TP%2317_f08 - CE 3102 Fall 2008 Thought Problem 17 Signal...

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CE 3102 Fall 2008 Thought Problem 17 Signal Timing, and Estimating Means In timing a traffic signal, the duration of a yellow interval should be such that it eliminates the dilemma zone, where a driver can neither stop without entering the intersection, nor clear the intersection before the signal turns red. The yellow interval ( τ ) which accomplishes this sets the stopping distance d s equal to the maximum clearing distance d c , or L W v a v vt p = + 2 2 where v is vehicle speed, a is the braking deceleration, t p is the driver's reaction time, W is the width of the cross street and L is the vehicle's length. v L W a v t p + + + = 2 For a particular intersection, the width of the cross-street ( W ) is easy to obtain, as are recommended default values for reaction time, deceleration, and vehicle length , and it is often recommended that the mean approach speed be used for v . A traffic engineer has obtained the following individual vehicle speed measurements (in mph) on an approach for an intersection: 33, 32, 33, 30, 27, 35, 34, 38, 46, 39 (1) How should the engineer estimate the mean approach speed? (2) Can the engineer be "reasonably certain" that this estimate is within 1 mph of the true mean speed?
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CE 3102 Fall 2008 Thought Problem 17 Instructor’s Partial Solution, part 1 (1) We claim that the sample average mph x n x n i i 7 . 34 1 1 = = = is a "good" estimate of the underlying mean speed. What does this mean? First, let's distinguish between an "estimate" which is a real number obtained by performing the estimating operation on the actual data in our sample, and an "estimator", which is a random variable obtained by performing the estimating operation on a sequence of random variables. That is Observations: x 1 x 2 …. x n Random Variables: X 1 X 2 X n Our sample average as an estimator would be = = n i i X n X 1 1 We then say an estimate is good if it comes from using a "good" estimator. So what makes X a good estimator? Suppose that the X i are independent, and identically (iid) distributed random variables, with common mean (expected value) μ and common variance σ 2 . Since X is also a random variable, it too has an expected value and a variance. More particularly, = = = = = = n i i n i i n n X E n X n E X E 1 1 ) ( 1 1 ) ( μ In words, the expected value of the sample average (as a random variable) equals the
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TP%2317_f08 - CE 3102 Fall 2008 Thought Problem 17 Signal...

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