Question

A school district is trying to decrease the uncertainty concerning pick-up times on a particular school bus route. Due to uncertainty in traffic patterns, the school district examines two alternative routes. In a sample of 15 days on alternative route #1, the average driving time for the school bus is 40 minutes with a standard deviation of 15 minutes. In a sample of 16 days using alternative route #2, the average drive time is 44 minutes with a standard deviation of 11 minutes.

At the 5% level of significance, is there a difference between the driving times of the two bus routes. Show all work on how you come to your conclusion and provide a clear explanation as if you were explaining the results to the school district.

Answer 1

Solution

The solution is based on two-sample t-test.

Let X = pick-up time (minutes)in route #1

      Y = pick-up time (minutes)in route #2

Let mean and standard deviation of X be respectively µ₁ and σ₁ and those of Y be µ₂ andσ₂, where σ₁² = σ₂² = σ², say and σ² is unknown.

Hypotheses:

Null: H₀: µ1 = µ2 Vs Alternative: H_A: µ1 ≠ µ2

Test Statistic:

t = (Xbar - Ybar)/[s√{(1/n₁) + (1/n₂)}]

where

s² = {(n₁ – 1)s₁² + (n₂ – 1)s₂²}/(n₁ + n₂ – 2);

Xbar and Ybar are sample averages and

s₁,s₂ are sample standard deviations based on n₁ observations on X and n₂ observations on Y respectively.

Calculations

Summary of Excel calculations is given below:

n1	15
n2	16
Xbar	40
Ybar	44
s1	15
s2	11
s2	171.2069
s	13.08461
tcal	0.850598
α	0.05
tcrit	2.04523
p-value	0.401962
(n1 + n2 - 2)	29
2α	0.1
n1 -1	14
n2 -1	15
s12	225
s22	121
(n1 - 1)s12	3150
(n2 - 1)s22	1815
sum	4965
n1 + n2-2	29
1/n1	0.066667
1/n2	0.0625
sum	0.129167
sqrt	0.359398
s.sqrt (SE)	4.702576

Distribution, Significance Level, α Critical Value and p-value:

Under H₀, t ~ t_{n1 + n2 - 2}. Hence, for level of significance α%,

Critical Value = upper (α/2)% point of t_{n1 + n2 - 2} and

p-value = P(t_{n1 + n2 - 2} > | tcal |)

Using Excel Function: Statistical TINV andTDIST, these are found to be as shown in the above table.

Decision:

Since | tcal | < tcrit, or equivalently since p-value > α, H₀ is accepted.

Conclusion:

There is not sufficient evidence to suggest that the mean pick-up times differ.

Hence, conclude that

there is no difference between the driving times of the two bus routes. Answer 1

DONE