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

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

 

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Sample Questions Are Posted Below

 

CHAPTER 5

 

The Normal Curve and

Standard Scores

 

 

 

 

 

LEARNING OBJECTIVES

 

After completing Chapter 5, students should be able to:

 

1. Describe the typical characteristics of a normal curve.

 

2. Define a z score.

 

3. Compute the z score for a raw score, given the raw score, the mean and standard deviation of the distribution.

 

4. Compute the z score for a raw score, given the raw score and the distribution of raw scores.

 

5. Explain the three main features of z distributions.

 

6. Use z scores with a normal curve to find: a) the percentage of scores falling below any raw score in the distribution; b) the percentage of scores falling above any raw score in the distribution and c) the percentage of scores falling between any two raw scores in the distribution.

 

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

 

 

DETAILED CHAPTER SUMMARY

 

 

1. The Normal Curve

 

1. Important in behavioral sciences.

 

 

1. Many variables of interest are approximately normally distributed.
2. Statistical inference tests have sampling distributions which become normally distributed as sample size increases.
3. Many statistical inference tests require sampling distributions that are normally distributed.

 

1. Characteristics.

 

1. Symmetrical, bell-shaped curve.
2. Equation.

 

 

 

 

Shows that the curve is asymptotic to the abscissa, i.e., it approaches the X axis and gets closer and closer but never touches it.

 

3. Area contained under the normal curve:

 

34. Area under the curve represents the percentage of scores contained within the area.                       b.         34.13% of scores between mean (µ) and +1s;  13.59% of area contained between a score equal to µ + 1s and a score of µ + 2s;  2.15% of area is between µ + 2s and µ + 3s;  and 0.13% falls beyond µ + 3s.

 

1. Since the curve is symmetrical, the same percentages hold for scores below the mean.

 

1. Standard Scores (z Scores)

 

1. Symbol.  Symbolized as z.

 

1. Definition.  A standard score is a transformed score which designates how many standard deviation units the corresponding raw score is above or below the mean.

 

C   Equation.

 

 

                                       z = ( – µ)/s    for population data

 

                                       z = (X – )/s    for sample data

 

 

1. Comparisons between different distributions.

 

1. Allows comparisons even when the units of the distributions are different.
2. Percentile ranks are possible.

 

1. Characteristics of z scores.

 

1. z scores have the same shape as the set of raw scores from which they were transformed.
2. µz = 0.  The mean of z scores equals zero.
3. sz = 1.00.  The standard deviation of z scores equals 1.00.

 

1. Using z scores.

 

1. Finding the area given the raw score.

 

 

z = (X – µ)/s

 

 

Use above formula to calculate z score. Then use table to determine the area under the normal curve for the various values of z.

 

2. Finding the raw score given the area.

 

 

X = µ + sz

 

 

Use above formula substituting the value of z that designates the area under the curve one wishes, and solve for X, the raw score.

 

 

TEACHING SUGGESTIONS AND COMMENTS

 

This is a short and easy chapter.  For computing z scores from raw scores or from a problem where they are given the distribution mean and standard deviation, I recommend that you use your own examples.  For determining areas under a normally distributed set of scores, I recommend you use the textbook examples.  Conceptually, students seem to easily understand that μ_z = 0, but have a little more difficulty with notion that σ_z = 1.  To help them understand the latter, I recommend going through the algebraic proof offered on p. 109, and show them a demonstration by transforming a short set of raw scores and then computing σ_z.  One other point needs emphasizing.  Since the chapter uses z scores in conjunction with the normal distribution, students sometimes get the wrong idea that all z distributions are normally shaped.  The truth is, of course, that the z distribution has the same shape at the untransformed raw scores.  Even though this point is stated in italics in the text, it needs emphasis in your lecture.

 

 

DISCUSSION QUESTIONS

 

1. Recently, the progressives of a particular country have complained that governmental policies have favored the rich, making a small percentage of citizens much richer, while leaving the rest of the citizens either unaffected or even slightly poorer. Data are available giving the mean and median incomes of all citizens prior to and after the current governmental policies were initiated.  If the progressives are correct, how might one use these data to support their assertion?  Explain.

 

2. Suppose you randomly took 10000 samples, each of size N, from a population of scores and computed the mean, median, and mode of each sample. Next, you computed the variance of the 10000 mean values, the 10000 median values, and the 10000 mode values.  If you rank ordered the resulting three variances, what order would you expect?  What property of the mean and median did you use in answering this question?

 

3. Both the range and standard deviation are measures of variability. If the variability of a set of scores changed, would you always expect the values of the range and standard deviation to change too?  If so, would you expect these values to always change in the same direction?  Explain.

 

4. Suppose that you wanted to compare two sets of 50 scores each. Discuss how you might do so, and state the advantages and disadvantages of each method you propose.

 

5. Why might anyone be interested in comparing two or more sets of scores? Illustrate, giving examples.

 

6. Discuss the advantages and disadvantages of using SPSS to do your homework problems.

 

 

TEST QUESTIONS

 

Multiple Choice

 

1. A stockbroker has kept a daily record of the value of a particular stock over the years and finds that prices of the stock form a normal distribution with a mean of $8.52 with a standard deviation of $2.38.

 

The percentile rank of a price of $13.87 is _________.

48. 48.78%
49. 1.22%
50. 98.78%
51. 51.22%

 

 

 

2. A stockbroker has kept a daily record of the value of a particular stock over the years and finds that prices of the stock form a normal distribution with a mean of $8.52 with a standard deviation of $2.38.

 

What percentage of the distribution lies between $5 and $11?

21. 21.48%
22. 78.41%
23. 49.41%
24. 57.98%

 

 

 

3. A stockbroker has kept a daily record of the value of a particular stock over the years and finds that prices of the stock form a normal distribution with a mean of $8.52 with a standard deviation of $2.38.

 

What percentage of the distribution lies below $7.42.

17. 17.72%
18. 32.28%
19. 82.28%
20. 31.92%

 

 

 

4. A stockbroker has kept a daily record of the value of a particular stock over the years and finds that prices of the stock form a normal distribution with a mean of $8.52 with a standard deviation of $2.38.

 

The stock price beyond which 0.05 of the distribution falls is _________.

4. $ 4.60
5. $12.47
6. $  4.57
7. $12.44

 

 

 

5. A stockbroker has kept a daily record of the value of a particular stock over the years and finds that prices of the stock form a normal distribution with a mean of $8.52 with a standard deviation of $2.38.

 

The percentage of scores that lie between $9.00 and $10.00 is _________.

15. 15.31%
16. 31.17%
17. 23.24%
18. 7.93%

 

 

 

6. A testing bureau reports that the mean for the population of Graduate Record Exam (GRE) scores is 500 with a standard deviation of 90. The scores are normally distributed.

 

The percentile rank of a score of 667 is _________.

3. 3.14%
4. 96.78%
5. 3.22%
6. 96.86%

 

 

 

7. A testing bureau reports that the mean for the population of Graduate Record Exam (GRE) scores is 500 with a standard deviation of 90. The scores are normally distributed.

 

The proportion of scores that lie above 650 is _________.

1. 0.4535
2. 0.9535
3. 0.0475
4. 0.0485

 

 

 

8. A testing bureau reports that the mean for the population of Graduate Record Exam (GRE) scores is 500 with a standard deviation of 90. The scores are normally distributed.

 

 

The proportion of scores that lie between 460 and 600 is _________.

1. 0.4394
2. 0.5365
3. 0.4406
4. 0.4635

 

 

 

9. A testing bureau reports that the mean for the population of Graduate Record Exam (GRE) scores is 500 with a standard deviation of 90.  The scores are normally distributed.

 

The raw score that lies at the 90th percentile is _________.

615. 615.20
616. 384.80
617. 616.10
618. 383.90

 

 

 

10. A testing bureau reports that the mean for the population of Graduate Record Exam (GRE) scores is 500 with a standard deviation of 90. The scores are normally distributed.

 

The proportion of scores between 300 and 400 is _________.

1. 0.3665
2. 0.4868
3. 0.8533
4. 0.1203

 

 

 

11. The standard deviation of the z distribution equals _________.
12. 1
13. 0
14. S X
15. N

 

 

 

 

12. The mean of the z distribution equals _________.
13. 1
14. 0
15. S X
16. N

 

 

 

13. The z score corresponding to the mean of a raw score distribution equals _________.
14. the mean of the raw scores
15. 0
16. 1
17. N

 

 

 

14. The normal curve is _________.
15. linear
16. rectangular
17. bell-shaped
18. skewed

 

 

 

15. In a normal curve, the inflection points occur at _________.
16. µ ± 1s
17. ±1s
18. µ ± 2s
19. µ

 

 

 

16. The z score corresponding to a raw score of 120 is _________.
17. 1.2
18. 2.0
19. 1.0
20. impossible to compute from the information given

 

 

 

 

17. An economics test was given and the following sample scores were recorded:

 

Individual	A	B	C	D	E	F	G	H	I	J
Score	12	12	7	10	9	12	13	8	9	8

 

The mean of the distribution is _________.

12. 12.00
13. 10.00
14. 9.00
15. 8.00

 

 

 

18. An economics test was given and the following sample scores were recorded:

 

Individual	A	B	C	D	E	F	G	H	I	J
Score	12	12	7	10	9	12	13	8	9	8

 

The standard deviation of the distribution is _________.

10. 10.20
11. 2.10
12. 2.11
13. 10.74

 

 

 

19. An economics test was given and the following sample scores were recorded:

 

Individual	A	B	C	D	E	F	G	H	I	J
Score	12	12	7	10	9	12	13	8	9	8

 

The z score for individual D is _________.

1. 1
2. 0
3. 10

 

 

 

 

20. An economics test was given and the following sample scores were recorded:

 

Individual	A	B	C	D	E	F	G	H	I	J
Score	12	12	7	10	9	12	13	8	9	8

 

The z score for individual E is _________.

1. 0.47
2. 9.00
3. 4.27
4. -0.47

 

 

 

21. An economics test was given and the following sample scores were recorded:

 

Individual	A	B	C	D	E	F	G	H	I	J
Score	12	12	7	10	9	12	13	8	9	8

 

The z score for individual G is _________.

1. -1.42
2. 13.00
3. 1.42
4. 6.16

 

 

 

22. A distribution has a mean of 60.0 and a standard deviation of 4.3. The raw score corresponding to a z score of 0.00 is _________.
23. 64.3
24. 14.0
25. 4.3
26. 60.0

 

 

 

23. A distribution has a mean of 60.0 and a standard deviation of 4.3. The raw score corresponding to a z score of -1.51 is _________.
24. 53.5
25. 66.5
26. 66.4
27. 53.6

 

 

 

24. A distribution has a mean of 60.0 and a standard deviation of 4.3. The raw score corresponding to a z score of 2.02 is _________.
25. 51.3
26. 68.7
27. 51.4
28. 68.6

 

 

 

25. If a population of scores is normally distributed, has a mean of 45 and a standard deviation of 6, the most extreme 5% of the scores lie beyond the score(s) of _________.
26. 35.13
27. 45.99
28. 56.76 and 33.24
29. 45.99 and 35.13

 

 

 

26. If a distribution of raw scores is negatively skewed, transforming the raw scores into z scores will result in a _________ distribution.
27. normal
28. bell-shaped
29. positively skewed
30. negatively skewed

 

 

 

27. The mean of the z distribution equals _________.
28. 0
29. 1
30. N
31. depends on the raw scores

 

 

 

28. The standard deviation of the z distribution equals _________.
29. 0
30. 1
31. the variance of the z distribution
32. b and c

 

 

 

 

29. S(z – µz) equals _________.
30. 0
31. 1
32. the variance
33. cannot be determined

 

 

 

30. The proportion of scores less than z = 0.00 is _________.
31. 0.00
32. 0.50
33. 1.00
34. -0.50

 

 

 

31. In a normal distribution the z score for the mean equals _________.
32. 0
33. the z score for the median
34. the z score for the mode
35. all of the above

 

 

 

32. In a normal distribution approximately _________ of the scores will fall within 1 standard deviation of the mean.
33. 14%
34. 95%
35. 70%
36. 83%

 

 

 

33. Would you rather have an income (assume a normal distribution and you are greedy) _________.
34. with a z score of 1.96
35. in the 95th percentile
36. with a z score of -2.00
37. with a z score of 0.000

 

 

 

 

34. How much would your income be if its z score value was 2.58?
35. $10,000
36. $ 9,999
37. $ 5,000
38. cannot be determined from information given

 

 

 

35. Which of the following z scores represent(s) the most extreme value in a distribution of scores assuming they are normally distributed?
36. 1.96
37. 0.0001
38. -0.0002
39. -3.12

 

 

 

36. Assuming the z scores are normally distributed, what is the percentile rank of a z score of -0.47?
37. 31.92
38. 18.08
39. 50.00
40. 47.00
41. 0.06

 

 

 

37. A standardized test has a mean of 88 and a standard deviation of 12. What is the score at the 90th percentile?  Assume a normal distribution.
38. 90.00
39. 112.00
40. 103.36
41. 91.00

 

 

 

38. On a test with a population mean of 75 and standard deviation equal to 16, if the scores are normally distributed, what is the percentile rank of a score of 56?
39. 58.30
40. 0.00
41. 25.27
42. 38.30
43. 11.70

 

 

 

39. On a test with a population mean of 75 and standard deviation equal to 16, if the scores are normally distributed,, what percentage of scores fall below a score of 83.8?
40. 55.00
41. 79.12
42. 20.88
43. 29.12
44. 70.88

 

 

 

40. On a test with a population mean of 75 and standard deviation equal to 16, if the scores are normally distributed, what percentage of scores fall between 70 and 80?
41. 75.66
42. 70 23
43. 24.34
44. 23.57
45. 12.17

 

 

 

41. You have just received your psychology exam grade and you did better than the mean of the exam scores. If so, the z transformed value of your grade must
42. be greater than 1.00
43. must be greater than 0.00
44. have a percentile rank greater than 50%
45. can’t determine with information given
46. b and c.

 

 

 

42. You have just taken a standardized skills test designed to help you make a career choice. Your math skills score was 63 and your writing skills score was 45.  The standardized math distribution is normally distributed, with m = 50, and s = 8.  The writing skills score distribution is also normally distributed, with m = 30, and s = 10.  Based on this information, as between pursuing a career that requires good math skills or one requiring good writing skills, you should chose _________.
43. neither, your skills are below average in both
44. the career requiring good math skills.
45. neither, this approach is bogus; dream interpretation should be used instead.
46. the career requiring good writing skills.

 

 

 

43. A distribution of raw scores is positively skewed. You want to transform it so that it is normally distributed.  Your friend, who fancies herself a statistics whiz, advises you to transform the raw scores to z scores; that the z scores will be normally distributed.  You should _________.
44. ignore the advice because your friend flunked her last statistics test
45. ignore the advice because z distributions have the same shape as the raw scores.
46. take the advice because z distributions are always normally distributed
47. take the advice because z distributions are usually normally distributed

 

 

 

44. All bell-shaped curves _________.
45. are normal curves
46. have means = 0
47. are symmetrical
48. a and c

 

 

 

45. If you transformed a set of raw scores, and then added 15 to each z score, the resulting scores _________.
46. would have a standard deviation = 1
47. would have a mean = 0
48. would have a mean =15
49. would have a standard deviation > 1
50. a and c

 

 

 

46. A set of raw scores has a rectangular shape. The z transformed scores for this set of raw scores has a _________ shape.
47. rectangular
48. normal (bell-shaped)
49. it depends on the number of scores in the distribution
50. none of the above

 

 

 

 

47. Makaela took a Spanish exam; her grade was 79. The distribution was normally shaped with = 70 and s = 12.  Juan took a History exam; his grade was 86.  The distribution was normally shaped with = 80 and s = 8.  Which did better on their exam relative to those taking the exam?
48. Makaela.
49. Juan.
50. Neither, they both did as well as each other.
51. Makaela, because her exam was harder
52. a and d.

 

 

 

48. Table A in your textbook has no negative z values, this means _________.
49. the table can only be used with positive z values
50. the table can be used with both positive and negative z values because it is symmetrical
51. the table can be used with both positive and negative z values because it is skewed
52. none of the above

 

 

 

49. A testing service has 1000 raw scores. It wants to transform the distribution so that the mean = 10 and the standard deviation = 1.  To do so, _________.
50. do a z transformation for each raw score and add 10 to each z score.
51. do a z transformation for each raw score and multiply each by 10
52. divide the raw scores by 10
53. compute the deviation score for each raw score. Divide each deviation score by the standard deviation of the raw scores.  Take this result for all scores and add 10 to each one.
54. a and d

 

 

 

50. Given the following set of sample raw scores, X: 1, 3, 4, 6, 8. What is the z transformed value for the raw score of 3?
51. -0.18
52. -0.48
53. -0.15
54. -0.52

 

 

 

 

 

True/False

 

1. A z distribution always is normally shaped.

 

 

 

2. All standard scores are z scores.

 

 

 

3. A z score is a transformed score.

 

 

 

4. A z score designates how many standard deviations the raw score is above or below the mean.

 

 

 

5. The z distribution takes on the same shape as the raw scores.

 

 

 

6. z scores allow comparison of variables that are measured on different scales.

 

 

 

7. In a normal curve, the area contained between the mean and a score that is 2.30 standard deviations above the mean is 0.4893 of the total area.

 

 

 

8. The normal curve reaches the horizontal axis in 4 standard deviations above and below the mean.

 

 

 

9. For any z distribution of normally distributed scores, P50 is always equal to zero.

 

 

 

10. If the original raw score distribution has a mean that is not equal to zero, the mean of the z transformed scores will not equal zero either.

 

 

 

11. It is impossible to have a z score of 30.2.

 

 

 

12. The area under the normal curve represents the proportion of scores that are contained in the area.

 

 

 

13. If the raw score distribution is very positively skewed, the standard deviation of the z transformed scores will not equal 1.

 

 

 

14   The area beyond a z score of –1.12 is the same as the area beyond a z score of 1.12.

 

 

 

15   A raw score that is 1 standard deviation above the mean of the raw score distribution will have a z score of 1.

 

 

 

16. The normal curve is a symmetric, bell-shaped curve.

 

 

 

17. The area under the normal curve represents the percentage of scores contained within the area.

 

 

 

18. It is impossible to have a z score of 23.5.

 

 

 

19. A z transformation will allow comparisons to be made when units of distributions are different.

 

 

 

20. If the original raw score distribution is not normally distributed, the mean of the z transformation scores of the raw data will not equal 0.

 

 

 

21. In the standard normal curve, 13.59% of the scores will always be contained between the mean (µ) and +1s.

 

 

 

22. In a plot of the normal curve, frequency is plotted on the X axis.

 

 

 

23. The standard deviation of the z distribution is always equal to 1.0.

 

 

 

24. The area beyond a z score of +2.58 is 0.005.

 

 

 

25. One cannot reasonably do z transformations on ratio data.

 

 

 

26. The normal curve never touches the X axis.

 

 

 

27. To do a z transformation, one must know only the population mean and the value of the raw score to be transformed.

 

 

 

28. To calculate the score at the 97.5th percentile, one would apply the formula X = µ + (s)(1.96).

 

 

 

29. The z score and the z distribution are the same thing.

 

 

 

 

Short Answer

 

1. Define asymptotic.

 

 

 

2. Define normal curve.

 

3. Define standard (z) scores.

 

 

 

4. List three characteristics of a z distribution.

 

 

 

5. Is a z distribution always normally shaped?  Explain.

 

6. Does the z transformation result in a score having the same units of measurement as the raw score?  Explain.  Why is this advantageous?

 

 

 

7. Are all bell-shaped curves normal curves?  Explain.

 

8. What is meant by a transformed score?  Give an example.

 

9. If a score is at the mean of a set of raw scores, where will it be if the set of raw scores is transformed to z scores?  Why?

 

For problems 10 through 17 use the following information:

 

In a population survey of patients in a rehabilitation hospital, the mean length of stay in the hospital was 12.0 weeks with a standard deviation equal to 1.0 week.  The distribution was normally distributed.

 

10. Out of 100 patients how many would you expect to stay longer than 13 weeks?

 

 

 

11. What is the percentile rank of a stay of 11.3 weeks?

 

 

 

12. What percentage of patients would you expect to stay between 11.5 weeks and 13.0 weeks?

 

 

 

13. What percentage of patients would you expect to be in longer than 12.0 weeks?

 

 

 

14. How many times out of 10,000 would you expect a patient selected at random to remain in the hospital longer than 14.6 weeks?

 

 

 

15. What proportion of patients are likely to be in less than 9.7 weeks?

 

 

 

16. What is the length of stay at the 90th percentile?

 

 

 

17. What is the length of stay at the 50th percentile?

 

 

 

18. On one college aptitude test with a mean of µ = 100 and a standard deviation of s = 16, a student achieved a score of 124. The same student took a different test which had a mean of µ = 50 and a standard deviation of s = 10.  On the second test the student achieved a score of 65.  On which test did the student do better?

 

 

 

19. If the mean height of college males is 70 inches with a standard deviation of 3 inches, what percentage of college males would be between 6′ and 6’4″? Assume a normal distribution.

 

 

 

20. Using the information in problem 19, what height would someone have to be in order to be in the 99th percentile?

 

 

 

21. Using the data in problem 19, what is the height below which the shortest 2.5% of the college males fall and what is the height above which the tallest 2.5% fall?

 

 

 

 

22. A surgeon is experimenting with a new technique for implanting artificial blood vessels. Using this technique with a great many operations, the mean time before clotting of an artificial blood vessel has been 32.5 days with a standard deviation of 2.6 days.  The following data were obtained on four operations.

 

 

 

 

1. What are the z scores for the four operations?
2. What is the percentile rank for each of the four operations?  Assume a normal distribution.
3. How long would a vessel have to stay open to be in the 95th percentile?  Assume a normal distribution.

 

 

 

23. Given the following z scores, find the area below z:
24. 1.68
25. -0.45
26. -1.96
27. -0.52
28. 2.58

 

 

 

24. Assuming that you wished to have the highest possible score on an exam relative to the other scores; would you rather have a score of 70 on a test with a mean of 60 and a standard deviation of 5.2 or a score of 81 on a test with a mean of 70 and a standard deviation of 7.1?

 

 

 

 

25. What is the percentile rank for each of the following z scores? Assume a normal distribution.
26. 1.23
27. 0.89
28. -0.46
29. -1.00

 

 

 

26. What z scores correspond to the following percentile ranks? Assume the scores are normally distributed.
27. 50
28. 46
29. 96
30. 75
31. 34
32.   4