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)
• it seems logical to usex 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 of
• it will also indicate how accurate is the point estimate,, 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)
x ~ N(μ, 3.68 = 0.5257)
49
You might think the appropri

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