Section 9: Confidence Intervals

Introduction to Probability & Statistics

Dr. Oliver Russell

201 - SN1

Lectures 1 & 2: Sections 9.1-9.2

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Target parameter

Definition
The unknown population parameter (e.g., mean or proportion) that we are
interested in estimating is called the target parameter.

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Point vs. interval estimator

Definition
A point estimator of a population parameter is a rule or formula that
tells us how to use the sample data to calculate a single number that can
be used as an estimate of the target parameter.

Definition
An interval estimator is a formula that tells us how to use the sample
data to calculate an interval that estimates the target parameter.

Definition
A confidence interval is an interval estimator with a level of reliability (or
certainty) attached to it.

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zα/2

Definition
The value zα/2 is defined as the value of the standard normal random
variable such that the area α/2 will lie to its right. In other words,
P(Z > zα/2 ) = α/2.

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Confidence coefficient and confidence interval

Definition
The confidence coefficient, (1 − α), is the probability that an interval
estimator encloses the population parameter—that is, the relative
frequency with which the interval estimator encloses the population
parameter when the estimator is used repeatedly a very large number of
times. The confidence level, (1 − α)% is the confidence coefficient
expressed as a percentage.

For example, a confidence interval for µ with confidence coefficient

(1 − α) could look like
x̄ ± zα/2 σx̄ .

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Example: confidence interval for µ

Example: if our confidence level is 90%, then in the long run, 90% of our
sample confidence intervals will contain µ. In this case, α = 0.10. Thus,
x̄ ± 1.645σx̄ is a 90% confidence interval for µ.

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Important values of zα/2

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RECAP: Central Limit Theorem (CLT)

Theorem
Consider a random sample of n observations selected from any population
with mean µ and standard deviation σ. Then, when n is sufficiently large,
the sampling distribution of X̄ will be approximately normal with mean

µX̄ = µ

and standard deviation

σ
σX̄ = √ .
n

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RECAP: Sampling distribution of X̄

Thus, the CLT says that for large enough n, approximately,

σ2
 
X̄ ∼ N µ, .
n

or, equivalently,
X̄ − µ
√ = Z ∼ N (0, 1) .
σ/ n
The larger n is, the closer X̄ becomes to a true normal distribution. For
most sampled populations, sample sizes of n ≥ 30 will suffice for the
normal approximation to be reasonable.

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RECAP: Small samples: problems and solutions
We face 2 problems when dealing with small sample sizes:
1 If the sample size, n, is not large enough, then the normal

approximation of X̄ may not hold.

Solution: if we assume the population is approximately normal, then
X̄ will also be approximately normal.

2 When estimating µ, we must often approximate σ using s, but for

small samples s may not be a good approximation for σ.
X̄ −µ
Solution: instead of using the standard normal statistic Z = σ
√ ,
n
which relies on either knowing σ or having a close approximation to
σ, we will define and use a new statistic,
X̄ − µ
Tn−1 =
√S
n

where we choose to use the sample standard deviation, S.

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RECAP: Student’s T -statistic

Definition
For any sample of size n randomly drawn from an approximately normal
population with mean µ, the Student’s T -statistic with (n − 1)
degrees of freedom (df) is defined as

X̄ − µ
Tn−1 =
√S
n
q Pn
2
i=1 (Xi −X̄ )
where we recall that S = n−1 is the sample standard deviation.

Note: this also applies if the population is not normally distributed, but
the sample size n is large enough.

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RECAP: T probabilities: table (See Appendix B)

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Quick quiz
Suppose you wish to estimate the mean µ of a population of interest by
taking a random sample of size n. Using the Student’s T -statistic, derive
a 100(1 − α)% confidence interval for µ.

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Quick quiz

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Summary: confidence interval for µ

Assuming we have either an approximately normal population or the

sample size n is large enough, we can compute a

100(1 − α)% confidence interval for µ based on a T -statistic:

√
x̄ ± (tα/2 )(s/ n)

where tα/2 is the T -value corresponding to an α/2 area in the upper tail
of the T -distribution with (n − 1) df, and s is the standard deviation of
the sample.

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Quick quiz
A manufacturer of printers wishes to estimate the mean number of
characters printed before the printhead fails. The printer manufacturer
tests n = 15 printheads and finds x̄ = 1.24 million characters printed until
failure and s = 0.19 million characters printed until failure. Form a 99%
confidence interval for the mean number of characters printed before the
printhead fails. Interpret the result.

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RECAP: population vs. sample proportions

Definition
When discussing data which only have 2 potential outcomes (say, success
or failure), the binomial proportion of a population, p, is the
population’s proportion of successes.

Definition
The sample proportion, P̂, is a random variable representing the
proportion of successes in a randomly drawn sample.

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RECAP: corollary of CLT for proportions

Corollary
By the CLT, if a sample size is large enough, then it turns out that the
random variable P̂ is also approximately normally distributed with mean

µP̂ = p

and standard deviation r

p(1 − p)
σP̂ = .
n

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RECAP: sampling distribution of P̂

In other words, if n large enough, then approximately,

 
p(1 − p)
P̂ ∼ N p, .
n

or, equivalently,
P̂ − p
p ∼ N (0, 1) .
p(1 − p)/n
Here, large enough means n ≥ 30, np ≥ 10 and n(1 − p) ≥ 10.

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Quick quiz
Suppose you wish to estimate the binomial proportion p of a population of
interest by taking a random sample of size n. Using the standard normal
Z -statistic, derive a 100(1 − α)% confidence interval for p.

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Quick quiz

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Summary: large-sample (n ≥ 30, np ≥ 10 and
n(1 − p) ≥ 10) confidence interval for p
Large-sample 100(1 − α)% confidence interval for p, based on a
standard normal (Z ) statistic:
r
p̂(1 − p̂)
p̂ ± zα/2
n

where p is the binomial proportion of the population and p̂ is the binomial

proportion of the sample.

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Quick quiz
The Bureau of Economic and Business Research (BEBR) conducts
quarterly surveys to gauge consumer sentiment in Florida. Suppose that
BEBR randomly samples 484 consumers and finds that only 157 are
optimistic about the state of the economy. Use a 90% confidence interval
to estimate the proportion of all consumers in Florida who are optimistic
about the state of the economy. What is the 90% margin of error?

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Margin of error

Definition
The 100(1 − α)% margin of error (or sampling error), ∆, is half of
the width of the confidence interval.

Examples:
For a confidence interval for µ, the 95% margin of error is
∆ = t0.025 √sn .
For a confidence
q interval for p, the 90% margin of error is
p̂(1−p̂)
∆ = z0.05 n .

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Quick quiz (Example 9.1.1)

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Quick quiz (Example 9.1.1)

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Quick quiz (Example 9.2.1)

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Quick quiz (Example 9.2.1)

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Quick quiz (Example 9.1.2)

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Quick quiz (Example 9.1.2)

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Quick quiz

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Quick quiz

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