Page 100 - Design for Six Sigma for Service (Six SIGMA Operational Methods)

Customer Survey Design, Administration, and Analysis 81

Example 4.5
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, and the population size is N = 2500. What sample size is needed if
we want the accuracy of the survey to be within ±3 percent of the true pro-
portion, with 95% confidence?
Using Eq. (4.5)

×
2
.
.
p − )]1
.
Z 2 [( p N 1 96 × ( 0 5 0 5 2500)
)(
n = a /2 = = 749
×
×
2
2
p − )] (1
.
.
.
.
Z a /2 [( p + N − )1 ∆ 2 p 1 96 × ( 0 5 0 5) + 2499 ( 0 03) 2
This sample size is smaller than that of Example 4.4.
Determination of Sample Size for Interval-Scale Variables
In survey analysis, some variables are interval-scale variables. For example,
personal income, age, and evaluation scores based on the Likert scale are all
interval-scale variables. The population means of these interval-scale variables
m are usually of interest. The sample mean of the interval-scale variable x is
often used as the statistical estimate of population mean m. Similarly, we would
like m as close to m as possible. From the properties of the normal distribution,
and if the random sampling method is used, the probability distribution of x is
⎛ s ⎞
2
x ~ N m , ⎟ (4.6)
⎜
⎝ n ⎠
The 100(1 − a)% confidence interval for µ is
s
x ± Z = x ± ∆
a/2 m (4.7)
n
where ∆ is the margin of error for µ.
m
By using the relationship
s
∆ = Z a/2 (4.8)
m
n
we can derive the sample size rule:
Z s 2
2
n = a (4.9)
∆ 2 m