Chapter 8
Chapter 8
Confidence Intervals
Aims
In this chapter we deal with:
Constructing confidence intervals by hand
The level of confidence
The width of the confidence interval
Sample size issues
The margin of error
Interpreting confidence intervals
By the end of this chapter you should:
be able to state the 4 parameters that we build confidence intervals for in this chapter
know how to construct a confidence interval by hand for each of these 4 parameters
be able to interpret a confidence interval in plain English
understand the nature of confidence levels
what is meant by the level of confidence
what affects the width of a confidence interval
what is meant by the margin of error
Page 1
Single Mean,
μ
Part Time Work I
What is the average number of hours worked per week by all those Stage 1 Statistics students
who have a part time job through the semester?
Hours worked per week:
25
12
10
20
13
12
13
4
8
6
15
12
(12 students)
(Source:
Stage 1 Statistics online survey
)
Method:
Use the sample data, to produce a range of believable values for
μ
, i.e., an interval estimate.
Assume the 12 observations form a random sample from a population with mean
μ
.
Use the sample mean,
x
= 12.5 hours as a (point) estimate for
μ
.
This estimate is based upon a particular sample; another sample with 12 different
students would give a different value for the estimate, i.e., this estimate value varies
from sample to sample.
We say ‘this estimate is subject to sampling variability’.
Place a
t-standard-error
interval around the estimate to allow for the uncertainty due to
the sampling variability:
estimate
±
t
×
se(
estimate
)
Set this method’s success rate at
95%
(i.e.,
very likely
to work)
Assume the data come from a Normal distribution, i.e,, the population has a Normal
distribution
Use the Student’s
t
-distribution to determine
how many
standard errors (the
‘
t
-multiplier’) to use in constructing the interval.
Summary statistics:
x
=
12.5
hours
s
=
5.73
hours
n
= 12
Population
(Stage 1 Statistics students
who have part time work)
Sample
(12 students)
X
= hours worked
μ
X
?
(parameter)
x
(estimate)

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