1. Spring 2013

2. Sampling Frame Sampling frame: the sampling frame is the list of the population (this is a general term) from which the sample is drawn. It is important to understand how the sampling frame defines the population represented. Example: If the study seeks to identify the safety effects of traffic signals, the sample frame should include a sample of signalized intersections in a given geographical area. If a control group is included, the sampling frame will include sites categorized under this group. Signalized Unsignalized Sig Int #1 Sig Int #2 Unsig Int #1 Unsig Int #2 Sig Int #9 Unsig Int #7

3. Sampling Frame Map crashes for Year 1 Map crashes for Year 2

4. Sampling Frame 03105 2071 1420 11263 Number of Crashes for Year 1 Number of Crashes for Year 2 10810 5120 4613 6037

5. Sampling Frame Intersection Number Crashes/YearTraffic Flow – Major Other Site Characteristics* Year 1011,5001 2312,0001 31010,0001 …………1 966,3001 1112,0002 2012,2002 …………2 936,1002 Signalized Intersections Database * ex: Nb of lanes, actuated signals, exclusive left-turn lane, etc.

6. Sampling Frame Signalized Intersections Database 0 1 Crash Count Year 1 2 Intersection 1 6 3 Crash Count Year 1 2 Intersection 9

7. Sampling Frame Intersection Number Crashes/YearTraffic Flow – Major Other Site Characteristics* Year 128,4001 209,0001 318,5001 …………1 737,9001 158,6002 219,4002 …………2 977,8002 Unsignalized Intersections Database * ex: Nb of lanes, actuated signals, exclusive left-turn lane, etc.

9. Source: Washington et al. (2003)

14. Crash Severity / Flow Range < 5,0005,000-9,999≥ 10,000 Fatal101215 Non-Fatal Injury 100120135 PDO550700900

15. Maps

16. Maps – GIS Information http://www.saferoadmaps.org/home/

17. Confidence Intervals Statistics are usually calculated from samples, such as the sample average X, variance s 2, the standard deviation s, are used to estimate the population parameters. For instance: X is used as an estimate of the population μ x s 2 is used as an estimate of the population variance σ 2 Interval estimates, defined as Confidence Intervals, allow inferences to be drawn about the population by providing an interval, a lower and upper value, within which the unknown parameter will lie with a prescribed level of confidence. In other words, the true value of the population is assumed to be located within the estimated interval.

18. Confidence Intervals Confidence Interval for μ and known σ 2 95% CI 90% CI Any CI

19. Confidence Intervals Compute the 95% confidence interval for the mean vehicular speed. Assume the data is normally distributed. The sample size is 1,296 and the sample mean X is 58.86. Suppose the population standard deviation (σ) has previously been computed to be 5.5.

20. Confidence Intervals Compute the 95% confidence interval for the mean vehicular speed. Assume the data is normally distributed. The sample size is 1,296 and the sample mean X is 58.86. Suppose the population standard deviation (σ) has previously been computed to be 5.5. Answer

21. Confidence Intervals Confidence Interval for μ and unknown σ 2 95% CI 90% CI Any CI Only valid if n > 30

22. Confidence Intervals Same example: Compute the 95% confidence interval for the mean vehicular speed. Assume the data is normally distributed. The sample size is 1,296 and the sample mean X is 58.86. Now, suppose a sample standard deviation (s) has previously been computed to be 4.41. Answer

23. Confidence Intervals Confidence Interval for a Population Proportion The relative frequency in a population may sometimes be of interest. The confidence interval can be computed using the following equation: Where, p is an estimator of the proportion in a population; and, q = 1 – p. Normal approximation is only good when np > 5 and nq > 5. ^ ^ ^

24. Confidence Intervals A transportation agency located in a small city is interested to know the percentage of people who were involved in a collision during the last calendar year. A random sample is conducted using 1000 drivers. From the sample, it was found that 110 drivers were involved in at least one collision. Compute the 90% CI.

25. Confidence Intervals A transportation agency located in a small city is interested to know the percentage of people who were involved in a collision during the last calendar year. A random sample is conducted using 1,000 drivers. From the sample, it was estimated that 110 drivers were involved in at least one collision. Compute the 90% CI. Answer

27. Confidence Intervals Confidence Interval Population Variance When the population variance is of interest, the confidence interval can be computed using the following equation: Where, X 2 is Chi-Square with n-1 degrees of freedom Assumption: the population is normally distributed.

28. Confidence Intervals Taking the same example before on the vehicular speed, compute the confidence interval (95%) for variance for the speed distribution. A sample of 100 vehicles has shown a variance equal to 19.51 mph.

29. Confidence Intervals Taking the same example before on the vehicular speed, compute the confidence interval (95%) for variance for the speed distribution. A sample of 100 vehicles has shown a variance equal to 19.51 mph. Answer Taken from Chi- Square Table

30. The Chi-Square Goodness-of -fit Non-parametric test useful for observations that are assumed to be normally distributed. Need to have more than 5 observations per cell. The test statistic is If the value on the right-hand side is less than the Chi-Square with n-1 degrees of freedom, the observed and estimated values are the same. If not, the observed and estimated values are not the same. You can also perform this test for two-way contingency tables.