. Define probability in conjunction with a continuous variable and solve problems when the variable is continuous and normally distributed.

 

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

 

 

DETAILED CHAPTER OUTLINE

 

 

  1. Inferential Statistics. Uses the sample scores to make a statement about a characteristic of the population.

 

  1. Hypothesis testing.  Data collected in an experiment in an attempt to validate some hypothesis involving a population.

 

  1. Parameter estimation.  The experimenter is interested in determining magnitude of a population characteristic;  e.g., population mean.

 

  1. Methodology of Inferential Statistics

 

  1. Random sampling.  A random sample is defined as a sample which has been selected from the population by a process which assures that each possible sample of a given size has an equal chance of being selected, and all the members of the population have an equal chance of being selected into the sample.

 

  1. Reasons for random sampling.

 

  1. Required in order to apply laws of probability to sample.
  2. It helps assure that the sample is representative of the population.

 

  1. Random number table.  Use of random number table is one method of assuring random sampling.

 

  1. Types of sampling.

 

  1. Sampling with replacement.  A method of sampling in which each member of the population selected for the sample is returned to the population before the next member is selected.
  2. Sampling without replacement.  A method of sampling in which the members of the sample are not returned to the population prior to selecting subsequent members.

 

III.    Probability

 

  1. Classical or a priori view of probability.  That which can be deduced from reason alone.  No actual data required.

 

 

where p(A) is read The probability of occurrence of event A.

 

  1. Empirical or a posteriori view of probability.  Meaning after-the-fact or after some data has been collected.

 

 

  1. Probability values.

 

  1. Range from 0.00 to 1.00 (i.e., from an event is certain not to occur to an event is certain to occur).
  2. Generally expressed as fraction or decimal.

 

  1. Computing probability.

 

  1. Addition rule.  This rule is concerned with determining the probability of occurrence of any of several possible events.  Rule for two events:

 

The probability of occurrence of A or B is equal to the probability of occurrence of A plus the probability of occurrence of B minus the probability of occurrence of both A and B, or

 

p(A or B) = p(A) + p(B) p(A and B)

 

  1. Multiplication rule.  This rule is concerned with the joint or successive occurrence of several events.  Rule for two events:

 

The probability of occurrence of both A and B is equal to the probability of occurrence of A times the probability of occurrence of B given A has occurred, or

p(A and B) = p(A)p(B|A)

 

Both rules can be extended to more than two events.

 

  1. Additional concepts.

 

  1. Mutually exclusive events.  Two events are mutually exclusive if they both cannot occur together (e.g., a coin coming up both heads and tails on one flip).
  2. Exhaustive events.  A set of events is exhaustive if the set includes all of the possible outcomes (e.g., for the flip of a coin, head and tail is the exhaustive set of possible outcomes).
  3. Mutually exclusive and exhaustive events.  When a set of events is both mutually exclusive and exhaustive the sum of the individual probabilities of each event in the set must equal 1.00.  Thus, under these conditions:

 

p(A) + p(B) + p(C) ++ p(Z) = 1.00

 

  1. When there are two events and they are mutually exclusive and exhaustive, then:

 

P + Q = 1.00

 

  1. More on the Multiplication Rule

 

  1. Mutually exclusive events.  If A and B are mutually exclusive, then:

 

p(A and B) = 0

 

  1. Independent events.  Two events are independent if the occurrence of one has no effect on the probability of occurrence of the other.  In this case the rule becomes:

 

p(A and B) = p(A)p(B|A) = p(A)p(B)

 

  1. Dependent events.  Two events are dependent if the probability of occurrence of B is affected by the occurrence of A.  In this case the rule becomes:

 

p(A and B) = p(A)p(B|A)

 

  1. Multiplication and addition rules.  Multiplication and addition rules can be combined to solve problems.

 

  1. Probability and Continuous Variables

 

  1. Equation.

 

 

 

  1. Solution.

 

  1.    Convert raw score to its transformed value (its standard score).
  2. Look up area in Table A.

 

 

TEACHING SUGGESTIONS AND COMMENTS

 

This chapter covers the topics of random sampling and probability.  It works well so I recommend that follow it in your lectures.

 

  1. Random Sampling.  The material on random sampling is straightforward and easy.  It is important that students know why random sampling is recommended for doing inference, that they can define random sampling, have an idea how to form a random sample, and know the difference between sampling with replacement and sampling without replacement.  Following the textbook presentation on random sampling works well to accomplish these goals.

 

  1. General Comments on Teaching Probability.  Mathematically, probability can be a very difficult topic.  When I was pursuing an undergraduate electrical engineering degree (in another life) mathematics was an important part of the curriculum.  Courses on probability were among the most difficult at the undergraduate level.  The opportunity therefore exists in teaching about probability to really press students to the wall, particularly when they are not usually that interested or skillful in math.  I imagine there are some professors when teaching a basic course in statistics, which use the probability material as an opportunity to create a grade separation between the more mathematically skillful students and the others.  I do not.

I dont view this course as the usual mathematics course, but rather as an applied mathematics course with emphasis on applied and not on mathematics per se.  In my opinion, the primary reason this course in included in a psychology curriculum or other behavior science curriculum is not to produce students well rounded in mathematics, but to give them knowledge and skills useful in their specific or related major, and then only secondarily to deepen their mathematical ability.  Therefore, I believe the amount and level of the probability material that they are taught should be that necessary to form a good foundation for the inference testing part of the course, and no more.  That is the premise from which I wrote the probability material of this chapter.

 

Fortunately, to provide a good basis for inference, we do not need to delve deeply into probability theory and mathematics.  Some basic probability definitions, along with the addition and multiplication rules does the job quite well.  The textbook material is written at a basic level and the end-of-chapter questions are not intended to challenge bright, mathematically sharp students, but rather to give students practice in solving problems involving basic concepts. and to help them determine if they have successfully learned the material as presented in the chapter.  In my experience this works well for most of the students I teach. It is not overly complicated and yet it does the job of providing a foundation for the inference material that follows.  I believe the material contained in the chapter is presented at the right level and has clear and sufficient examples to acquire the needed foundation.  Therefore I recommend you follow the textbook presentation, using the same examples.  It is also possible to interject your own examples as well for some variety.

 

  1. Addition Rule.  Nothing special here; the section works well.  An important point to emphasize is that the addition rule is invoked to find the probability of A or B, not A and B.  If we want to determine p(A and B), we need to use the multiplication rule, not the addition rule.

 

  1. Multiplication Rule.  This section works well too.  Again, it is important to emphasize that the multiplication rule is used to solve problems that ask for the probability of both A and B, not just one of them, A or B.

 

  1. Definitions.  To understand the material on these two rules, knowing the definitions of mutually exclusive events and of exhaustive events is important.  However, these definitions are usually learned by students without requiring any special effort on the instructors part, other that presenting them in the lecture.

 

  1. Probability and continuous variables.  This is an easy section.  There is an obvious similarity to the material presented in conjunction with the z test and normal distributions presented in Chapter 5.  In fact, the main difference is that the problems in Chapter 8 ask for probability values rather than area under the curve.  I point out to the students this similarity and assert that if they can handle the material in Chapter 5, it should be very easy to this grasp this topic.  Then I go over the problems contained in this part of the chapter, using transparencies, and they quickly catch on.  If a student is having any difficulty with this material, it usually means that the student has the same difficulty with the material in Chapter 5.  Resolving the difficulty with the Chapter 5 issue usually clears up both.

 

 

DISCUSSION QUESTIONS

 

  1. The process of random sampling guarantees that the sample selected will be representative of the population. Is this statement true?  Discuss.

 

  1. For the laws of probability to apply to the sample, it is necessary for the sample to be a random sample. Is this statement true?  Discuss.

 

  1. If I were to roll a real die, it is unlikely that the a priori and a posteriori probability of rolling a 5 will be equal. Is this statement correct?  Discuss.

 

  1. Assume you are sampling one score from a rectangular distribution of population scores having a mean = 40 and a standard deviation = 15. You want to determine the probability of getting a score 48.  You compute the z transformation of 48 and look up the area corresponding to the z score in Table A (column C) of your textbook.  You conclude that this area gives the desired probability.  Is this conclusion correct?  Discuss.

 

 

TEST QUESTIONS

 

Multiple Choice

 

  1. If events A and B are independent, then p(A and B) = _________.
  2. p(A) + p(B) p(A and B)
  3. p(A)p(B)
  4. p(A or B)
  5. 0

 

 

 

 

  1. If A and B are mutually exclusive and exhaustive, then p(A and B) = _________.
  2. 1
  3. 0
  4. p(A) + p(B)
  5. p(A) + p(B) p(A and B)

 

 

 

  1. If the odds in favor of an event occurring are 9 to 1, the probability of the event occurring is _________.
  2. 9
  3. 1/9
  4. 9/10
  5. 8/9

 

 

 

  1. Probabilities vary between _________.
  2. 0 and 2
  3. 0 and 100
  4. -1 and 0
  5. 0 and 1

 

 

 

  1. The Addition Rule states _________.
  2. p(A or B) = p(A) + p(B) p(A and B)
  3. p(A and B) = p(A)p(B|A)
  4. P + Q = 1
  5. p(A)p(B|A) = p(A)p(B)

 

 

 

  1. A sample is random if _________.
  2. each possible sample of a given size has an equal chance of being selected
  3. all members of the population have an equal chance of being selected into the sample
  4. all members of the sample have an equal chance of being selected
  5. a, or a and b
  6. a and c

 

 

 

 

  1. The Multiplication Rule states _________.
  2. p(A or B) = p(A) + p(B) p(A and B)
  3. p(A or B) = p(A) + p(B)
  4. p(A and B) = p(A)p(B|A)
  5. p(A and B) = p(A) + p(B)

 

 

 

  1. A priori probability refers to _________.
  2. a probability value deduced from reason alone
  3. the highest priority probability
  4. a probability value determined after collecting data
  5. none of the above

 

 

 

  1. A posteriori probability refers to _________.
  2. a probability value deduced from reason alone
  3. low priority probability
  4. a probability value determined after collecting data
  5. none of the above

 

 

 

  1. Two events are mutually exclusive if _________.
  2. they are independent
  3. they both cannot occur together
  4. the occurrence of one slightly alters the probability of occurrence of the other
  5. the probability of their joint occurrence equals one

 

 

 

  1. A set of events is exhaustive if _________.
  2. the sum of their probabilities equals one
  3. the set includes all of the possible events
  4. they are mutually exclusive
  5. they are independent

 

 

 

 

  1. Two events are independent if _________.
  2. the occurrence of one has no effect on the probability of occurrence of the other
  3. the occurrence of one precludes the occurrence of the other
  4. the occurrence of one substantially alters the probability of occurrence of the other
  5. the sum of their probabilities equals one
  6. a and d

 

 

 

  1. Suppose you are going to randomly order individuals A, B, C, D, E and F. The probability the order will begin A B _ _ _ _ is _________.
  2. 1.000
  3. 0.033
  4. 0.027
  5. 0.000

 

 

 

  1. A famous hypnotist performs in Meany Hall before a crowd of 350 students and 180 non-students. The hypnotist knows from previous experience that one-half of the students and two-thirds of the non-students are hypnotizable. What is the probability that a randomly chosen person from the audience will be hypnotizable or will be a non-student?
  2. 0.330
  3. 0.340
  4. 0.869
  5. 0.670

 

 

 

  1. Captain Kirk and Mr. Spock are engaged in a 3-D backgammon playoff, a game employing 6 dice. Kirk asks Spock the probability of rolling the dice and observing 6 sixes.  Assume the dice are not biased.  Spocks correct a priori reply is _________.
  2. Insufficient data, Captain.
  3. One-sixth to the sixth power, Sir. Translation: (1/6)6
  4. One-thirtysixth, Sir.
  5. One-thirtysix to the sixth power, Sir. (1/36)6
  6. The probability is equal to (1/6)X(1/5)X(1/4)X(1/3)X (1/2)X(1/1), Sir. (1/720)

 

 


 

  1. A hungry undergraduate student was looking for a way of making some extra money. The student turned to a life of vice gambling.  To be a good gambler, he needed to know the probability of certain events.  Help him out by answering the following question.

 

The probability of drawing 3 aces in a row without replacement from a deck of 52 ordinary playing cards is _________.

  1. 0.00018
  2. 0.00046
  3. 0.00017
  4. 0.00045

 

 

 

  1. A hungry undergraduate student was looking for a way of making some extra money. The student turned to a life of vice gambling.  To be a good gambler, he needed to know the probability of certain events.  Help him out by answering the following question.

 

The probability of drawing a face card (king, queen or jack) of any suit from a deck of 52 ordinary playing cards in one draw is _________.

  1. 0.020
  2. 0.231
  3. 0.077
  4. 0.019

 

 

 

  1. A hungry undergraduate student was looking for a way of making some extra money. The student turned to a life of vice gambling.  To be a good gambler, he needed to know the probability of certain events.  Help him out by answering the following question.

 

The probability of drawing an ace, a king and a queen of any suit in that order is _________.  Sampling is without replacement from a deck of 52 ordinary playing cards.

  1. 0.00045
  2. 0.00046
  3. 0.00048
  4. 0.00018

 

 

 


  1. A hungry undergraduate student was looking for a way of making some extra money. The student turned to a life of vice gambling.  To be a good gambler, he needed to know the probability of certain events.  Help him out by answering the following question.

 

The probability of rolling boxcars (two sixes) with one roll of a pair of fair dice is _________.

  1. 0.167
  2. 0.333
  3. 0.033
  4. 0.028

 

 

 

  1. A hungry undergraduate student was looking for a way of making some extra money. The student turned to a life of vice gambling.  To be a good gambler, he needed to know the probability of certain events.  Help him out by answering the following question

 

A royal flush in poker is when you end up with the ace, king, queen, jack, and 10 of the same suit.  Its the most rare event in poker.  If you are playing with a well- shuffled, legitimate deck of 52 cards, what is the probability that if you are dealt 5 cards, you will have a royal flush?  Assume randomness.

  1. 0.0000000032
  2. 0.000000013
  3. 0.0000015
  4. 0.0000004

 

 

 

  1. Lets assume you are having a party and have stocked your refrigerator with beverages. You have 12 bottles of Coors beer, 24 bottles of Rainier beer, 24 bottles of Schlitz light beer, 12 bottles of Hamms beer, 2 bottles of Heineken dark beer and 6 bottles of Pepsi soda. You go to the refrigerator t

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