A transportation safety analyst found a relation between average daily traffic (

Question

A transportation safety analyst found a relation between average daily traffic (ADT) of highways and number of crashes per year. He selected four highways and developed a regression equation. The regression equation is: Number of crashes 2.5 + 9.2 X (your X is actually log^ADT_10) Use the data table for answering parts in below a) Find the 90% prediction interval for the predicted mean number of crashes on a highway X = 4. Assuming S_e = 0.99 and S_x = 0.59 b) Is this interval likely to be useful

Explanation / Answer

(a) 90% prediction interval for X= 4

so predicted value y or say number of crashes when X = 4 say average daily traffic ADT = 104 = 10000

y^ = 2.5+ 9.2 * 4 = 39.3

so 90% prediction interval = y^ +- tcrit .se

where se = standard error of prediction = sy.x sqrt [ 1 + 1/n + (x0 -xbar)2 /SSx]

Here x0 = 4 and xbar = 3.845 and SSx = 0.59 ; syx = 0.99

se = 0.99 * sqrt [ 1 +1/4 + (4 - 3.845)2 /0.59]

se = 0.99 * 1.136 = 1.125

here tcritical = 2.920 for alpha = 0.1 and dF = n-2 = 2

90% prediction interval = y^ +- tcrit .se = 39.3 +- 2.920 * 1.125

= (36.015, 42.585)

(b) Role of prediction interval

A prediction interval is an estimate of an interval in which future observations will fall, with a certain probability, given what has already been observed. So here there is total certainty not like a confidence interval which are certain for some perticular percentage.So, now we are sure that any future value of X will given number if crashes values in between this interval. So, numbe of traffic density if 10000, then numbe of crashes will be in between 36 and 42.

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