Mann-Whitney U Test

The Mann-Whitney U test is a non-parametric hypothesis test that is used as an alternative to the parametric t-test. The test is equivalent to the Wilcoxon Rank Sum test. Two independent samples are used to determine whether or not one population is stochastically larger than the other.

This non-parametric test is chosen because the data that has been

collected are not normally distributed, are highly skewed, and are not continuous.

As an attribute of a non-parametric test, Mann Whitney U does not require any assumptions regarding the underlying distribution. However, there are three assumptions regarding use of this test including random and independent

samples, along with an ordinal measurement scale.

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To calculate the value of Mann-Whitney U test, the following formula is applied:

𝑈 = 𝑛₁𝑛₂+ 𝑛2(𝑛2+ 1)

2 − � 𝑅_𝑖

𝑛₂

𝑖= 𝑛₁+1

Where: 𝑈 is Mann-Whitney U test, 𝑛₁ is the sample size one, 𝑛₂ is the sample size two and 𝑅_𝑖 is the rank of the sample size.

Two tables are presented in the results section for the Mann-Whitney U test. The first table includes the mean rank and sum of ranks. In order to produce sum of ranks, sample data are sorted in ascending order, without regard to which sample the data comes from. If two or more observations are identical, the ranks would be averaged and all tied observations assigned this averaged rank. Once the rank is assigned, the sum of ranks can be computed by adding the rank of all observations for each sample. With the sum of ranks computed, the mean rank can be obtained by dividing sum of ranks by the total number in that group.

The second table shows the Mann-Whitney U results. The 𝑈 value is approximated by the Z statistic, which has an asymptotic distribution, when the sample size is large (which is true in this case). The Z distribution has a mean of 0 and standard deviation of 1. The significance of the Mann- Whitney U test will be based on one-tailed test with a significance level of 95% (p < .05). In other words, for any Z value that is beyond 1.65, the null hypothesis of identical distributions would be rejected.

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Section 4: Results

The following tables present the results of the final survey. Responses from the survey were collected from February 12, 2009 until April 10, 2009. Two thousand two hundred fifty three appropriate responses were used to generate the results. A copy of the questionnaire is included in the Appendix section.

Section 4.1 represents the overall results for descriptive statistics for 2,253 college students, 2,052 responders that are not involve in any crashes and 201 students who were involved in crashes due to the usage of the cell phone while driving. Section 4.2 displays the frequency column of the three respective break down as demonstrate in Section 4.1. Lastly, Section 4.3 presents the analysis of Mann-Whitney T test by dividing the students into 2 groups – students involved in crashes and students without crashes.

4.1 Descriptive Statistics

Table 4.1.1 presents the abbreviations that are used in Table 4.1.2, Table 4.1.3, and Table 4.1.4. These abbreviations represent the 15 questions and responses that are being presented in the Descriptive Statistics Tables (Table 4.1.2, Table 4.1.3, and Table 4.1.4).

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Table 4.1.1: Explanation of the Abbreviations Used in Descriptive Statistics Table and Ranks Table

Abbreviation Description

dh Number of hours a student drives in a week

c Number of calls a student makes while driving in a week a Number of calls a student answers while driving in a week

hf Has student ever used a hands free device to talk on phone while driving (0 = No, 1 = Yes)

ab Has student ever browsed the address book to locate a contact number while driving (0 = No, 1 = Yes)

rms Number of text messages a student types and sends while driving in a week

sms Number of text messages a student reads while driving in a week rml Number of emails a student composes and sends while driving in

a week

sml Number of emails a student reads while driving in a week

p Has student ever taken pictures with a cell phone while driving (0

= No, 1 = Yes)

pf Number of pictures a student photographs while driving in a week cc Has student ever been involved in a close call situation involving

the use of a cell phone either by the student or another driver (0

= No, 1 = Yes)

ccf Number of close call situations involving a cell phone used a student faced in the last 30 days

ac Has student ever been involved in traffic accident involving the use of a cell phone either by the student or another driver (0 = No, 1 = Yes)

acf Number of crashes involving cell phone usage a student has faced

Table 4.1.2 presents descriptive analysis of the overall results. The 15 responses that are gathered from 2253 students were analyzed with SPSS 13.5.

The descriptive statistics that were gathered include range, minimum, maximum, mean, median, standard deviation and variance. Range reported as 1 indicates a

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yes or no response (0= No, 1 = Yes). Table 4.1.3 represents descriptive statistics for students who were not involved in crashes while Table 4.1.4 presents the remaining 201 students who were reported to have suffered crashes.

Table 4.1.2: Overall Descriptive Statistics for Different Cell Phone Usages (obtained from SPSS 13.5)

N = 2253 Range Minimum Maximum Mean Median Standard deviation Variance

dh 23 0 23 8.18 7 4.88 23.80

c 50 0 50 8.44 5 9.07 82.26

a 50 0 50 7.28 5 8.07 65.18

hf 1 0 1 0.16 0 0.37 0.36

ab 1 0 1 0.43 0 0.50 0.25

rms 200 0 200 13.49 2 27.11 734.95

sms 200 0 200 18.83 4 27.61 762.33

rml 100 0 100 0.58 0 4.2 17.72

sml 100 0 100 1.15 0 5.37 28.89

p 1 0 1 0.12 0 0.32 0.10

pf 35 0 35 0.24 0 1.28 1.64

cc 1 0 1 0.62 1 0.49 0.24

ccf 30 0 30 1.23 0 2.31 5.32

ac 1 0 1 0.09 0 0.29 0.08

acf 10 0 10 0.10 0 0.42 0.18

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Table 4.1.3: Students Not Involved In Crashes Descriptive Statistics for Different Cell Phone Usages (obtained from SPSS 13.5)

N = 2052 Range Minimum Maximum Mean Median Standard deviation Variance

dh 23 0 23 8.22 7 4.89 23.90

c 50 0 50 8.38 5 9.00 80.39

a 50 0 50 7.22 5 7.93 62.89

hf 1 0 1 0.16 0 0.37 0.14

ab 1 0 1 0.42 0 0.50 0.24

rms 200 0 200 13.21 2 27.02 729.84

sms 200 0 200 14.52 4 27.39 750.44

rml 100 0 100 0.57 0 4.3 18.77

sml 100 0 100 1.06 0 5.12 26.24

p 1 0 1 0.11 0 0.31 0.10

pf 35 0 35 0.23 0 1.31 1.72

cc 1 0 1 0.60 1 0.49 0.24

ccf 30 0 30 1.13 0 2.19 4.81

Note: All 2051 responders has ac = no

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Table 4.1.4: Students Involved in Crashes Descriptive Statistics for Different Cell Phone Usages (obtained from SPSS 13.5)

N = 201 Range Minimum Maximum Mean Median Standard deviation Variance

dh 23 0 23 7.80 7 4.77 22.71

c 50 0 50 9.06 6 10.07 101.42

a 50 0 50 7.92 5 9.41 88.55

hf 1 0 1 0.13 0 0.34 0.12

ab 1 0 1 0.47 0 0.50 0.25

rms 150 0 150 16.37 1.4 27.96 781.91

sms 150 0 150 18.00 5 29.61 876.92

rml 20 0 100 0.64 0 2.65 7.01

sml 75 0 100 2.01 0 7.37 54.28

p 1 0 1 0.19 0 0.39 0.15

pf 5 0 35 0.37 0 0.91 0.83

cc 1 0 1 0.83 1 0.38 0.15

ccf 20 0 30 2.23 1 3.08 9.47

acf 10 0 10 1.09 1 0.94 0.89

Note: All 201 responders has ac = yes