Introduction to Probability and Statistics for Engineers and Scientists

8.3Tests Concerning the Mean of a Normal Population 311

EXAMPLE 8.3j In a single-server queueing system in which customers arrive according to a
Poisson process, the long-run average queueing delay per customer depends on the service
distribution through its mean and variance. Indeed, ifμis the mean service time, andσ^2
is the variance of a service time, then the average amount of time that a customer spends
waiting in queue is given by

λ(μ^2 +σ^2 )

2(1−λμ)

provided thatλμ < 1, whereλis the arrival rate. (The average delay is infinite if
λμ≥1.) As can be seen by this formula, the average delay is quite large whenμis only
slightly smaller than 1/λ, where, sinceλis the arrivalrate,1/λis the average time between
arrivals.
Suppose that the owner of a service station will hire a second server if it can be shown
that the average service time exceeds 8 minutes. The following data give the service times
(in minutes) of 28 customers of this queueing system. Do they indicate that the mean
service time is greater than 8 minutes?

8.6, 9.4, 5.0, 4.4, 3.7, 11.4, 10.0, 7.6, 14.4, 12.2, 11.0, 14.4, 9.3, 10.5,

10.3, 7.7, 8.3, 6.4, 9.2, 5.7, 7.9, 9.4, 9.0, 13.3, 11.6, 10.0, 9.5, 6.6

SOLUTION Let us use the preceding data to test the null hypothesis that the mean service
time is less than or equal to 8 minutes. A smallp-value will then be strong evidence
that the mean service time is greater than 8 minutes. Running Program 8.3.2 on these
data shows that the value of the test statistic is 2.257, with a resultingp-value of .016.
Such a smallp-value is certainly strong evidence that the mean service time exceeds
8 minutes. ■

Table 8.2 summarizes the tests of this subsection.

TABLE 8.2 X 1 ,...,XnIs a Sample from aN(μ,σ^2 )Populationσ^2 Is UnknownX =

∑n

i= 1

Xi/n

S^2 =
∑n
i= 1

(Xi−X)^2 /(n−1)

Test Significance p-Value if

H 0 H 1 StatisticTS LevelαTest TS=t

μ=μ 0 μ=μ 0 √n(X−μ 0 )/S Reject if|TS|>tα/2,n− 1 2 P{Tn− 1 ≥|t|}

μ≤μ 0 μ>μ 0 √n(X−μ 0 )/S Reject ifTS>tα,n− 1 P{Tn− 1 ≥t}

μ≥μ 0 μ<μ 0 √n(X−μ 0 )/S Reject ifTS<−tα,n− 1 P{Tn− 1 ≤t}

Tn− 1 is a t-random variable withn− 1 degrees of freedom:P{Tn− 1 >tα,n− 1 }=α.