1. UNIT 4-A: DATA ANALYSIS and REPORTING 1.Frequency Distribution 2.Cross Tabulation 3.Hypothesis Testing GM07: RESEARCH METHODOLOGY

2. Data in raw form (as collected): 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Data in ordered array from smallest to largest: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 Stem-and-leaf display: Organizing Numerical Data 2 144677 3 028 4 1

3. Frequency Distribution In a frequency distribution, one variable is considered at a time. A frequency distribution for a variable produces a table of frequency counts, percentages, and cumulative percentages for all the values associated with that variable.

4. Organizing Numerical Data Numerical Data Ordered Array Stem and Leaf Display Histograms Ogive Tables 2 144677 3 028 4 1 41, 24, 32, 26, 27, 27, 30, 24, 38, 21 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 Frequency Distributions Cumulative Distributions Polygons

5. Tabulating Numerical Data: Frequency Distributions Sort raw data in ascending order: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Find range: 58 - 12 = 46 Select number of classes: 5 (usually between 5 and 15) Compute class interval (width): 10 (46/5 then round up) Determine class boundaries (limits): 10, 20, 30, 40, 50, 60 Compute class midpoints: 15, 25, 35, 45, 55 Count observations & assign to classes

6. Frequency Distributions, Relative Frequency Distributions and Percentage Distributions Class Frequency 10 but under 20 3.15 15 20 but under 30 6.30 30 30 but under 40 5.25 25 40 but under 50 4.20 20 50 but under 60 2.10 10 Total 20 1 100 Relative Frequency Percentage Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

7. Graphing Numerical Data: The Histogram Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 No Gaps Between Bars Class Midpoints Class Boundaries

8. Graphing Numerical Data: The Frequency Polygon Class Midpoints Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

9. Tabulating Numerical Data: Cumulative Frequency Cumulative Cumulative Class Frequency % Frequency 10 but under 20 3 15 20 but under 30 9 45 30 but under 40 14 70 40 but under 50 18 90 50 but under 60 20 100 Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

10. Graphing Numerical Data: The Ogive (Cumulative % Polygon) Class Boundaries (Not Midpoints) Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

11. Graphing Bivariate Numerical Data (Scatter Plot)

12. Tabulating and Graphing Categorical Data:Univariate Data Categorical Data Tabulating Data The Summary Table Graphing Data Pie Charts Pareto Diagram Bar Charts

13. Summary Table (for an Investor’s Portfolio) Investment Category AmountPercentage (in thousands ) Stocks 46.5 42.27 Bonds 32 29.09 CD 15.5 14.09 Savings 16 14.55 Total 110 100 Variables are Categorical

14. Graphing Categorical Data: Univariate Data Categorical Data Tabulating Data The Summary Table Graphing Data Pie Charts Pareto Diagram Bar Charts

15. Bar Chart (for an Investor’s Portfolio)

16. Pie Chart (for an Investor’s Portfolio) Percentages are rounded to the nearest percent. Amount Invested in K$ Savings 15% CD 14% Bonds 29% Stocks 42%

17. Pareto Diagram Axis for line graph shows cumulative % invested Axis for bar chart shows % invested in each category

18. Tabulating and Graphing Bivariate Categorical Data Contingency tables: investment in thousands Investment Investor A Investor B Investor C Total Category Stocks 46.5 55 27.5 129 Bonds 32 44 19 95 CD 15.5 20 13.5 49 Savings 16 28 7 51 Total 110 147 67 324

19. Tabulating and Graphing Bivariate Categorical Data Side by side charts

20. Principles of Graphical Excellence Presents data in a way that provides substance, statistics and design Communicates complex ideas with clarity, precision and efficiency Gives the largest number of ideas in the most efficient manner Almost always involves several dimensions Tells the truth about the data

21. “Chart Junk” Good Presentation 1960: $1.00 1970: $1.60 1980: $3.10 1990: $3.80 Minimum Wage 0 2 4 1960197019801990 $ Bad Presentation

22. No Relative Basis Good Presentation A’s received by students. Bad Presentation 0  200 300 FRSOJRSR Freq.  10  30 FRSOJRSR % FR = Freshmen, SO = Sophomore, JR = Junior, SR = Senior

23. Compressing Vertical Axis Good Presentation Quarterly Sales Bad Presentation 0 25 50 Q1Q2Q3 Q4 $ 0 100 200 Q1Q2 Q3 Q4 $

24. No Zero Point on Vertical Axis Good Presentation Monthly Sales Bad Presentation 0 39 42 45 J F MAMJ $ 36 39 42 45 JFMAMJ $ Graphing the first six months of sales. 36

25. Statistics for Frequency Distribution Measures of central tendency – Mean, median, mode, geometric mean Quartile Measure of variation – Range, Interquartile range, variance and standard deviation, coefficient of variation Measure of Shape – Symmetric, skewed, using box-and-whisker plots

26. Measures of Central Tendency Central Tendency AverageMedianMode Geometric Mean

27. Mean (Arithmetic Mean) Mean (arithmetic mean) of data values – Sample mean – Population mean Sample Size Population Size

28. Mean (Arithmetic Mean) The most common measure of central tendency Affected by extreme values (outliers) (continued) 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5Mean = 6

29. Median Robust measure of central tendency Not affected by extreme values In an ordered array, the median is the “middle” number – If n or N is odd, the median is the middle number – If n or N is even, the median is the average of the two middle numbers 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5

30. Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 0 1 2 3 4 5 6 No Mode

31. Geometric Mean Useful in the measure of rate of change of a variable over time Geometric mean rate of return – Measures the status of an investment over time

32. Quartiles Split Ordered Data into 4 Quarters Position of i-th Quartile and Are Measures of Noncentral Location = Median, A Measure of Central Tendency 25% Data in Ordered Array: 11 12 13 16 16 17 18 21 22

33. Measures of Variation Variation VarianceStandard DeviationCoefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range Interquartile Range

34. Range Measure of variation Difference between the largest and the smallest observations: Ignores the way in which data are distributed 7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 Range = 12 - 7 = 5

35. Measure of variation Also known as midspread – Spread in the middle 50% Difference between the first and third quartiles Not affected by extreme values Interquartile Range Data in Ordered Array: 11 12 13 16 16 17 17 18 21

36. Important measure of variation Shows variation about the mean – Sample variance: – Population variance: Variance

37. Standard Deviation Most important measure of variation Shows variation about the mean Has the same units as the original data – Sample standard deviation: – Population standard deviation:

38. Comparing Standard Deviations Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 s =.9258 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 4.57 Data C

39. Coefficient of Variation Measures relative variation Always in percentage (%) Shows variation relative to mean Is used to compare two or more sets of data measured in different units

40. Comparing Coefficient of Variation Stock A: – Average price last year = $50 – Standard deviation = $5 Stock B: – Average price last year = $100 – Standard deviation = $5 Coefficient of variation: – Stock A: – Stock B:

41. Shape of a Distribution Describes how data is distributed Measures of shape – Symmetric or skewed Mean = Median =Mode Mean < Median < Mode Mode < Median < Mean Right-Skewed Left-SkewedSymmetric

42. Exploratory Data Analysis Box-and-whisker plot – Graphical display of data using 5-number summary Median( ) 4 6 8 10 12 X largest X smallest

43. Distribution Shape and Box-and-Whisker Plot Right-SkewedLeft-SkewedSymmetric

44. Cross-Tabulation While a frequency distribution describes one variable at a time, a cross-tabulation describes two or more variables simultaneously. Cross-tabulation results in tables that reflect the joint distribution of two or more variables with a limited number of categories or distinct values

45. Gender and Internet Usage Gender Row Internet Usage MaleFemaleTotal Light (1) 5 10 15 Heavy (2) 10 5 15 Column Total 15 15

46. Two Variables Cross-Tabulation Since two variables have been cross-classified, percentages could be computed either columnwise, based on column totals, or rowwise, based on row totals. The general rule is to compute the percentages in the direction of the independent variable, across the dependent variable. The correct way of calculating percentages is as shown in next slide.

47. Internet Usage by Gender

48. Gender by Internet Usage

49. Introduction of a Third Variable in Cross-Tabulation Refined Association between the Two Variables No Association between the Two Variables No Change in the Initial Pattern Some Association between the Two Variables No Association between the Two Variables Introduce a Third Variable Original Two Variables

50. The introduction of a third variable can result in four possibilities: As can be seen from, 52% of unmarried respondents fell in the high-purchase category, as opposed to 31% of the married respondents. Before concluding that unmarried respondents purchase more fashion clothing than those who are married, a third variable, the buyer's sex, was introduced into the analysis. As shown in the table, in the case of females, 60% of the unmarried fall in the high-purchase category, as compared to 25% of those who are married. On the other hand, the percentages are much closer for males, with 40% of the unmarried and 35% of the married falling in the high purchase category. Hence, the introduction of sex (third variable) has refined the relationship between marital status and purchase of fashion clothing (original variables). Unmarried respondents are more likely to fall in the high purchase category than married ones, and this effect is much more pronounced for females than for males. Three Variables Cross-Tabulation Refine an Initial Relationship

51. Purchase of Fashion Clothing by Marital Status Purchase of Fashion Current Marital Status Clothing Married Unmarried High31%52% Low69%48% Column100% Number of respondents 700300

52. Purchase of Fashion Clothing by Marital Status Purchase of Fashion Clothing Sex Male Female Married Not Married Not Married High35%40%25%60% Low65%60%75%40% Column totals 100% Number of cases 400120300180

53. Table shows that 32% of those with college degrees own an expensive automobile, as compared to 21% of those without college degrees. Realizing that income may also be a factor, the researcher decided to reexamine the relationship between education and ownership of expensive automobiles in light of income level. In Table, the percentages of those with and without college degrees who own expensive automobiles are the same for each of the income groups. When the data for the high income and low income groups are examined separately, the association between education and ownership of expensive automobiles disappears, indicating that the initial relationship observed between these two variables was spurious. Three Variables Cross-Tabulation Initial Relationship was Spurious

54. Ownership of Expensive Automobiles by Education Level Own Expensive Automobile Education College DegreeNo College Degree Yes No Column totals Number of cases 32% 68% 100% 250 21% 79% 100% 750

55. Ownership of Expensive Automobiles by Education Level and Income Levels

56. Table shows no association between desire to travel abroad and age. When sex was introduced as the third variable, Table was obtained. Among men, 60% of those under 45 indicated a desire to travel abroad, as compared to 40% of those 45 or older. The pattern was reversed for women, where 35% of those under 45 indicated a desire to travel abroad as opposed to 65% of those 45 or older. Since the association between desire to travel abroad and age runs in the opposite direction for males and females, the relationship between these two variables is masked when the data are aggregated across sex as in Table. But when the effect of sex is controlled, as in Table, the suppressed association between desire to travel abroad and age is revealed for the separate categories of males and females. Three Variables Cross-Tabulation Reveal Suppressed Association

57. Desire to Travel Abroad by Age

58. Desire to Travel Abroad by Age and Gender

59. Consider the cross-tabulation of family size and the tendency to eat out frequently in fast-food restaurants as shown in Table. No association is observed. When income was introduced as a third variable in the analysis, Table was obtained. Again, no association was observed. Three Variables Cross-Tabulations No Change in Initial Relationship

60. Eating Frequently in Fast-Food Restaurants by Family Size

61. Eating Frequently in Fast Food-Restaurants by Family Size and Income

62. To determine whether a systematic association exists, the probability of obtaining a value of chi-square as large or larger than the one calculated from the cross-tabulation is estimated. An important characteristic of the chi-square statistic is the number of degrees of freedom (df) associated with it. That is, df = (r - 1) x (c -1). The null hypothesis (H 0 ) of no association between the two variables will be rejected only when the calculated value of the test statistic is greater than the critical value of the chi-square distribution with the appropriate degrees of freedom, as shown. Statistics Associated with Cross-Tabulation Chi-Square

63. The phi coefficient ( ) is used as a measure of the strength of association in the special case of a table with two rows and two columns (a 2 x 2 table). The phi coefficient is proportional to the square root of the chi-square statistic It takes the value of 0 when there is no association, which would be indicated by a chi-square value of 0 as well. When the variables are perfectly associated, phi assumes the value of 1 and all the observations fall just on the main or minor diagonal. Statistics Associated with Cross-Tabulation Phi Coefficient   =  2 n

64. While the phi coefficient is specific to a 2 x 2 table, the contingency coefficient (C) can be used to assess the strength of association in a table of any size. The contingency coefficient varies between 0 and 1. The maximum value of the contingency coefficient depends on the size of the table (number of rows and number of columns). For this reason, it should be used only to compare tables of the same size. Statistics Associated with Cross-Tabulation Contingency Coefficient C =  2  2 + n

65. Cramer's V is a modified version of the phi correlation coefficient,, and is used in tables larger than 2 x 2. or Statistics Associated with Cross-Tabulation Cramer’s V  V =  2 min (r-1), (c-1) V =  2 /n min (r-1), (c-1)

66. Asymmetric lambda measures the percentage improvement in predicting the value of the dependent variable, given the value of the independent variable. Lambda also varies between 0 and 1. A value of 0 means no improvement in prediction. A value of 1 indicates that the prediction can be made without error. This happens when each independent variable category is associated with a single category of the dependent variable. Asymmetric lambda is computed for each of the variables (treating it as the dependent variable). A symmetric lambda is also computed, which is a kind of average of the two asymmetric values. The symmetric lambda does not make an assumption about which variable is dependent. It measures the overall improvement when prediction is done in both directions. Statistics Associated with Cross-Tabulation Lambda Coefficient

67. Other statistics like tau b, tau c, and gamma are available to measure association between two ordinal-level variables. Both tau b and tau c adjust for ties. Tau b is the most appropriate with square tables in which the number of rows and the number of columns are equal. Its value varies between +1 and -1. For a rectangular table in which the number of rows is different than the number of columns, tau c should be used. Gamma does not make an adjustment for either ties or table size. Gamma also varies between +1 and -1 and generally has a higher numerical value than tau b or tau c. Other Statistics Associated with Cross-Tabulation

68. Cross-Tabulation in Practice While conducting cross-tabulation analysis in practice, it is useful to proceed along the following steps. 1.Test the null hypothesis that there is no association between the variables using the chi-square statistic. If you fail to reject the null hypothesis, then there is no relationship. 2.If H 0 is rejected, then determine the strength of the association using an appropriate statistic (phi-coefficient, contingency coefficient, Cramer's V, lambda coefficient, or other statistics), as discussed earlier. 3.If H 0 is rejected, interpret the pattern of the relationship by computing the percentages in the direction of the independent variable, across the dependent variable. 4.If the variables are treated as ordinal rather than nominal, use tau b, tau c, or Gamma as the test statistic. If H 0 is rejected, then determine the strength of the association using the magnitude, and the direction of the relationship using the sign of the test statistic.

69. Hypothesis Testing  In statistics, a hypothesis is a claim or statement about a property of a population.  A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property of a population. Rare Event Rule for Inferential Statistics If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct

70. Steps Involved in Hypothesis Testing Draw Research Conclusion Formulate H 0 and H 1 Select Appropriate Test Choose Level of Significance Determine Probability Associated with Test Statistic Determine Critical Value of Test Statistic TSCR Determine if TSCR falls into (Non) Rejection Region Compare with Level of Significance,  Reject or Do not Reject H 0 Collect Data and Calculate Test Statistic

71. Note about Identifying H 0 and H 1

72. Definitions Critical Region The critical region (or rejection region) is the set of all values of the test statistic that cause us to reject the null hypothesis. Significance Level The significance level (denoted by  ) is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. Common choices for  are 0.05, 0.01, and 0.10. Critical Value A critical value is any value that separates the critical region (where we reject the null hypothesis) from the values of the test statistic that do not lead to rejection of the null hypothesis, the sampling distribution that applies, and the significance level . The critical value of z = 1.96 corresponds to a significance level of  = 0.05.

73. Type I & Type II Errors  A Type I error is the mistake of rejecting the null hypothesis when it is true.  The symbol   (alpha) is used to represent the probability of a type I error.  A Type II error is the mistake of failing to reject the null hypothesis when it is false.  The symbol  (beta) is used to represent the probability of a type II error.

74. Controlling Type I and Type II Errors  For any fixed , an increase in the sample size n will cause a decrease in   For any fixed sample size n, a decrease in  will cause an increase in . Conversely, an increase in  will cause a decrease in .  To decrease both  and , increase the sample size.

75. Conclusions in Hypothesis Testing  We always test the null hypothesis. 1. Reject the H 0 2. Fail to reject the H 0

76. Two-tailed, Right-tailed, Left-tailed Tests The tails in a distribution are the extreme regions bounded by critical values.

77. Two-tailed Test  is divided equally between the two tails of the critical region H 0 : = H 1 :  Means less than or greater than

78. Right-tailed Test H 0 : = H 1 : > Points Right

79. Left-tailed Test H 0 : = H 1 : < Points Left

80. Decision Criterion Traditional method: Reject H0 if the test statistic falls within the critical region. Fail to reject H0 if the test statistic does not fall within the critical region. P-value method: Reject H0 if P-value   (where  is the significance level, such as 0.05). Fail to reject H0 if P-value > . Another option: Instead of using a significance level such as 0.05, simply identify the P-value and leave the decision to the reader.

81. Decision Criterion Confidence Intervals: Because a confidence interval estimate of a population parameter contains the likely values of that parameter, reject a claim that the population parameter has a value that is not included in the confidence interval.

82. P-Value The P-value (or p-value or probability value) is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true. The null hypothesis is rejected if the P-value is very small, such as 0.05 or less.

83. Example: Finding P-values.

84. Wording of Final Conclusion

85. Accept versus Fail to Reject  Some texts use “accept the null hypothesis.”  We are not proving the null hypothesis.  The sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect).

86. Comprehensive Hypothesis Test

87. Statistical Tests Parametric Interval or ratio Scaled Data Assumption about population probability distribution Example Z, t, F test etc. Non-Parametric Nominal or ordinal data No assumption about population probability distribution Example χ 2, sign, Wilcoxon Signed-Rank, Kruskal-Wallis Test etc.

88. A Classification of Hypothesis Testing Procedures for Examining Differences Independent Samples Paired Samples Independent Samples Paired Samples * Two-Group t test * Z test * Paired t test * Chi-Square * Mann-Whitney * Median * K-S * Sign * Wilcoxon * McNemar * Chi-Square Hypothesis Tests One Sample Two or More Samples One Sample Two or More Samples * t test * Z test * Chi-Square * K-S * Runs * Binomial Parametric Tests (Metric Tests) Non-parametric Tests (Nonmetric Tests)

89. A Broad Classification of Hypothesis Tests Median/ Rankings Distributions Means Proportions Tests of Association Tests of Differences Hypothesis Tests

90. Applications of Z – test (n>30) Test of significance for single mean Test of significance for difference of means Test of significance for difference of standard deviation (s.d.) Testing a Claim about a Proportion Testing difference of Two proportions

91. Test of significance for single mean Where Sample Mean  Population mean  Population standard deviation(s.d.) n Sample size NOTE: If  population standard deviation(s.d.) is unknown then estimated sample standard deviation s will be used. 95% confidence interval for  is 99% confidence interval for  is

92. Test of significance for difference of means Where, are the means of first and second sample, are standard deviations of samples, are the sample size of first sample & second sample IF and and are not known then

93. Test of significance for difference of standard deviation (s.d.) IF and are not known then

94. Testing a Claim about a Proportion p = population proportion (used in the null hypothesis) q = 1 – p  n = number of trials p = x (sample proportion) n

95. Testing difference of Two proportions Where Note that p* is the combined two sample proportion weighted by the two sample sizes

96. Applications of t – test(n≤30) Test the significance of the mean of a random sample Test the difference between means of two samples (Independent Samples) Test the difference between means of two samples (Dependent Samples) Test of significance of an observed correlation coefficients

97. Test the significance of the mean of a random sample Where Sample Mean  Population mean S standard deviation of the sample= n Sample size Degree of freedom(d.f.)= n-1 95% confidence interval for  is

98. Test of significance for difference of means Where S combined standard deviation are the mean, standard deviation and sample size of first and second sample IF s 1, s 2 and n 1 n 2 are given d.f. = n 1 + n 2 -2

99. Test the difference between means of two samples (Dependent Samples) Where is the mean of the differences n is the number of paired observations S is standard deviation of differences d.f. = n – 1

100. Test of significance of an observed correlation coefficients Where r is correlation coefficient n is sample size d.f. = n – 2

101. An F test of sample variance may be performed if it is not known whether the two populations have equal variance. In this case, the hypotheses are: H 0 : 1 2 = 2 2 H 1 : 1 2 2 2 Two Independent Samples F Test

102. The F statistic is computed from the sample variances as follows where n 1 = size of sample 1 n 2 = size of sample 2 n 1 -1= degrees of freedom for sample 1 n 2 -1= degrees of freedom for sample 2 s 1 2 = sample variance for sample 1 s 2 2 = sample variance for sample 2 suppose we wanted to determine whether Internet usage was different for males as compared to females. A two-independent- samples t test was conducted. Two Independent Samples F Statistic

103. Nonparametric Tests Nonparametric tests are used when the independent variables are nonmetric. Like parametric tests, nonparametric tests are available for testing variables from one sample, two independent samples, or two related samples.

104. Non-parametric Test Chi-Square Test Binomial Runs 1-Samples K-S 2-Independent samples K-Independent Samples 2-Dependent Samples K-Dependent Samples

105. Applications of χ 2 Test 1.Goodness of Fit 2.Contingency Analysis (or Test of Independence) 3.Test of population variance

106. Where O i and E i are Observed and expected frequencies Degree of freedom(d.f.) = n-1 Note: No E i should be less than 5, if so cell(s) must be combined and d.f. should be reduced accordingly If some parameters are calculated from O i to calculate E i e.g. mean or standard deviation etc then d.f. should be reduced by 1 for each such parameters. Goodness of Fit

107. Contingency Analysis (or Test of Independence) Where Oi and Ei are Observed and expected frequencies Degree of freedom = (rows-1)(column-1) Note: When Degree of freedom is 1 AND N<50, adjust χ2 by Yates's Correction Factor i.e. Unless in such case original (O i – E i ) 2 term is preserved

108. Test of population variance Where S 2 is sample variance σ 2 is population variance Degree of freedom = (n-1) n number of observations

109. Sometimes the researcher wants to test whether the observations for a particular variable could reasonably have come from a particular distribution, such as the normal, uniform, or Poisson distribution. The Kolmogorov-Smirnov (K-S) one-sample test is one such goodness-of-fit test. The K-S compares the cumulative distribution function for a variable with a specified distribution. A i denotes the cumulative relative frequency for each category of the theoretical (assumed) distribution, and O i the comparable value of the sample frequency. The K-S test is based on the maximum value of the absolute difference between A i and O i. The test statistic is Nonparametric Tests One Sample K = Max A i - O i

110. The decision to reject the null hypothesis is based on the value of K. The larger the K is, the more confidence we have that H 0 is false. For = 0.05, the critical value of K for large samples (over 35) is given by 1.36/ Alternatively, K can be transformed into a normally distributed z statistic and its associated probability determined. In the context of the Internet usage example, suppose we wanted to test whether the distribution of Internet usage was normal. A K-S one-sample test is conducted, yielding the data shown in Table indicates that the probability of observing a K value of 0.222, as determined by the normalized z statistic, is 0.103. Since this is more than the significance level of 0.05, the null hypothesis can not be rejected, leading to the same conclusion. Hence, the distribution of Internet usage does not deviate significantly from the normal distribution. Nonparametric Tests One Sample

111. K-S One-Sample Test for Normality of Internet Usage

112. The chi-square test can also be performed on a single variable from one sample. In this context, the chi-square serves as a goodness-of-fit test. The runs test is a test of randomness for the dichotomous variables. This test is conducted by determining whether the order or sequence in which observations are obtained is random. The binomial test is also a goodness-of-fit test for dichotomous variables. It tests the goodness of fit of the observed number of observations in each category to the number expected under a specified binomial distribution. Nonparametric Tests One Sample

113. When the difference in the location of two populations is to be compared based on observations from two independent samples, and the variable is measured on an ordinal scale, the Mann-Whitney U test can be used. In the Mann-Whitney U test, the two samples are combined and the cases are ranked in order of increasing size. The test statistic, U, is computed as the number of times a score from sample or group 1 precedes a score from group 2. If the samples are from the same population, the distribution of scores from the two groups in the rank list should be random. An extreme value of U would indicate a nonrandom pattern, pointing to the inequality of the two groups. For samples of less than 30, the exact significance level for U is computed. For larger samples, U is transformed into a normally distributed z statistic. This z can be corrected for ties within ranks. Nonparametric Tests Two Independent Samples

114. We examine again the difference in the Internet usage of males and females. This time, though, the Mann-Whitney U test is used. The results are given in Table. One could also use the cross-tabulation procedure to conduct a chi- square test. In this case, we will have a 2 x 2 table. One variable will be used to denote the sample, and will assume the value 1 for sample 1 and the value of 2 for sample 2. The other variable will be the binary variable of interest. The two-sample median test determines whether the two groups are drawn from populations with the same median. It is not as powerful as the Mann-Whitney U test because it merely uses the location of each observation relative to the median, and not the rank, of each observation. The Kolmogorov-Smirnov two-sample test examines whether the two distributions are the same. It takes into account any differences between the two distributions, including the median, dispersion, and skewness. Nonparametric Tests Two Independent Samples

115. Mann-Whitney U - Wilcoxon Rank Sum W Test Internet Usage by Gender SexMean RankCases Male20.9315 Female 10.0715 Total 30 Corrected for ties U W z 2-tailedp 31.000 151.000 -3.406 0.001 Note U = Mann-Whitney test statistic W = Wilcoxon W Statistic z = U transformed into normally distributedz statistic.

116. The Wilcoxon matched-pairs signed-ranks test analyzes the differences between the paired observations, taking into account the magnitude of the differences. It computes the differences between the pairs of variables and ranks the absolute differences. The next step is to sum the positive and negative ranks. The test statistic, z, is computed from the positive and negative rank sums. Under the null hypothesis of no difference, z is a standard normal variate with mean 0 and variance 1 for large samples. Nonparametric Tests Paired Samples

117. The example considered for the paired t test, whether the respondents differed in terms of attitude toward the Internet and attitude toward technology, is considered again. Suppose we assume that both these variables are measured on ordinal rather than interval scales. Accordingly, we use the Wilcoxon test. The sign test is not as powerful as the Wilcoxon matched-pairs signed-ranks test as it only compares the signs of the differences between pairs of variables without taking into account the ranks. In the special case of a binary variable where the researcher wishes to test differences in proportions, the McNemar test can be used. Alternatively, the chi-square test can also be used for binary variables. Nonparametric Tests Paired Samples

118. Wilcoxon Matched-Pairs Signed-Rank Test Internet with Technology

119. A Summary of Hypothesis Tests Related to Differences SampleApplicationLevel of ScalingTest/Comments One Sample DistributionsNonmetric K-S and chi-square for goodness of fit Runs test for randomness Binomial test for goodness of fit for dichotomous variables One SampleMeansMetricttest, if variance is unknown z test, if variance is known Proportion Metric Z test

120. A Summary of Hypothesis Tests Related to Differences