Chapter 7(7b): Statistical Applications in Traffic Engineering

1. Chapter 7(7b): Statistical Applications in Traffic Engineering Chapter objectives: By the end of these chapters the student will be able to (We spend 3 lecture periods for this chapter. We do skip simple descriptive stats because they were covered in CE361.):

2. 7.7 The Poisson distribution (“counting distribution” or “Random arrival” discrete probability function) With mean µ = m and variance 2 = m. If the above characteristic is not met, the Poisson theoretically does not apply. • The binomial distribution tends to approach the Poisson distribution with parameter m = np. Also, the binomial distribution approaches the normal distribution when np/(1-p)>=9 • When time headways are exponentially distributed with mean  = 1/, the number of arrivals in an interval T is Poisson distributed with mean = m = T. Note that the unit  is veh/unit time (arrival rate). (Read the sample problem in page 144, table 7.5)

3. 7.8 Hypothesis testing Two distinct choices: Null hypothesis, H0 Alternative hypothesis: H1 E.g. Inspect 100,000 vehicles, of which 10,000 vehicles are “unsafe.” This is the fact given to us. H0: The vehicle being tested is “safe.” H1: The vehicle being tested is “unsafe.” In this inspection, 15% of the unsafe vehicles are determined to be safe Type II error (bad error) and 5% of the safe vehicles are determined to be unsafe  Type I error (economically bad but safety-wise it is better than Type II error.)

4. Types of errors We want to minimize especially Type II error. Decision Reality Steps of the Hypothesis Testing Reject H0 Accept H0 H0 is true Type I error Correct • State the hypothesis • Select the significance level • Compute sample statistics and estimate parameters • Compute the test statistic • Determine the acceptance and critical region of the test statistics • Reject or do not reject H0 Correct Type II error H1 is true Fail to reject a false null hypothesis Reject a correct null hypothesis (see the binary case in p. 145/146. to get a feel of Type II error.) P(type I error) =  (level of significance) P(type II error ) = 

5. Dependence between , , and sample size n There is a distinct relationship between the two probability values  and  and the sample size n for any hypothesis. The value of any one is found by using the test statistic and set values of the other two. • Given  and n, determine . Usually the  and n values are the most crucial, so they are established and the value is not controlled. • Given  and , determine n. Set up the test statistic for  and  with H0 value and an H1 value of the parameter and two different n values. Here we are comparing means; hence divide σ by sqrt(n). The t (or z) statistics is: t or z 7.8.1 Before-and-after tests with two distinct choices

6. 7.8.2 Before-and-after tests with generalized alternative hypothesis • The significance of the hypothesis test is indicated by , the type I error probability.  = 0.05 is most common: there is a 5% level of significance, which means that on the average a type I error (reject a true H0) will occur 5 in 100 times that H0 and H1 are tested. In addition, there is a 95% confidence level that the result is correct. 0.025 each • If H1 involves a not-equal relation, no direction is given, so the significance area is equally divided between the two tails of the testing distribution. Two-sided • If it is known that the parameter can go in only one direction, a one-sided test is performed, so the significance area is in one tail of the distribution. 0.05 One-sided upper

7. Two-sided or one-sided test These tests are done to compare the effectiveness of an improvement to a highway or street by using mean speeds. • If you want to prove that the difference exists between the two data samples, you conduct a two-way test.(There is no change.) • If you are sure that there is no decrease or increase, you conduct a one-sided test. (There was no decrease) Null hypothesis H0: 1 = 2 (there is no change) Alternative H1: 1 = 2 Null hypothesis H0: 1 = 2 (there is no increase) Alternative H1: 1  2

8. Example The decision point (or typically zc: • For two-sided: 1.96*1.53 = 2.998 • For one-sided: • 1.65*1.53 =2.525 |µ1 - µ2| = |60-55| = 5 > zc By either test, H0 is rejected. At significance level  = 0.05 (See Table 7-3.)

9. 7.8.3 Other useful statistical tests The t-test (for small samples, n<=30) – Table 7.6: The F-test (for small samples) – Table 7.7: In using the t-test we assume that the standard deviations of the two samples are the same. To test this hypothesis we can use the F-test. (By definition the larger s is always on top.) (See the samples in pages 149 and 151.

10. 7.8.3 Other useful statistical tests (cont) The F-Test to test if s1=s2 When the t-test and other similar means tests are conducted, there is an implicit assumption made that s1=s2. The F-test can test this hypothesis. The numerator variance > The denominator variance when you compute a F-value. If Fcomputed≥ Ftable(n1-1, n2-1, a), then s1≠s2 at a asignificance level. If Fcomputed < Ftable (n1-1, n2-1,a), then s1=s2 at a asignificance level. Discuss the problem in p.151.

11. Paired difference test You perform a paired difference test only when you have a control over the sequence of data collection. e.g. Simulation  You control parameters. You have two different signal timing schemes. Only the timing parameters are changed. Use the same random number seeds. Then you can pair. If you cannot control random number seeds in simulation, you are not able to do a paired test. Table 7-8 shows an example showing the benefits of paired testing  The only thing changed is the method to collect speed data. The same vehicle’s speed was measure by the two methods.

12. Paired or not-paired example (table 7.8) H0: No increase in test scores (means one-sided or one-tailed) Not paired: Paired: |56.9 – 61.2| = 4.3 < 4.54 (=1.65*2.74) Hence, H0 is NOT rejected. 4.3 increase > 0.642 (=1.65*0.388) Hence, H0 is clearly rejected.