Chapter 7 – Estimation of Population Means
Confidence Intervals for µ
We now move into a statistical topic called
“statistical inference”
• the goal of inference is to draw
conclusions (make inferences) about a
population from a sample
• often, we are interested in obtaining
information about µ (and sometimes σ) For example, we might want to know:
1. The mean tar content of a certain brand
of cigarette
2. The mean lifetime of a newly developed
steel-belted radial tire
3. The mean gas mileage of a new model
car
4. The mean annual income of liberal-arts
graduates
5. The mean height of male UWO
undergraduate students If the population is small, then we can
calculate µ directly from the small number
of observations in the population
• however, if the population if large, as it
often is in practice, it is not practical to
calculate µ (sometimes it is impossible)
What Do We Do?
• we gather data
• take a random sample from the population
that we wish to draw conclusions about
• calculate x (and s)
x
• it seems logical to use to estimate µ
(and s to estimate σ) We will start with the simplest case:
Inference about µ from a normal population,
where σ is known (section 7.1)
Example 7.1
A large hospital wishes to estimate the
average length of time patients remain in the
hospital.
The hospital’s administrators randomly
select the records of 49 previous patients
and calculate x to be 4.53 days.
Previous research on this subject has shown
σ = 3.68 days and that the lengths of stay are
normally distributed.
What can you say about µ? Note
For now, we will assume that σ is known
(either from a previous study or from
historical records)
• we will cover the more usual case, where
σ is unknown, later
Solution to 7.1
1. x = 4.53 is called a point estimate of µ
2. Why is x not equal to µ? Suppose the administrators took several
samples:
Sample x
1 4.53
2 4.82
3 5.45
4 3.98
All four of these numbers are point
estimates of µ
• but they vary, they are not all the same
However, in real life, you only get to take
one sample
• since you know that x is not equal to µ, it
may be better to calculate a range, or
interval, of values, centred around
• this interval is called a confidence interval
(CI) for µ • it allows us to take into consideration the
variability ofx
• it will also indicate how accurate is the
point estimate, x , that we have calculated
We hope that the CI contains the true, but
unknown, value of µ
• we will state how confident we are that µ
lies in the interval
• this is given by the confidence level
(typical levels are 90%, 95%, 99%) Let’s look for a 95% CI for µ
We have:
x = length of stay for a patient
x ~ N(µ, 3.68)
3.68
x ~ N(µ, 49 = 0.525)
You might think the appropriate interval
is:
x ± std. error ofx
→ 4.53 ± 0.5257
But we want to be 95% certain that th

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