NONPARAMETRIC STATISTICS In general, a statistical technique is categorized as NPS if it has at least one of the following assumptions: The method is used on nominal data The method is used in ordinal data The method is used in interval scale or ratio scale data but there is no assumption regarding the probability distribution of the population where the sample is selected. Sign Test Mann-Whitney Test

Nonparametric versus Parametric Statistics Parametric statistical procedures are inferential procedures that rely on testing claims regarding parameters such as In some circumstances, the use of parametric procedures required some certain assumptions that need to be satisfied such as check for the normality distribution. Nonparametric statistical procedures are inferential procedures that are not based on parameters and required fewer assumptions that need to satisfy to perform tests. They do not required the population to follow a specific type of distribution or in other words known as distribution free procedures. Inferences : Parametric makes inference regarding the mean. Nonparametric makes inference regarding the median.

Advantages of Nonparametric Most of the tests required very few assumptions. The computation is fairly easy. The procedures can be used for count data or rank data. Disadvantages of NonParametric Less sensitive. Therefore, larger differences are needed before the null hypothesis can be rejected. Less efficient than parametric procedures. Required a larger sample size for nonparametric procedure.

Sign Test The sign test is used to test the value of median for a specific sample. When applying the sign test, the data values are converted to plus ‘+’ sign and minus ‘-’ sign. Any data value(s) that is (are) equal to hypothesized median M0 is (are) denoted as 0 and ignored. It based on the direction of the ‘+’ and ‘–’ sign of the observation and not their numerical magnitude. The number of ‘+’ sign and ‘-’ sign are compared to determine for any significance difference.

There are two types of sign test : 1. One sample sign test 2. Paired sample sign test One sample sign test: For single samples Used to test the values of a median for a specific sample The sign test for a single sample is a nonparametric test used to test the values of a population median. * The paired-sample sign test is a nonparametric test used to test the difference between two population medians when the samples are dependent.

One Sample Sign Test STEP 1: Identify the claim and state the hypotheses. STEP 2: Put a ‘+’ sign for a value greater than the hypothesized median value Put a ‘-’ sign for a value less than the hypothesized median value Put a ‘0’ as the value equal to the hypothesized median value Two Tailed Left-Tailed Right-Tailed

k = minimum number between ‘+’ and ‘-’ signs STEP 3: Compute test statistic. STEP 4: Obtain the critical value: The critical value will be obtained from the sign test table. This value is based on the chosen significance level and the sample size n where n is the total number of ‘+’ and ‘-’ signs. STEP 5: Make a decision. If the test statistic, k ≤ critical value, we reject STEP 6: Conclusion. Type of Test Left-Tailed Two-Tailed k = minimum number between ‘+’ and ‘-’ signs k = number of ‘+’ signs Right-Tailed k = number of ‘-’ signs

Example: The following data constitute a random sample of 15 measurement of the octane rating of a certain kind gasoline: 99.0 102.3 99.8 100.5 99.7 96.2 99.1 102.5 103.3 97.4 100.4 98.9 98.3 98.0 101.6 Test whether the median of octane rating is greater than 98.0 at the 0.05 level of significance. Solution: + + + + + - + + + - + + + 0 +

1. Compare each data with the hypothesized median. 99.0 102.3 99.8 100.5 99.7 96.2 99.1 102.5 103.3 97.4 100.4 + + + + + - + + + - + 98.9 98.3 98.0 101.6 + + 0 + 3. Compute the test statistic. Number of ‘+’ sign = 12 Number of ‘-’ sign = 2 Total number of ‘+’ signs and ‘-’ sign = 14 k = 2. The number of ‘-’ was chosen because this is right-tailed. Obtain the critical value. α= 0.05 and n=14, the critical value: k = 3.

5. Make decision. Since the test statistic, k = 2 < 3, we reject Conclusion. We can conclude that the median octane rating of the given kind of gasoline exceeds 98.0

Exercise: Snow Cone Sales A convenience store owner hypothesizes that the median number of snow cones she sells per day is 40. A random sample of 20 days yields the following data for the number of snow cones sold each day. At α=0.05, test the owner’s hypothesis. Answer: Reject the claim (H0) 18 43 40 16 22 30 29 32 37 36 39 34 45 28 52

Exercise: Clean Air A researcher suggest that the median of the number of days per month that a larger city getting an unhealthy air is 11 days per month. A random sample of 20 months shows the number of days per month that have unhealthy air as below: By using sign test, test whether the median number of days that getting unhealthy air per month is 11 at α=0.05. Answer: Fail to reject H0 15 6 14 16 1 21 9 22 3 19 5 10 23 8 13

Mann-Whitney Test Mann-Whitney Test is a nonparametric procedure that is used to test the equality of two population medians from the independent samples. To determine whether a difference exist between two populations Sometimes called as Wilcoxon rank sum test The equivalent parametric test to Mann-Whitney test is the t-test for two independent samples.

STEP 1: Identify the claim and state the hypotheses. Two independent random samples are required from each population. 1. Null and alternative hypothesis * The Mann-Whitney test is nonparametric test that uses ranks to determine if two independent samples were selected from populations that have the same distributions. Two tail test Left tail test Right tail test Rejection area

Hypotheses to be tested are:

STEP 2: Rank the data values: Combine all data values from the two independent samples and regard them as a single sample. Rank the combined data value as if they were from a single group. Rank all the data from the smallest to the largest. The smallest data value gets a rank 1 and so on. In the event of tie, each of the tied get the average rank that the values are occupying.

STEP 3: Compute the test statistic T: Designate the smaller size of the two sample as ‘Sample 1’. If the sample are equal, either one or more may be designated as ‘Sample 1’. List the ranks for data values from sample 1 and find the sum of the rank for ‘Sample 1’. Repeat the same thing to ‘Sample 2’. The test statistic is given by: Type of Test Test Statistic Two-Tailed Left-Tailed Right-Tailed

The critical value is given by: STEP 4: Obtain the Critical value of T The Mann-Whitney test/Wilcoxon rank sum table list lower and upper critical value, TL with n1 and n2 are the sample size for ‘Sample 1’ and ‘Sample 2’. The critical value is given by: Where TL is obtain from Table of Critical Values for Mann-Whitney Test:

Two-Tailed Left-Tailed Right-Tailed STEP 5: Make a decision either to reject or fail to reject H0 STEP 6: Conclusion Two-Tailed Left-Tailed Right-Tailed Rejection Region

Example: Data below show the marks obtained by electrical engineering students in an examination: Can we conclude the achievements of male and female students are different at significance level Gender Marks Male Female 60 62 78 83 40 65 70 88 92

Solution: 1. 2. 3. Gender Marks Rank Male Female 60 62 78 83 40 65 70 88 92 2 3 6 7 1 4 5 8 9

4. From the table of Mann Whitney test for Reject Since , thus we fail to reject 6. Thus we can conclude that there is not enough evidence to support the claim that there is a difference in the achievements between male and female.

Exercise: School Lunch A nutritionist decided to see if there was a difference in the number of calories served for lunch in elementary and secondary schools. She selected a random sample of eight elementary schools and another random sample of eight secondary schools in Pennsylvania. The data are shown. (Test using Mann-Whitney test at 0.05 level of significance.) Answer: Reject H0 Elementary Secondary 648 694 589 730 625 750 595 810 789 860 727 702 657 564 761

Exercise: Using high school records, Johnson High school administrators selected a random sample of four high school students who attended Garfield Junior High and another random sample of five students who attended Mulbery Junior High. The ordinal class standings for the nine students are listed in the table below. By using Mann-Whitney test, can you conclude that the median of ranking record of Garfield Junior High is greater than Mulbery Junior High at 0.05 level of significance. Answer: Reject H0 Garfield J. High Mulbery J. High Student Class standing Fields 8 Hart 70 Clark 52 Phipps 202 Jones 112 Kirwood 144 TIbbs 21 Abbott 175 Guest 146