Transcript Confidence Intervals

STAT 111 Introductory Statistics
Lecture 9: Inference and Estimation
June 2, 2004
Today’s Topics
• Introduction to statistical inference
• Point Estimation
• Confidence Intervals
Introduction
• The application of the methods of probability to
the analysis and interpretation of empirical data
is known as statistical inference.
• More specifically, it is the process by which we
generalize from a particular sample to the
theoretical population from which the sample
came.
Introduction
• The precise form of the generalization can vary
considerably from situation to situation.
• Possible forms of statistical inference:
– Single numerical estimate
– Range of numerical estimates
– Simple “yes” or “no”
Example
• Suppose the chief programming executive at
ABC is trying to decide which shows to cancel
and which to renew.
• Data might be the day-by-day logs of programs
that are watched by a random sample of families.
• Task: use sample information to estimate total
number of viewers tuned to ABC programs.
Example
• Suppose instead that a zoologist would like
whether a particular species of vampire bat
prefers blood at room temperature or at body
temperature.
• Equal numbers of similar bats into 2 cages; cage
A has blood at room temperature, B at body
temperature.
• He finds bats in A consumed 3% more blood.
Estimation and Hypothesis Testing
• The two previous examples highlight the two
broad areas into which statistical inference is
traditionally divided.
• In the first example, inference is numerical. This
is the area referred to as estimation.
• In the second example, the inference is instead a
“yes” or “no” decision between two conflicting
theories. This is what we call hypothesis testing.
• Both areas have wide applicability.
Parameter Estimation
• In many situations, the family of probability
models describing a phenomenon may be known
(or at least assumed to be known), but the
particular member of the family that best
describes that phenomenon may be unknown.
• Hence, estimating the unknown parameter or
parameters of a presumed data model is usually
one of the steps we have to take in an inference
problem.
Parameter Estimation
• Our usual goal then is to estimate the value of the
(unknown) population parameter based on an
appropriate statistic we observe from a random
sample of that population.
• Two types of estimation exist:
– Point estimation – This is what we meant by a single
numerical estimate.
– Confidence interval – This is what we meant by a
range of numerical estimates.
Point Estimation
• A point estimator draws inference about a
population by estimating the value of an unknown
parameter using a single value or a point.
Population
distribution
Sampling
distribution
parameter
Point
estimator
Point Estimation
• A point estimate summarizes up the value of the
population parameter using a single value.
• Naturally, then, we have some properties for a
point estimate that we would desire in order to
feel comfortable using it.
• What sort of properties should a good (point)
estimator have?
Desirable Properties of Estimators
• Certainly, it seems reasonable to ask as a first
condition that the sampling distribution of our
estimator be somehow “centered” with respect to
the population parameter.
• If this condition is not met, then our point
estimator will tend to be consistently
overestimating or underestimating the value of
the parameter, something that we typically do not
desire.
Desirable Properties of Estimators
• This first condition is what we call unbiasedness.
• In other words, on the average, a good estimator
will be equal to the population parameter it is
estimating.
• Mathematically, if W is an estimator, and θ is the
population parameter being estimated by W, then
W is unbiased if
E (W )   , for all 
Desirable Properties of Estimators
• A second property of a good estimator is
precision. An estimator is said to be precise if its
distribution’s dispersion is small.
• The idea of precision leads to the concept of
efficiency.
• Suppose we have multiple unbiased estimators
for the population parameter. Which one should
we use? Are they all equivalent, or are some
better than others?
Desirable Properties of Estimators
• Formally, let W1 and W2 be two unbiased
estimators for a population parameter θ with
variances Var(W1) and Var(W2), respectively.
• Then W1 is said to be more efficient than W2 if
Var(W1) is less than Var(W2).
• We define the relative efficiency of W1 with
respect to W2 as the ratio Var(W2) / Var(W1).
• Which is the more efficient estimator if this ratio
is less than 1? Greater than 1?
Desirable Properties of Estimators
• Unbiasedness and efficiency lead to the most
basic characterizations of point estimates, but
there are other properties of a statistic and its
sampling distribution that merit examination.
• The first concerns the limiting behavior of the
statistic as the sample size n gets large.
• In some cases, it is possible that the sampling
distribution has some very desirable properties in
the limit that it fails to possess for any finite n.
Desirable Properties of Estimators
• Consistency is one such property of the sampling
distribution that appears in the limit.
• Roughly speaking, an estimator is consistent if, as
n gets large, the probability that our statistic W
lies arbitrarily close to the parameter being
estimated becomes arbitrarily close to 1.
• Two immediate implications of consistency:
– W is asymptotically unbiased
– Var(W) converges to 0
Desirable Properties of Estimators
• The last property we might desire from an
estimator is sufficiency.
• If we draw a sample of size n from some
population with a given distribution, we know
that the sample space is all possible n-tuples.
• An estimator W, then, has the effect of
partitioning this sample space into a set of
mutually exclusive subsets.
Desirable Properties of Estimators
• As an example, suppose we draw two
observations from a discrete distribution on the
non-negative integers, and we define our statistic
W as the mean of these two observations.
• Then, W is observed to be 3 for any one of the
following pairs of observations: (0,6), (1,5),
(2,4), (3,3), (4,2), (5,1). And similarly, W will
equal 2.5 if the outcome of our draws is (0,5),
(1,4), (2,3), (3,2), (4,1), or (5,0).
Desirable Properties of Estimators
• So, in this example, knowing the sample mean W
of our outcome provides the same amount of
information as the actual outcome itself does.
• In other words, W is sufficient for the population
parameter we are trying to estimate.
• A statistic is sufficient if knowing its value gives
us just as much information about the parameter
of interest as knowing the actual sample itself
does.
Example
• Let X1, …, Xn be a simple random sample from a
population with mean µ and variance σ2.
• Suppose the sample size is larger than 1, and let
m be an integer between 1 and n (i.e., 1 < m < n).
• Consider these three estimators for µ:
X1
X1    X m
Xm 
m
X1    X n
Xn 
n
Example
• Which of these estimators is unbiased for µ?
• What are the relative efficiencies of the three
estimators (pairwise comparisons)?
– Based on these results, which estimator is the most
efficient? The least?
Interval Estimation
• An interval estimator draws inference about a
population by estimating the value of an unknown
parameter using a interval
Population
distribution
Sampling
distribution
parameter
Interval
estimator
Confidence Intervals
• A confidence interval has the form
estimate ± margin of error
• The estimate is our guess for the value of the
unknown population parameter.
• The margin of error shows how accurate we
believe our guess is, based on the variability of
the estimate.
Example
• The heights of American female students aged 18
to 24 are approximately normal with mean µ and
standard deviation 2.5. We repeatedly select 100
female students at random. The sample mean X
follows the normal distribution with mean µ and
standard deviation
2.5
 0.25
100
Example
• According to 68-95-99.7 rule, the probability is
about 0.95 that X will be within 0.5 inches(two
standard deviations) of the population mean µ.
• To say that X lies within 0.5 inches of µ is the
same as saying that µ lies within 0.5 inches of X
• So 95% of all samples we take will capture the
true µ in the interval from X  0.5 to X  0.5
Example
• Suppose now we observe a sample with X  63
• Then, for the interval [63 – 0.5, 63 + 0.5] = [62.5,
63.5], we have two possibilities:
– The interval between 62.5 and 63.6 contains the true
µ.
– Our SRS was one of the few samples for which X
is not within 0.5 inches of the true µ. Only 5% of all
samples will give such inaccurate results.
Example
• We say that we are 95% confident that the
unknown mean height of American female
students lies between 62.5 and 63.5.
• This is shorthand for saying “we arrived at these
numbers by a method that gives correct results
95% of the time.”
• It is incorrect to say that there is probability 0.95
that the unknown mean height of American
female students lies between 62.5 and 63.5
Confidence Intervals
• Recall that the sampling distribution of the
sample mean is, for large enough sample sizes,
always at least approximately normal regardless
of the actual probability distribution.
• Suppose we choose an SRS of size n from a
population with unknown mean µ and standard
deviation σ. A level C confidence interval for µ
is
* 
xz
n
Confidence Intervals
• Here, z* is the value on the standard normal curve
with area C between –z* and z*.
• The confidence interval will be exact when the
population distribution is normal, and thanks to
the Central Limit Theorem, it will be
approximately correct for large n in other cases.
Example
• Assume that the helium porosity (in percentage)
of coal samples taken from any particular seam is
normally distributed with true standard deviation
σ = 0.75
– Compute a 90% confidence interval for the true
average porosity of a certain seam if the average
porosity for 20 specimens from the seam was 4.85
– Compute a 95% confidence interval for the true
average porosity of that same seam using the
information above.
Confidence Intervals
• Generally speaking, the margin of error is
determined by the choice of C for the confidence
interval.
• High confidence and small margin of error are
desirable.
• High confidence – method almost always gives
correct answers.
• Small margin of error – parameter is pinned
down quite precisely.
Confidence Intervals
• Suppose you calculate a margin of error and
decide that it is too large.
• How to reduce it:
– Use a lower level of confidence (smaller C)
– Increase the sample size (larger n)
– Reduce σ
• In our last example, how would the 95%
confidence interval change if our sample
consisted of 200 specimens instead of 20?
Confidence Intervals
• The confidence interval for a population mean
will have a specified margin of error m when the
sample size is
2
*
z 

n  
 m 
• In surveys for determining proportions, this tends
to explain why for a survey sample of about 1000
people gives a margin of error of approximately
.03
Confidence Intervals
• Remember:
– Data must be an SRS from the population.
– Formula is incorrect for complex sampling designs.
– No correct method for inference from data
haphazardly collect with unknown bias.
– Outliers can have a large effect on the interval.
– For small sample size and non-normal populations,
the true confidence level is different from the value C.
– Standard deviation σ must be known.
– Margin of error covers only random sampling errors.