Page 98 - Design for Six Sigma for Service (Six SIGMA Operational Methods)

Customer Survey Design, Administration, and Analysis 79

of population proportion p, where p is the real proportion of customers who
like ABC Bank’s service. Of course, we would like p ˆ as close to p as
possible. From the properties of the normal distribution, and if the random
sampling method is used, the probability distribution of p ˆ is

⎛ ( p 1− ) p ⎞
p ˆ ~ N p , n ⎠ (4.1)
⎝

The 100(1 − α)% confidence interval for p is
( p − ) p ˆ( p − ˆ) p
1
1
ˆ p ± Z ≈ ˆ p ± Z (4.2)
a
/2
n a /2 n
ˆ(1
We can use ∆ = Z a/2 p −(1 n p) ≈ Z a/2 p − ˆ p) to represent the half width of
p
n
the confidence interval for p. The magnitude of ∆ represents the accuracy
p
of ˆ p as an estimator of p, because
=
Pp ˆ − ∆ p ≤ p ≤ p ˆ + ∆ p )( −a )100 % (4.3)
1
(
∆ is also called the margin of error.
p
Example 4.4
In a customer satisfaction survey, the preliminary results indicate that the pro-
portion of unsatisfied customers is very close to the proportion of satisfied
customers. What sample size is needed if we want the accuracy of the survey to
be within ±3 percent of the true proportion, with 95% confidence?
In this case, clearly p ≈ 50%, from the problem, statement, we want

p − p)
(1
∆ = Z = %3
p a /2
n
Therefore
p ⎞
⎛ Z p − ) 2
1 (
n = ⎜ a 2 ⎟ (4.4)
⎝ ∆ p ⎠
is the sample size formula for this case. Specifically,
⎛ 196 × 0 5 1 05 . ) ⎞ 2
.
. (
−
n = ⎜ ⎟ = 1067
.
⎝ 003 ⎠
for this example, where Z 0.025 = 1.96.
So a sample of 1067 or more people is needed to ensure the accuracy of ±3 percent.