Question

. In his quest to test whether interval times (x minutes) between arrivals of vehicles at a busy transport terminal are exponentially distributed with parameter β, an investigator collected one hundred such interval times as shown in the table below Time Interval (mins) 0 – 4.99 5 – 9.99 10 – 14.99 15 – 24.99 25 – 39.99 40 Cars 22 15 18 16 11 18 Note that x¯ = 10 a. State the null and the alternative hypotheses for this problem. [2 Marks] b. What is the maximum likelihood estimate of β under the null hypothesis?. [2 Marks] c. Determine the class probabilities and the expected frequencies under the null hypothesis. [8 Marks] d. Perform an appropriate test for testing your null hypothesis specified in (a) above at significance level of 0.05

Answer 1

mean = = 10

a) Hypothesis:

H₀ : Time Interval (min) Exponential distribution

H₁ : H₀  is Not true

b)

maximum likelihood estimate of  β = 1 /   = 1 / 10 = 0.10

c)

d) Test statistic =

= (22 - 39.32)² / 39.32 + (15 - 23.88)² / 23.88 + (18 - 14.48)² / 14.48 + (16 - 14.11)² / 14.11 +(29 - 8.21)² / 8.21

= 64.68627

(the expected frequency in the last class = 1.83 < 5 , so we combined last 2 classes)

P -value at

0.0000 < 0.05

P - value <

so, we reject the H₀.

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