Lec 7, Ch4, pp83-99: Spot Speed Studies (Objectives) Know when you need to conduct a speed study and where you should do it Learn how spot speed data can be collected (from the reading) Know how to determine the sample size needed Know how to reduce the collected data Know how to compare mean speeds What we cover in class today… Use of spot speed studies and time and locations to choose Methods for conducting spot speed studies The formula to determine the sample size How to compute descriptive statistics How to compare two mean speeds When are spot speed studies needed? Spot speed studies are conducted to estimate the distribution of speeds of vehicles in a stream of traffic at a particular location on a highway. Speed Limit Used for: 50 Establish speed zones Determine whether complaints about speeding are valid Establish passing and no-passing zones Design geometric alignment Analyze accident data Evaluate the effects of physical improvements, etc., etc. Location, time of day, and duration… The objective and scope of the study dictate these. Basic data collection Like deciding speed limits Find locations where system characteristics change and TWTh Speed trend analyses Avoid external influences such as traffic lights, busy access roads; offpeak, TWTh mid blocks of streets, straight,level sections of highways Specific traffic All other specific purposes engineering problems Conduct it at the location of interest and time of day At least 1 hour and at least 30 data (if you want to assume normal distribution) Descriptive statistics of speed data Once data are collected, the first thing you do is to compute several descriptive statistics to get some ideas about the distribution of the speed data. (Note that many statistical analyses used in traffic engineering assume data are normally distributed. So, the goal is to check whether they are really normally distributed. Typical descriptive statistics are: Average speed Variance and standard deviation Median speed Modal speed (or Modal speed range Needs a histogram) The ith-percentile spot speed Pace Usually a 10-mph interval that has the greatest number of observations. Descriptive statistics (cont.) Average speed Speed data Grouped Not grouped u = uj/N Standard deviation Speed data Grouped s= f(ui – u)2 N-1 Variance s2 Not grouped Descriptive statistics (cont) Median speed The speed at the middle value in a series of spot speeds. Or, 50th-percentile speed Modal speed The speed value that occurs most frequently in a sample of speeds ith-percentile speed The spot speed below which i percent of the vehicles travel, e.g. 85th-percentile speed Pace The range of speed that has the greatest number of observations; usually 10-mph range 85% 50% Determining the sample size… (p.89) Need to know a bit of statistical principles here… It’s all based on the normal distribution curve. d N = Min. sample size Z α/2 = 1.96 for 95% conf. Level 68.3% 2 Frequency N= Zα/2 95.0% = Estimated standard deviation Rural 2-lane: 5.3 mph Rural 4-lane: 4.2 mph -/sqrt(n) +/sqrt(n) -1.96/sqrt(n) +1.96/sqrt(n) Urban 2-lane: 4.8 mph Urban 4-lane: 4.9 mph d = Precision level (depends on the study) At 95% Confidence level, Z = 1.96 (This is the most important number to remember.) Example 4.2 by spreadsheet (Click Example 4.2 in the lecture schedule.) Comparing two mean speeds This test is 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 twoway test. We discuss only this one in this class. Null hypothesis H0: 1 = 2 Alternative H1: 1 = 2 If you are sure that the improvement would improve the performance of the highway or street, you conduct a one-way test. This will be discussed in CE562. Alternative H1: 1 2 Comparing two mean speeds (cont) Step 1: Find mean and SD of the two samples u1 = 35.5 mph S1 = 7.5 mph n1 = 250 u2 = 38.7 mph S2 = 7.4 mph n2 = 280 Step 2: Compute the standard deviation of the difference in means (Assumes S12 and S22 are similar) Sd = SQRT(S12/n1 + S22/n2) = SQRT(7.52/250 + 7.42/280) = 0.65 Step 3: Test the hypothesis. If |u1- u2| > ZSd, the mean speeds are significantly different at the confidence level of Z. |35.5 – 38.7| = 3.2 mph > 1.96*0.65 = 1.3mph It can be concluded that the difference in mean speeds is significant at the 95% confidence level.