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INSTANT DOWNLOAD WITH ANSWERS
UNDERSTANDING STATISTICS IN THE BEHAVIORAL SCIENCES 10TH EDITION BY PAGANO – TEST BANK
CHAPTER 5
The Normal Curve and
Standard Scores
LEARNING OBJECTIVES
After completing Chapter 5, students should be able to:
- Describe the typical characteristics of a normal curve.
- Define a z score.
- Compute the z score for a raw score, given the raw score, the mean and standard deviation of the distribution.
- Compute the z score for a raw score, given the raw score and the distribution of raw scores.
- Explain the three main features of z distributions.
- 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.
- Understand the illustrative examples, do the practice problems and understand the solutions.
DETAILED CHAPTER SUMMARY
- The Normal Curve
- Important in behavioral sciences.
- Many variables of interest are approximately normally distributed.
- Statistical inference tests have sampling distributions which become normally distributed as sample size increases.
- Many statistical inference tests require sampling distributions that are normally distributed.
- Characteristics.
- Symmetrical, bell-shaped curve.
- 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.
- Area contained under the normal curve:
- 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.
- Since the curve is symmetrical, the same percentages hold for scores below the mean.
- Standard Scores (z Scores)
- Symbol. Symbolized as z.
- 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
- Comparisons between different distributions.
- Allows comparisons even when the units of the distributions are different.
- Percentile ranks are possible.
- Characteristics of z scores.
- z scores have the same shape as the set of raw scores from which they were transformed.
- µz = 0. The mean of z scores equals zero.
- sz = 1.00. The standard deviation of z scores equals 1.00.
- Using z scores.
- 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.
- 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
- 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.
- 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?
- 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.
- 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.
- Why might anyone be interested in comparing two or more sets of scores? Illustrate, giving examples.
- Discuss the advantages and disadvantages of using SPSS to do your homework problems.
TEST QUESTIONS
Multiple Choice
- 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 _________.
- 48.78%
- 1.22%
- 98.78%
- 51.22%
- 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?
- 21.48%
- 78.41%
- 49.41%
- 57.98%
- 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.
- 17.72%
- 32.28%
- 82.28%
- 31.92%
- 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 _________.
- $ 4.60
- $12.47
- $ 4.57
- $12.44
- 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 _________.
- 15.31%
- 31.17%
- 23.24%
- 7.93%
- 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 _________.
- 3.14%
- 96.78%
- 3.22%
- 96.86%
- 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 _________.
- 0.4535
- 0.9535
- 0.0475
- 0.0485
- 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 _________.
- 0.4394
- 0.5365
- 0.4406
- 0.4635
- 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 _________.
- 615.20
- 384.80
- 616.10
- 383.90
- 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 _________.
- 0.3665
- 0.4868
- 0.8533
- 0.1203
- The standard deviation of the z distribution equals _________.
- 1
- 0
- S X
- N
- The mean of the z distribution equals _________.
- 1
- 0
- S X
- N
- The z score corresponding to the mean of a raw score distribution equals _________.
- the mean of the raw scores
- 0
- 1
- N
- The normal curve is _________.
- linear
- rectangular
- bell-shaped
- skewed
- In a normal curve, the inflection points occur at _________.
- µ ± 1s
- ±1s
- µ ± 2s
- µ
- The z score corresponding to a raw score of 120 is _________.
- 1.2
- 2.0
- 1.0
- impossible to compute from the information given
- 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 _________.
- 12.00
- 10.00
- 9.00
- 8.00
- 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 _________.
- 10.20
- 2.10
- 2.11
- 10.74
- 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
- 0
- 10
- 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 _________.
- 0.47
- 9.00
- 4.27
- -0.47
- 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 _________.
- -1.42
- 13.00
- 1.42
- 6.16
- 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 _________.
- 64.3
- 14.0
- 4.3
- 60.0
- 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 _________.
- 53.5
- 66.5
- 66.4
- 53.6
- 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 _________.
- 51.3
- 68.7
- 51.4
- 68.6
- 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 _________.
- 35.13
- 45.99
- 56.76 and 33.24
- 45.99 and 35.13
- If a distribution of raw scores is negatively skewed, transforming the raw scores into z scores will result in a _________ distribution.
- normal
- bell-shaped
- positively skewed
- negatively skewed
- The mean of the z distribution equals _________.
- 0
- 1
- N
- depends on the raw scores
- The standard deviation of the z distribution equals _________.
- 0
- 1
- the variance of the z distribution
- b and c
- S(z – µz) equals _________.
- 0
- 1
- the variance
- cannot be determined
- The proportion of scores less than z = 0.00 is _________.
- 0.00
- 0.50
- 1.00
- -0.50
- In a normal distribution the z score for the mean equals _________.
- 0
- the z score for the median
- the z score for the mode
- all of the above
- In a normal distribution approximately _________ of the scores will fall within 1 standard deviation of the mean.
- 14%
- 95%
- 70%
- 83%
- Would you rather have an income (assume a normal distribution and you are greedy) _________.
- with a z score of 1.96
- in the 95th percentile
- with a z score of -2.00
- with a z score of 0.000
- How much would your income be if its z score value was 2.58?
- $10,000
- $ 9,999
- $ 5,000
- cannot be determined from information given
- Which of the following z scores represent(s) the most extreme value in a distribution of scores assuming they are normally distributed?
- 1.96
- 0.0001
- -0.0002
- -3.12
- Assuming the z scores are normally distributed, what is the percentile rank of a z score of -0.47?
- 31.92
- 18.08
- 50.00
- 47.00
- 0.06
- 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.
- 90.00
- 112.00
- 103.36
- 91.00
- 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?
- 58.30
- 0.00
- 25.27
- 38.30
- 11.70
- 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?
- 55.00
- 79.12
- 20.88
- 29.12
- 70.88
- 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?
- 75.66
- 70 23
- 24.34
- 23.57
- 12.17
- 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
- be greater than 1.00
- must be greater than 0.00
- have a percentile rank greater than 50%
- can’t determine with information given
- b and c.
- 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 _________.
- neither, your skills are below average in both
- the career requiring good math skills.
- neither, this approach is bogus; dream interpretation should be used instead.
- the career requiring good writing skills.
- 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 _________.
- ignore the advice because your friend flunked her last statistics test
- ignore the advice because z distributions have the same shape as the raw scores.
- take the advice because z distributions are always normally distributed
- take the advice because z distributions are usually normally distributed
- All bell-shaped curves _________.
- are normal curves
- have means = 0
- are symmetrical
- a and c
- If you transformed a set of raw scores, and then added 15 to each z score, the resulting scores _________.
- would have a standard deviation = 1
- would have a mean = 0
- would have a mean =15
- would have a standard deviation > 1
- a and c
- A set of raw scores has a rectangular shape. The z transformed scores for this set of raw scores has a _________ shape.
- rectangular
- normal (bell-shaped)
- it depends on the number of scores in the distribution
- none of the above
- 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?
- Makaela.
- Juan.
- Neither, they both did as well as each other.
- Makaela, because her exam was harder
- a and d.
- Table A in your textbook has no negative z values, this means _________.
- the table can only be used with positive z values
- the table can be used with both positive and negative z values because it is symmetrical
- the table can be used with both positive and negative z values because it is skewed
- none of the above
- 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, _________.
- do a z transformation for each raw score and add 10 to each z score.
- do a z transformation for each raw score and multiply each by 10
- divide the raw scores by 10
- 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.
- a and d
- 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?
- -0.18
- -0.48
- -0.15
- -0.52
True/False
- A z distribution always is normally shaped.
- All standard scores are z scores.
- A z score is a transformed score.
- A z score designates how many standard deviations the raw score is above or below the mean.
- The z distribution takes on the same shape as the raw scores.
- z scores allow comparison of variables that are measured on different scales.
- 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.
- The normal curve reaches the horizontal axis in 4 standard deviations above and below the mean.
- For any z distribution of normally distributed scores, P50 is always equal to zero.
- 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.
- It is impossible to have a z score of 30.2.
- The area under the normal curve represents the proportion of scores that are contained in the area.
- 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.
- The normal curve is a symmetric, bell-shaped curve.
- The area under the normal curve represents the percentage of scores contained within the area.
- It is impossible to have a z score of 23.5.
- A z transformation will allow comparisons to be made when units of distributions are different.
- 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.
- In the standard normal curve, 13.59% of the scores will always be contained between the mean (µ) and +1s.
- In a plot of the normal curve, frequency is plotted on the X axis.
- The standard deviation of the z distribution is always equal to 1.0.
- The area beyond a z score of +2.58 is 0.005.
- One cannot reasonably do z transformations on ratio data.
- The normal curve never touches the X axis.
- To do a z transformation, one must know only the population mean and the value of the raw score to be transformed.
- To calculate the score at the 97.5th percentile, one would apply the formula X = µ + (s)(1.96).
- The z score and the z distribution are the same thing.
Short Answer
- Define asymptotic.
- Define normal curve.
- Define standard (z) scores.
- List three characteristics of a z distribution.
- Is a z distribution always normally shaped? Explain.
- Does the z transformation result in a score having the same units of measurement as the raw score? Explain. Why is this advantageous?
- Are all bell-shaped curves normal curves? Explain.
- What is meant by a transformed score? Give an example.
- 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.
- Out of 100 patients how many would you expect to stay longer than 13 weeks?
- What is the percentile rank of a stay of 11.3 weeks?
- What percentage of patients would you expect to stay between 11.5 weeks and 13.0 weeks?
- What percentage of patients would you expect to be in longer than 12.0 weeks?
- 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?
- What proportion of patients are likely to be in less than 9.7 weeks?
- What is the length of stay at the 90th percentile?
- What is the length of stay at the 50th percentile?
- 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?
- 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.
- Using the information in problem 19, what height would someone have to be in order to be in the 99th percentile?
- 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?
- 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.
- What are the z scores for the four operations?
- What is the percentile rank for each of the four operations? Assume a normal distribution.
- How long would a vessel have to stay open to be in the 95th percentile? Assume a normal distribution.
- Given the following z scores, find the area below z:
- 1.68
- -0.45
- -1.96
- -0.52
- 2.58
- 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?
- What is the percentile rank for each of the following z scores? Assume a normal distribution.
- 1.23
- 0.89
- -0.46
- -1.00
- What z scores correspond to the following percentile ranks? Assume the scores are normally distributed.
- 50
- 46
- 96
- 75
- 34
- 4
CHAPTER 17
Chi-Square and
Nonparametric Tests
LEARNING OBJECTIVES
After completing Chapter 17, students should be able to:
- Specify the distinction between parametric and nonparametric tests, when to use each, and give an example of each.
- Specify the level of variable scaling that Chi-square requires for its use; understand that chi-square uses sample frequencies and predicts to population proportions.
- Define a contingency table; specify the H_{1} and H_{0} for chi-square analyses.
- Understand that chi-square basically computes the difference between f_{e} and f_{o}, and the larger this difference, the more likely we can reject H_{0}.
- Solve problems using chi-square, and specify the assumptions underlying this test.
The following objectives apply to the Wilcoxon matched-pairs signed ranks test, the Mann-Whitney U test, and the Kruskal-Wallis test.
- Specify the parametric test that each substitutes for; solve problems using each test; and specify the assumptions underlying each test;
- Rank order the sign test, the Wilcoxon match-pairs signed ranks test and the t test for correlated groups with regard to power.
- Understand the illustrative examples, do the practice problems and understand the solutions.
DETAILED CHAPTER SUMMARY
- Distinctions Between Parametric and Nonparametric Tests
- Parametric tests. Parametric tests (e.g., t, z, F) depend substantially on population characteristics or parameters for their use.
- Nonparametric tests. Nonparametric tests (e.g., sign test) depend minimally on population characteristics.
- Distribution free tests. Whereas parametric tests may require that samples be random from normally distributed populations, nonparametrics require that samples be random from populations with the same distributions, hence the term distribution free tests.
- Advantages for parametric tests.
- Parametric tests are generally more powerful and versatile.
- Parametric tests are generally robust to violations of the test assumptions.
- Examples of nonparametric tests.
- Sign test
- Mann-Whitney U test
- Chi-square test
- Wilcoxon matched-pairs signed ranks test
- Kruskal-Wallis test
- Chi-Square (c2) Single Variable Experiments
- Use. Often used with nominal data.
- What is tested. Tests if the observed results differ significantly from the results expected if H_{0} were true.
- Computational formula.
where fo = the observed frequency in the cell
fe = the expected frequency in the cell (if H_{0} were true)
S = summation over all cells
- Evaluation of c2obt.
- Family of curves
- Vary with df
- Lower df curves are positively skewed
- k – 1 degrees of freedom where k equals the number of groups or categories
- The larger the discrepancy between the observed and expected results the larger the value of c2obt and therefore the more unreasonable that H_{0} is true.
- If c2obt ³ c2crit, reject H_{0}
III. Chi-square: Test of Independence Between Two Variables
- Use. Used to determine whether two variables are related.
- Contingency table. This is a two-way table showing the contingency between two variables where the variables have been classified into mutually exclusive categories and the cell entries are frequencies.
- Null Hypothesis. Null hypothesis states that the observed frequencies are due to random sampling from a population in which the proportions in each category of one variable are the same for each category of the variable.
- Alternative Hypothesis. Alternative hypothesis. Alternative hypothesis is that these proportions are different.
- Calculation of c2 for contingency tables.
- fe can be found by multiplying the marginals (i.e. row and column totals lying outside the table) and dividing by N.
- Sum (fo – fe)2/fe for each cell.
- Evaluation of c2obt.
- Degrees of freedom for experiments involving the contingency between two variables are equal to the number of fo scores that are free to vary while at the same time keeping the column and row marginals the same. In equation form:
df = (r – 1)(c – 1)
where r = number of rows in the contingency table
c = number of columns in the contingency table
- If c2obt ³ c2crit, reject H_{0}
- Assumptions underlying c2_{.}
- Independence exists between each observation in the contingency table.
- Sample size is large enough so that the expected frequency in each cell is at least 5 for tables where r or c is greater than 2.
- If table is 1 x 2 or 2 x 2 then each expected frequency should be at least 10.
- c2 can be used with any type of scaling if the data are reduced to mutually exclusive categories and frequency entries.
- Wilcoxon Matched-Pairs Signed Ranks Test
- Use.
- Used in correlated groups designs with data that are at least of ordinal scaling.
- Used when assumptions of t test for correlated groups are seriously violated.
- Power. Relatively powerful. More powerful than sign test, less powerful than t test.
- Data. Considers both magnitude and direction of the rank order of the difference scores.
- Alternative Hypothesis. Alternative hypothesis stated with no population parameters; e.g. independent variable affects dependent variable.
- Null Hypothesis. Null hypothesis stated with no population parameters; e.g. independent variable has no effect on dependent variable.
- Calculation of statistic Tobt.
- Calculate the difference between each pair of scores.
- Rank the absolute values of the difference scores from the smallest to the largest.
- Assign to the resulting ranks the sign of the difference score whose absolute value yielded that rank.
- Compute the sum of the ranks separately for the positive and negative signed ranks. The lower sum is Tobt.
- As a check, the sum of the unsigned ranks should equal n(n + 1)/2.
- If rows scores are tied such that the difference of the paired scores equals zero, then these scores are discarded and N reduced by one.
- If ties occur in the difference scores, the ranks are given a value equal to the mean of the tied ranks.
- Evaluation of Tobt.
- If Tobt < Tcrit, reject H_{0}. Tcrit depends on a and N.
- Assumptions of the signed ranks test.
- Raw scores must be of at least ordinal scaling.
- Difference scores must also be of at least ordinal scaling.
- Mann-Whitney U Test
- Use. Used in a two group, independent groups design as a substitute for the t test when its assumptions are seriously violated. Measures the degree of separation between the two sets of sample scores.
- Requirements. It is a nonparametric test that requires only ordinal scaling of the dependent variable. Does not require population normality.
- Analysis. Rank orders the scores, computes the sum of ranks for each group, and tests whether these sums are significantly different. Makes no prediction about population means.
- Calculation of U_{obt} or U′_{obt}. Computes the statistic U_{obt} or U′_{obt}. To calculate U_{obt} or U′_{obt}
- Combine all the scores and rank order them, beginning with 1 for the lowest score.
- Sum the ranks for each group.
- Substitute these values into the equations and compute U_{obt} and U′_{obt}. U_{obt} is always the smaller of the two results.
- Equations:
where n_{1} = Number of scores in group 1
n_{2} = Number of scores in group 2
R1 = sum of the ranks for group 1
R2 = sum of the ranks for group 2
- Evaluation of U_{obt}. Since U_{obt} and U′_{obt} give the same information regarding degree of separation, it is only necessary to evaluate one of them. The textbook always evaluates U_{obt}.
If U_{obt} £ U_{crit}, reject H_{0}
with U_{crit} found in Tables C.1-C.4 using a, n_{1} and n_{2}. U_{crit} is the upper of the two entries found in the appropriate cell of the appropriate table.
- Kruskal-Wallis Test
- Use. Used in independent groups design as a substitute for parametric ANOVA when its assumptions are seriously violated. Like parametric ANOVA, Kruskal-Wallis is a nondirectional test.
- Requirements. It is a nonparametric test which requires only ordinal scaling of the dependent variable. Does not require population normality.
- Analysis. Computes the sum of ranks for each group and tests whether these sums are significantly different. Makes no prediction about population means.
- Calculation of statistic Hobt. Statistic computed is Hobt. To compute Hobt
- Combine all the scores and rank order them, beginning with 1 for the lowest score.
- Sum the ranks for each group.
- Substitute these values into the equation and compute Hobt.
- Equation:
where R1 = sum of the ranks for sample 1
R2 = sum of the ranks for sample 2
R3 = sum of the ranks for sample 3
Rk = sum of the ranks for sample k
k = number of samples or groups
- Evaluation of Hobt.
If Hobt ³ Hcrit, reject H_{0}
with Hcrit found in Table H using df = k – 1.
TEACHING SUGGESTIONS AND COMMENTS
This chapter covers Chi-square and nonparametric tests. Nonparametric tests are easier to teach and easier for students to understand. Of all the nonparametric tests discussed in the chapter, Chi-square far and away is the most important and most frequently encountered in the research literature. I always lecture on the Chi-square material. I seldom lecture on any of the remaining tests because of time limitations. If time permits, I prefer to go next with the Mann-Whitney U test because it is more frequently used in research and is a powerful, alternate test to the t test of independent groups. Specific suggestions and comments follow.
- Introduction: distinction between parametric and nonparametric tests. This section sets the stage for the nonparametric tests that follow. It makes the points that nonparametric tests are used as substitutes for parametric tests when parametric tests can’t be used due to violations of assumptions, and that parametric tests are the tests of choice because they are more powerful. Nothing difficult here. The section works well and I recommend you follow it.
- Chi-square (). This is one of the most often used inference tests in social psychology. Students find this material very easy to understand and interesting because of the interesting examples that one can use.
- Single-variable experiments. There are two important concepts to understand. The first is that although the null hypothesis evaluates population proportions, the cell entries must be frequencies. The second is to understand that the greater the difference between f_{o} and f_{e} is, the more reasonable H_{1} becomes. This understanding is best developed in conjunction with the equation for . Computation of is easy and the decision rule is straight forward. Since f_{o} is given in the problem, the challenge is to determine f_{e} for each cell. It is worth making the point that always has a value that is positive because in the equation for computing it, the difference between f_{o} and f_{e} is squared (). It is also worth pointing out that chi-square is a nondirectional test because it doesn’t matter if f_{o} is smaller or larger than f_{e}, since the difference between the two is squared.
- Test of independence between two variables. In this section the use of chi-square to investigate whether two categorical variables are related or independent. Students need to learn the definition of contingency table and how to determine f_{e} for the cells in a contingency tables. Once this is understood, computation of is easy and straight forward. Explaining degrees of freedom is a little tricky, but the explanation given on p. 492 works well and therefore, I suggest you follow it. The assumption underlying chi-square that specifies the minimum value of f_{e} required in each cell is a little too detailed to require students remember it. All I require is that they know there is a required minimum value, and I add they can look it up if they ever need to for any research they engage in.
DISCUSSION QUESTIONS
- Is it true that parametric tests are generally more powerful than nonparametric tests? If so, give two reasons why do we might choose to use a nonparametric test instead of a parametric test.
- The section, “What Is The Truth-Statistics and Applied Social Research-Useful or Abuseful’?” p. 512, raises some important issues. After reading that section, please answer the following questions.
- Do you think it ethical if social scientists with strong political views go out deliberately and do research biasing their questionnaires so that data will confirm their political views? How do you justify your answer?
- If an organization conducts socially relevant research and the findings turn out to be against the interest of the organization, does the company have the moral obligation to inform the public? How do you justify for your answer?
- Do drug companies have an ethical responsibility to report the outcomes of experiments they fund involving their drugs when the outcomes show their product to be inferior or no better than competing drugs? How do you justify your answer?
- What is the relationship between (f_{o} – f_{e}) and the magnitude of real effect? Given this relationship, does the equation for make sense? Explain.
- How do you make sense of the term “contingency” as used in “contingency table” when testing for the independence between two variables with ?
- The Wilcoxon matched-pairs signed ranks test is a substitute for what parametric test? Compare the power of the Wilcoxon matched-pairs signed ranks test with that of the t test for correlated groups and the sign test. How can you explain the relative power of each test?
- The Mann-Whitney U test is a substitute for what parametric test? Compare the power of the Mann-Whitney U test with that of the t test for independent groups and explain the difference.
- The Kruskal-Wallis test is a substitute for what parametric test? Compare the power of the Kruskal-Wallis test with the one-way parametric ANOVA and explain the difference.
TEST QUESTIONS
Multiple Choice
- Chi-square is used to test differences between _________.
- proportions
- means
- variances
- none of the above
- The larger the discrepancy between foand fe for each cell, _________.
- the more likely the results will not be significant
- the more likely H0 will be rejected
- the more likely the population proportions are the same
- the more likely the population proportions are different
- a and c
- b and d
- For any given alpha level, c2crit_________.
- increases with increases in N
- decreases with increases in N
- increases with increases in degrees of freedom
- a and c
- Chi-square should not be used if _________.
- df = 1
- fe is below 5
- fo is below 5
- fe = fo
- Chi-square may be used with _________
- nominal data
- ordinal data
- interval data
- ratio data
- all of the above
- To compute c2, the entries in the contingency table should be _________.
- frequencies
- means
- variances
- degrees of freedom
- The degrees of freedom for a contingency table equal _________.
- rc – 1
- (r – 1)(c – 1)
- (r – 1)(c)
- (c – 1)(r)
- N – 1
- In most situations, parametric tests _________.
- have the same power as nonparametric tests
- are less powerful than nonparametric tests
- are more powerful than nonparametric tests
- are less sensitive than nonparametric tests
- b and d
- Which of the following are examples of parametric tests?
- t test
- sign test
- Mann-Whitney U test
- Chi-square test
- F test
- a and e
- Which of the following are examples of nonparametric tests?
- t test
- sign test
- Mann-Whitney U test
- Chi-square test
- F test
- b, c and d
- all of the above
- Which of the following are true?
- fo is the symbol for the observed frequency
- fe is the symbol for the expected frequency
- c2 is the symbol for chi-square
- all of the above
- The sampling distribution of chi-square is _________.
- skewed
- varies with df
- is a theoretical distribution
- all of the above
- a and b
- The c2test is _________.
- always directional
- never directional
- generally nondirectional
- generally directional
- When evaluating c2obt, the critical region for rejection of H0_________.
- lies under both tails of the distribution
- lies under the right hand tail of the distribution
- lies under the left hand tail of the distribution
- lies in the middle of the distribution
- A contingency table _________.
- is a two-way table
- involves two variables
- involves two mutually exclusive variables
- all of the above
- a and b
- The computation of fe_________.
- is based on population proportion estimates
- is based on known population proportions
- is based on population means
- none of the above
- In a 2 x 2 contingency table, if we keep the marginals at their observed values, how many foscores are free to vary?
- 3
- 0
- 1
- all of them
- The Wilcoxon signed ranks test _________.
- is used with a correlated groups design
- is used with data that is nominal in scaling
- uses both the magnitude and direction of the data
- all of the above
- a and b
- If N = 18 and a = 0.05_{2 tailed}, the value of T_{crit} is _________.
- 40
- ±40
- -40
- 47
- If a = 0.05, and df = 4, the value of c2crit= _________.
- 9.488
- 0.711
- 7.815
- 11.070
- Prior to a recent gubernatorial election, a survey was conducted to determine whether there was a relationship between sexual gender and preference for the Democratic or Republican candidate. The following data were recorded. Assume the data will be analyzed with Chi-square.
The value of c2obt = _________.
- 2.06
- 2.09
- 1.80
- 1.75
- Prior to a recent gubernatorial election, a survey was conducted to determine whether there was a relationship between sexual gender and preference for the Democratic or Republican candidate. The following data were recorded. Assume the data will be analyzed with Chi-square.
The value of df = _________.
- 2
- 1
- 3
- need more information
- Prior to a recent gubernatorial election, a survey was conducted to determine whether there was a relationship between sexual gender and preference for the Democratic or Republican candidate. The following data were recorded. Assume the data will be analyzed with Chi-square.
Using a = 0.05, c2crit = _________.
- 3.841
- 5.412
- 2.706
- -3.841
- Prior to a recent gubernatorial election, a survey was conducted to determine whether there was a relationship between sexual gender and preference for the Democratic or Republican candidate. The following data were recorded. Assume the data will be analyzed with Chi-square.
Using a = 0.05, what is your conclusion?
- accept H0; there is no relationship between sex and candidate preference
- reject H0; there is a significant relationship between sex and candidate preference
- retain H0; the study does not show a significant relationship between sex and candidate preference
- retain H0; this study shows a significant relationship between sex and candidate preference
- A study is conducted to determine whether sunshine affects depression. Eight individuals are given a questionnaire measuring depression immediately following a run of 10 consecutive days when the sun shone for over 80% of the daylight hours. The same individuals have their depression measured immediately following 10 consecutive days without any sunshine. The following data are collected. The higher the score the greater the depression.
Individuals | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Sunshine | 10 | 12 | 14 | 11 | 12 | 10 | 14 | 15 |
No sunshine | 20 | 21 | 17 | 14 | 18 | 8 | 18 | 14 |
Using the Wilcoxon signed ranks test to evaluate the data, the value of Tobt is _________.
- -3
- 3
- 33
- 4
- A study is conducted to determine whether sunshine affects depression. Eight individuals are given a questionnaire measuring depression immediately following a run of 10 consecutive days when the sun shone for over 80% of the daylight hours. The same individuals have their depression measured immediately following 10 consecutive days without any sunshine. The following data are collected. The higher the score the greater the depression.
Individuals | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Sunshine | 10 | 12 | 14 | 11 | 12 | 10 | 14 | 15 |
No sunshine | 20 | 21 | 17 | 14 | 18 | 8 | 18 | 14 |
Using the Wilcoxon signed ranks test to evaluate the data, with a = 0.052 tail, Tcrit = _________.
- 2
- 5
- ±3
- 3
- A study is conducted to determine whether sunshine affects depression. Eight individuals are given a questionnaire measuring depression immediately following a run of 10 consecutive days when the sun shone for over 80% of the daylight hours. The same individuals have their depression measured immediately following 10 consecutive days without any sunshine. The following data are collected. The higher the score the greater the depression.
Individuals | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Sunshine | 10 | 12 | 14 | 11 | 12 | 10 | 14 | 15 |
No sunshine | 20 | 21 | 17 | 14 | 18 | 8 | 18 | 14 |
Using the Wilcoxon signed ranks test to evaluate the data with a = 0.052 tail, what do you conclude? Assume for the purpose of this question, that sunshine was the only systematic difference between the conditions.
- reject H0; sunshine appears to affect depression
- reject H0 ; sunshine has no effect on depression
- retain H0; we cannot conclude that sunshine affects depression
- accept H0; sunshine affects depression
- Which of the following are nonparametric tests?
- sign test
- Wilcoxon test
- Mann-Whitney U test
- Kruskal-Wallis test
- all of the above
- a, b and c
- The Mann-Whitney U test can be used with _________.
- nominal data
- interval data
- ordinal data
- ratio data
- all of the above
- b, c and d
- Generally, if R1is greater than R2, _________.
- the equation yields the U value
- the equation yields the U value
- the power of the test is necessarily low
- we must reject
- As the separation between the two groups of scores increases, Uobt_________.
- increases
- decreases
- stays the same
- approaches U‘obt
- In the Mann-Whitney U test, the U value of 0 _________.
- represents a low degree of separation between the two groups
- implies that the two groups are identical
- represents the greatest degree of separation between the two groups
- indicates that the power of the experiment is very low.
- The statistics (U or U’) used in the Mann-Whitney U test, measure _________.
- the differences between the means of the two groups
- the direction of the differences between pairs of scores
- the power of the experiment
- the separation between the two groups
- A Mann-Whitney U value of zero indicates _________.
- you can reject the null hypothesis
- low degree of separation between the scores of each group
- zero difference between the two groups
- high degree of separation between the scores of each group
- U + U’ equals _________.
- n
- n1 + n2
- n1 x n2
- none of the above
- The Mann-Whitney U test is used with _________.
- an independent groups design
- a replicated measures design
- a correlated groups design
- b and c
- The Mann-Whitney U test uses _________.
- only the direction of the scores
- only the magnitude of the scores
- both the magnitude and direction of the scores
- none of the above
- Consider the following set of scores: 25, 28, 28, 28, 30. In assigning ranks to the scores, a score of 28 receives a rank of _________.
- 2.5
- 2
- 4
- 3
- Using the data in question 38, a score of 30 would receive a rank of _________.
- 4
- 5
- 3
- 0
- If n1= 5, n2 = 4, and a = 0.012 tail, the power of the Mann-Whitney U test equals _________.
- 1
- 0
- 0.01
- need more information
- If n1= 6, n2 = 8 and a = 0.052 tail, Ucrit value is _________.
- 8
- 40
- 6
- 42
- A student at a Midwest college is interested in whether Psychology majors spend more or less time studying than English majors. She randomly selects 8 Psychology majors and 8 English majors and determines their weekly studying time. The following are the scores. Note one person dropped out of the study.
Psychology Majors | 16 | 12 | 13 | 10 | 9 | 10 | 8 | |
English Majors | 10 | 25 | 15 | 17 | 23 | 14 | 19 | 18 |
An analysis is being conducted using the Mann-Whitney U test. The value of Uobt is _________.
- 6
- 50
- 14
- 42
- A student at a Midwest college is interested in whether Psychology majors spend more or less time studying than English majors. She randomly selects 8 Psychology majors and 8 English majors and determines their weekly studying time. The following are the scores. Note one person dropped out of the study.
Psychology Majors | 16 | 12 | 13 | 10 | 9 | 10 | 8 | |
English Majors | 10 | 25 | 15 | 17 | 23 | 14 | 19 | 18 |
An analysis is being conducted using the Mann-Whitney U test. If a = 0.052 tail, Ucrit = _________.
- 7
- 10
- 46
- 49
- A student at a Midwest college is interested in whether Psychology majors spend more or less time studying than English majors. She randomly selects 8 Psychology majors and 8 English majors and determines their weekly studying time. The following are the scores. Note one person dropped out of the study.
Psychology Majors | 16 | 12 | 13 | 10 | 9 | 10 | 8 | |
English Majors | 10 | 25 | 15 | 17 | 23 | 14 | 19 | 18 |
An analysis is being conducted using the Mann-Whitney U test. Using a = 0.052 tail, what do you conclude?
- reject H0; there is a significant difference in the amount of time Psychology and English majors study
- reject H0; there is no difference in the amount of time Psychology and English majors study
- accept H0; there is no difference in the amount of time Psychology and English majors study
- retain H0; there is no difference in the amount of time Psychology and English majors study
- The statistic used with the Kruskal-Wallis test is _________.
- Fobt
- Hobt
- Uobt
- tobt
- A researcher conducts a one-way ANOVA involving three groups. When she analyzes the data she realizes the data seriously violate the assumptions underlying parametric ANOVA. Therefore, she decides to use the Kruskal-Wallis test to conclude with regard to H0. The data are given below.
(1) | (2) | (3) |
6
10 12 16 14 7 |
8
12 9 15 17 11 |
12
13 20 18 22 15 |
Hobt = _________.
- 4.25
- 5.43
- 2.68
- 6.86
- A researcher conducts a one-way ANOVA involving three groups. When she analyzes the data she realizes the data seriously violate the assumptions underlying parametric ANOVA. Therefore, she decides to use the Kruskal-Wallis test to conclude with regard to H0. The data are given below.
(1) | (2) | (3) |
6
10 12 16 14 7 |
8
12 9 15 17 11 |
12
13 20 18 22 15 |
Using a = 0.05, Hcrit = _________.
- 9.210
- 7.815
- 5.991
- 7.824
- A researcher conducts a one-way ANOVA involving three groups. When she analyzes the data she realizes the data seriously violate the assumptions underlying parametric ANOVA. Therefore, she decides to use the Kruskal-Wallis test to conclude with regard to H0. The data are given below.
(1) | (2) | (3) |
6
10 12 16 14 7 |
8
12 9 15 17 11 |
12
13 20 18 22 15 |
What is the appropriate conclusion concerning H0?
- reject H0; the independent variable has a real effect
- accept H0; the independent variable has no effect
- retain H0; the independent variable has a real effect
- retain H0; we cannot conclude the independent variable has a real effect
- The c2test can be used for variables with _________ scaling as long as the categories are mutually exclusive.
- nominal
- ordinal
- interval
- ratio
- all the above
- The _________ test is the most powerful test for a repeated measures design.
- sign
- t
- Wilcoxin signed ranks
- all the tests are equally powerful
- Which of the following tests are parametric statistical tests:
- sign test
- chi-square test
- Wilcoxin signed ranks test
- none of the above
- If an experiment using frequency data tested the preference for 6 brands of soup, there would be _________ degrees of freedom.
- 1
- N – 1
- 5
- 6
- The value of c2obtfor the table below is _________. (Assume equal probabilities for fe in each cell.)
- 96.00
- 9.42
- 8.13
- 13.28
- The value of c2critfor the data in Question 53 with a = 0.01 is _________.
- 6.635
- 15.086
- 13.277
- 11.668
- The conclusion for the data in Question 53 is _________.
- reject H_{0}
- reject H_{1}
- retain H_{0}
- retain H_{1}
- The value of fefor the cell in row W, column A is _________.
- 24.9
- 3.2
- 16.0
- 63.0
- The value of c2obtfor the table in question 56 is _________.
- 19.01
- 21.38
- 24.87
- 16.82
- The value of c2critfor the table in question 56 with a = 0.05 is _________.
- 13.277
- 7.779
- 3.841
- 9.488
- The conclusion for the data in question 56 is _________.
- reject H_{0}
- reject H_{1}
- retain H_{0}
- fail to accept H_{1}
- For the following table there is(are) _________ degrees of freedom.
- k – 1
- 1
- 2
- 4
- For the following table there is(are) _________ degrees of freedom.
The value of c2obt for the table is _________.
- 47.43
- 17.96
- 4.07
- 9.71
- For a low value of df the c2distribution is _________.
- normally distributed
- positively skewed
- negatively skewed
- none of the above
- The value of Tobtfor the following data is _________.
- 21
- -6
- 15
- 6
- If there are 16 subjects in a repeated measures design then the sum of the unsigned ranks equals _________.
- 136
- 68
- 272
- 32
- If Tobt= 12 and Tcrit = 10, one would _________.
- reject H_{0}
- retain H_{0}
- accept H_{0}
- reject H_{1}
- The statistics used for the Mann-Whitney U test measure _________.
- the mean differences between the two groups
- the direction of the differences between pairs of scores
- the power of the experiment
- the separation between the two sets of scores
- Consider the following set of scores: 81, 83, 84, 84, 87. What rank would you give to a score of 84?
- 3
- 3.5
- 4
- 4.5
- To answer this question, refer to the following data. Assume the data are being analyzed with the Mann-Whitney U test.
The value of U_{obt} (not U′_{obt}) is _________.
- 14.5
- 20.5
- 21.5
- 27.5
- To answer this question, refer to the following data. Assume the data are being analyzed with the Mann-Whitney U test.
The value of U_{crit} using a = 0.05_{2 tail} is _{ }_________.
- 36
- 5
- 6
- 30
- To answer this question, refer to the following data. Assume the data are being analyzed with the Mann-Whitney U test.
The conclusion using a = 0.05_{2 tail} is _________.
- reject H_{0}
- reject H_{1}
- retain H_{0}
- retain H_{1}
- In Chapter 16, we presented the data from an independent groups design and asked if it was appropriate to use parametric ANOVA. The data are presented again below. The correct answer was that it was not appropriate to use parametric ANOVA because of unequal n’s and homogeneity of variance assumption violation.
Is it possible to analyze the data with an alternate test?
- yes
- no
- If your answer to question 71 is yes, what is the name of the test?
- t test for independent groups
- F test
- Kruskal-Wallis
- Mann-Whitney U test
- In Chapter 16, we presented the data from an independent groups design and asked if it was appropriate to use parametric ANOVA. The data are presented again here. The correct answer was that it was not appropriate to use parametric ANOVA because of unequal n’s and homogeneity of variance assumption violation. Instead, analyze the data with the Kruskal-Wallis test.
Hobt = _________ .
- 10.25
- 10.63
- 15.96
- 5.96
- In Chapter 16, we presented the data from an independent groups design and asked if it was appropriate to use parametric ANOVA. The data are presented again below. The correct answer was that it was not appropriate to use parametric ANOVA because of unequal n’s and homogeneity of variance assumption violation. Instead, analyze the data with the Kruskal-Wallis test.
What are the df?
- 1
- 2
- 3
- need more information
- In Chapter 16, we presented the data from an independent groups design and asked if it was appropriate to use parametric ANOVA. The data are presented again below. The correct answer was that it was not appropriate to use parametric ANOVA because of unequal n’s and homogeneity of variance assumption violation. Instead, analyze the data with the Kruskal-Wallis test.
Using a = 0.05, Hcrit = _________.
- 3.841
- 7.815
- 5.991
- 7.824
- In Chapter 16, we presented the data from an independent groups design and asked if it was appropriate to use parametric ANOVA. The data are presented again here. The correct answer was that it was not appropriate to use parametric ANOVA because of unequal n’s and homogeneity of variance assumption violation. Instead, analyze the data with the Kruskal-Wallis test.
What do you conclude, using a = 0.05?
- retain H_{0}. There is no difference in the populations.
- accept H_{0}. There is no difference in the populations.
- reject H_{0}. At least one of the population means differs from at least one of the s.
- reject H_{0}. At least one of the distributions differs from at least one of the s.
True/False
- All inference tests depend on population characteristics.
- Parametric tests depend less on population characteristics than nonparametric tests.
- Parametric tests are more versatile than nonparametric tests.
- c2obt cannot be negative.
- To find fe for any cell, multiply the marginals for that cell and divide by N.
- The Wilcoxin signed ranks test is less powerful than the sign test.
- To use the Wilcoxin signed ranks test, the difference scores must be at least of ordinal scaling.
- An assumption of c2 is that the scores in each cell are independent. (
- The Wilcoxin signed ranks test is more powerful than the t test for correlated groups.
- Using c2, the closer the observed frequency of each cell is to the expected frequency for that cell, the higher the probability of rejecting H0.
- In order to reject the null hypothesis, c2obt³ c2crit.
- The c2 distribution is a family of curves that vary with degrees of freedom.
- The c2test answers questions about population proportions.
- For valid use of chi-square, each subject can only have one entry in the table, and the table entries must be frequencies.
- Parametric tests are always more desirable than nonparametric tests.
- The Mann-Whitney U test makes no assumption about the shape of the population scores.
- The Mann-Whitney U test is used with a repeated measures design.
- If Uobtand U’obt are equal, there is little overlap between the groups.
- U = 0 is the lowest possible U value.
- U = 0 indicates the greatest degree of separation between the groups.
- Generally, U’obt< Uobt
- The Mann-Whitney U test can only be used to analyze directional alternative hypotheses.
- The Mann-Whitney U test analyzes the separation between the groups.
- The data must be at least interval in scaling to use the Mann-Whitney U test.
- The Mann-Whitney U test uses both the magnitude and direction of the scores.
- Uobtand U’obt yield the same information with regard to the degree of separation.
- The Kruskal-Wallis test is used as a substitute for parametric one-way ANOVA.
- The Kruskal-Wallis test assumes population normality.
- The Kruskal-Wallis test requires only ordinal scaling of the dependent variable.
- When using the Kruskal-Wallis test, tied scores between conditions are thrown out.
- The Kruskal-Wallis test requires there are at least 5 scores in each sample.
- Nonparametric tests are generally more powerful than parametric tests.
- Anytime it is appropriate to use a nonparametric statistic it is appropriate to use a parametric statistic.
- A c2test can only be applied to nominally scaled variables.
- In general, nonparametric tests have fewer requirements or assumptions about population characteristics than parametric tests do.
- As a general rule an investigator should use parametric tests whenever possible to help minimize the probability of making a Type II error.
- In a single variable c2experiment there are N – 1 degrees of freedom.
- c2is basically a measure of the overall discrepancy between fe and fo.
- In any specific case fo- fe should equal zero if H_{0} is true.
- To use the c2test, the categories in the contingency table must always be mutually exclusive.
- The value of focan be found by multiplying the marginals for that cell, and dividing by N.
- If c2is negative then H_{0} must be false.
- In a 2 x 2 table there are (r – 1)(c – 1) degrees of freedom.
- The Kruskal-Wallis test is a nonparametric alternate for parametric one-way ANOVA, independent groups design.
- In general, the Kruskal-Wallis test is as powerful as parametric ANOVA.
- The theoretical sampling distribution of c2assumes that the distribution is discrete.
- In order to properly use the c2test each cell should have a value of fe equal to or greater than 10.
- The sampling distribution of c2is normally distributed.
- The Wilcoxin signed ranks test is used only with ordinal data.
- The sign test is less powerful than the Wilcoxin signed ranks test.
- Both the c2test and the Wilcoxin signed ranks test are one-tailed tests.
- If the ranking has been done correctly for the Wilcoxin signed ranks test, the sum of the unsigned ranks should equal n(n+1)/2.
- If Tobt³ Tcrit, reject H_{0}.
- When using the Wilcoxin signed ranks test, if the raw scores are tied, the scores are disregarded and N is reduced by 1.
- The proper use of the Wilcoxin signed ranks test requires that both the raw scores and the difference between the raw scores be of at least ordinal scaling.
- The Mann-Whitney U test tests the difference between sample means.
- In an independent groups experiment involving two groups, if chance alone were operating one would expect a great deal of overlap between the two sets of scores.
- To use the Mann-Whitney U test, n1must equal n2.
- Even though for a given experiment U_{obt} and U′_{obt} have different values, they still indicate the same degree of separation.
- The Kruskal-Wallis test is used with a correlated groups design.
- The Kruskal-Wallis test analyzes the difference between sample means.
- Both the Mann-Whitney U test and the Kruskal-Wallis test analyze differences between sums of ranks.
Short Answer
- Define chi-square ().
- Define contingency table.
- Define degree of separation.
- Define expected frequency (f_{e}).
- Define Kruskal-Wallis test.
- Define Mann-Whitney U test.
- Define marginals.
- Define observed frequency (f_{o}).
- Define Wilcoxon matched pairs signed ranks test.
- What distinguishes parametric from nonparametric tests? Give some examples.
- When might we use a nonparametric test? Give an example?
- What are the assumptions underlying the Kruskal-Wallis test?
- What are the assumptions underlying the chi-square test?
- What are the assumptions underlying the Mann-Whitney U test?
- What are the assumptions underlying the Wilcoxon signed ranks test?
- What is a contingency table?
- In analyzing the data from a two-way contingency table, involving variables A and B, what is the null hypothesis. Be specific using variables A and B, and the terms “proportions”, frequency, and independent.
- Identify the most sensitive, alternate nonparametric test for the following: t test for correlated groups, t test for independent groups, one-way, independent groups ANOVA.
- What variable does the Mann-Whitney U test measure to determine if the IV has had a real effect? What is the relationship between this variable and the real effect of the IV that makes this variable legitimate to use?
- The section, “What Is The Truth-Statistics and Applied Social Research-Useful or Abuseful’?” p. 512, raises some important issues. After reading that section, please answer the following questions.
- Do you think it ethical if social scientists with strong political views go out deliberately and do research biasing their questionnaires so that data will confirm their political views? How do you justify your answer?
- If an organization conducts socially relevant research and the findings turn out to be against the interest of the organization, does the company have the moral obligation to inform the public? How do you justify for your answer?
- Do drug companies have an ethical responsibility to report the outcomes of experiments they fund involving their drugs when the outcomes show their product to be inferior or no better than competing drugs? How do you justify your answer?
- Is it true that parametric tests are generally more powerful than nonparametric tests? If so, give two reasons why do we might choose to use a nonparametric test instead of a parametric test.
- A designer of electronic equipment wants to develop a calculator which will have market appeal to high school students. Past marketing surveys have shown that the color of the numeric display is important in terms of market preference. The designer makes up 210 sample calculators and then has a random sample of students from the area high schools rate which calculator they prefer. The calculators are identical except for the color of the display. The results of the survey were that 96 students preferred red, 82 preferred blue, and 32 preferred green.
- State H_{1} for this experiment.
- State H_{0} for this experiment.
- What is the value of c2obt
- What do you conclude using a = 0.01?
- One of the important assumptions underlying the use of parametric statistics is that the sample is randomly selected from a normally distributed population. Consider a sample of N = 500. A sample mean and standard deviation is calculated and we find that the following is true. Between the mean and -1s there are 150 scores. Between the mean and +1s there are 130 scores. Between -1s and -2s there are 70 scores. Between +1s and +2s there are 82 scores. Beyond +2 standard deviations are 30 scores. Beyond -2 standard deviations are 38 scores.
- If the population from which this sample was selected were normally distributed, what would the expected frequencies be for each cell in a sample of size 500?
- Using a = .01 what would you conclude about the population from which this sample was selected?
- A family therapist in a hospital wanted to know if patients with a terminal illness wanted to be informed of their true medical condition. The therapist also wondered if a person age had an effect on their attitude. Because of ethical constraints the therapist asked a healthy sample of subjects who were visitors to the hospital whether they would wish to be told if they had a terminal illness. The age of the respondents was also recorded. The results are shown in the table below.
- Draw a table with the values of fe in each of the appropriate cells.
- What is the value of c2obt
- What is the value of c2crit for a = 0.01?
- What do you conclude?
- A neuropsychologist wants to determine if people who have a dominant right cerebral hemisphere differ from people with a dominant left cerebral hemisphere in their choice of either music or reading as a preferred activity. He surveyed 127 subjects with the following results.
- What is the value of c2obt?
- What is the value of c2crit for a = 0.05?
- What do you conclude?
- What type error might one be making?
- A social scientist wants to know if education and socioeconomic status (SES) are independent. He collects the following data.
What do you conclude using a = 0.05?
- Consider the following table.
What is the appropriate statistical test to use to analyze this data if it were all nominal data.
- A group of pain researchers want to test the hypothesis that different religious groups have different pain complaints. The following data were collected from a review of the patient charts from a hospital pain clinic.
- State H_{1}.
- State H_{0}.
- What do you conclude using a = 0.01?
- Given the following data:
a Do you think hair color and eye color are independent (use a = 0.01)?
- What type of error might you be making?
- A psychologist wants to investigate whether there might be a relationship between birth complications and the development of schizophrenia. In a longitudinal study she gathers the following data.
- State H_{1}.
- State H_{0}.
- What do you conclude using a = 0.05?
- A new drug is supposed to be effective in reducing motion sickness in people who are prone to such illness. A group of subjects are given a placebo and taken for a ride in a car over a preplanned route. At the end of the trip the subjects are asked to rate their illness on a 20-point scale. A week later the same subjects are given the new drug and taken for an identical ride and asked to rate their degree of illness again. The data are:
- State H_{1}.
- State H_{0}.
- What is the value of Tobt?
- What do you conclude using a = 0.052 tail?
- A group of clients requesting marital therapy were given communication skills training and then rated by independent observers before and after therapy on their ability to resolve problems in a series of hypothetical conflict situations. The results are shown below. A higher score indicates better performance on the task.
- What is the value of Tobt?
- What do you conclude using a = 0.052 tail?
- A political advisor believes that his candidate should not spend time addressing groups of voters who have a low opinion of him. The advisor reasons if they have a low opinion the voter won’t change his mind anyway. To test this idea the advisor gets a group from an audience to rate the candidate before and after a speech. From this group he selects a sample of voters who initially rated the candidate poorly and then analyzes the effect of the speech. Here are the data. A higher rating indicates a higher opinion.
- What is the value of Tobt?
- What do you conclude using a = 0.051 tail?
- What type error might one be making?
- An animal geneticist is trying to pick an appropriate species of fish for repopulating a lake. He wants to compare how long certain types of fish live. Species A is used for a control group and Species B serves as the experimental group. A random sample of fish from both species is drawn and the following ages are recorded for life span in months.
- State the nondirectional alternative hypothesis.
- State the null hypothesis
- Analyze the data with a nonparametric test. What do you conclude, using a = 0.052 tail?
- Someone has told you that left-handed people have different spatial reasoning abilities than right-handed people. You are skeptical, so you decide to test the idea. You randomly select 15 people from your class and administer a spatial reasoning test to them. A higher score reflects better spatial reasoning. You obtain the following results.
- State the null hypothesis.
- State the alternative hypothesis.
- Assume the data do not allow analysis with the t test. Analyze the data with the most powerful alternative test. What is your conclusion using a = 0.052 tail?
- The following data were collected in an independent groups experiment to test the effect of different levels of a drug on blood pressure (mmHg). Assume the data seriously violate the assumptions underlying parametric ANOVA. Therefore, you will have to use an alternative test to analyze the data.
- What test will you use?
- What is your conclusion? Use a = 0.01
- In an independent groups experiment, four wines are rated by individuals according to taste preference. The resulting data are shown below. The rating scale is from 1 to 20, with 20 representing the highest possible score. Assume the data preclude use of parametric ANOVA because of assumption violations. Analyze these results using an alternative test.
- What test will you use?
- Using a = 0.05, what is your conclusion?
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