Transcript Section 6.1 () - People Server at UNCW

6.1 Inference for a Single Proportion
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Statistical confidence
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Confidence intervals
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How confidence intervals behave
Sampling Distribution of a Sample
Proportion
Sampling Distribution of a Sample Proportion
Choose an SRS of size n from a population of size N with proportion p
of successes. Let pˆ be the sample proportion of successes. Then:
The mean of the sampling distribution is p.
The standard deviation of the sampling distribution is
s p̂ =
p(1- p)
n
As n increases, the sampling distribution becomes approximately Normal.
For large n, p̂ has approximately the N(p, p(1- p) / n distribution.
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Statistical Inference
After we have selected a sample, we know the responses of the
individuals in the sample. However, the reason for taking the sample is to
infer from that data some conclusion about the wider population
represented by the sample.
Statistical inference provides methods for drawing conclusions about a
population from sample data.
Population
Sample
Collect data from a
representative sample...
Make an inference
about the population.
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Methods for drawing conclusions about a population
from sample data are called statistical inference
So we’ll use data to make these inferences; i.e., draw
conclusions about populations from data in our samples
or from our experiments
We'll consider two types of inference:
 Confidence interval estimation
 Tests of significance
In both of these cases, we'll consider our data as either
being a random sample from a population or as data
from a randomized experiment
Start with estimation… there are two situations we'll
consider
 estimating the mean m of a population of
measurements
 estimating the proportion p of Ss in a population of
Ss and Fs
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In either case, we'll construct a confidence interval of the
form estimate +/- M.O.E., where M.O.E. = margin of
error of the estimator.
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The MOE gives information on how good the estimate is
through the variation in the estimator (its standard error)
and through the level of confidence in the confidence
interval (through a tabulated value).
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The standard error of an estimator is its estimated
standard deviation (treating the estimator as a statistic
with a sampling distribution…)
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Best estimator of m is X and we will learn that
approximately N(m,s / n)
X
is
Best estimator of p is phat and we’ve learned that phat
is approx. N( p, p(1- p) ) . We’ll start here…
n
In case of inference, we’ll try to make sure that n
is a fairly large sample… this will assure normality
of the sampling distribution of p-hat The mean
and standard deviation of p-hat will be given by
these formulas:
We did a simulation using Table B and can use
our results to show the formulas make sense…
I’ve modified Example 6.4 on page 320:
Assume p = 0.60; i.e., that 60% of the population are “Success”. We will simulate
drawing a random sample of size 20 from the population
We can imitate the population by Table B, with each entry standing for a person. Six
of the 10 digits (say 0 to 5) stand for people who are “Success”. The remaining four
digits, 6 to 9, stand for “Failure”. Because all digits in a random number table are
equally likely, this assignment produces a population proportion of “Success” equal
to p = 0.60. We then imitate an SRS of 20 students from the population by taking 20
consecutive digits from Table B. The statistic is the proportion of 0s to 5s in the
sample of size n = 20.
Here are the first 100 entries in Table B, with digits 0 to 5 highlighted –
What are the first 5 p-hats?? Continue with JMP…
These samples show the sampling variability of p-hat: because the samples are
random, we don’t expect to get the same proportion of S’s in each sample of
n=20… but notice that the variability in the p-hats can be characterized as
normal… I used the “Random -> Binomial Formula in JMP & divided by 20.
Sampling Distribution of a Sample
Proportion
Sampling Distribution of a Sample Proportion
Choose an SRS of size n from a population of size N with proportion p
of successes. Let pˆ be the sample proportion of successes. Then:
The mean of the sampling distribution is p.
The standard deviation of the sampling distribution is
s p̂ =
p(1- p)
n
As n increases, the sampling distribution becomes approximately Normal.
For large n, p̂ has approximately the N(p, p(1- p) / n distribution.
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Large-Sample Confidence Interval
for a Proportion
To construct a confidence interval for an unknown population proportion
p we’ll use our best estimator p-hat and construct the CI as
estimate +/- M.O.E. … here the MOE is (value from Table) * (SE of
estimator)
estimator ± (critical value) × (standard deviation of estimator)
The sample proportion p̂ is the statistic (or estimator) we use to
estimate p. The standard deviation of the sampling distibution of p̂ is:
s p̂ =
p(1- p)
n
Since we don’t know p, we replace it with the sample proportion p̂.
This gives us the standard error (SE) of the sample proportion:
p̂(1- p̂)
n
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Large-Sample Confidence Interval
for a Proportion
How do we find the critical value for our confidence interval?
statistic ± (critical value) × (standard deviation of statistic)
If the Normal condition is met, we can use a Normal curve. To find a
level C confidence interval, we need to catch the central area C
under the standard Normal curve.
For example, to find a 95%
confidence interval, we use a critical
value of 2 based on the 68-95-99.7
rule. Using a standard Normal table
or a calculator, we can get a more
accurate critical value.
Note, the critical value z* is actually
1.96 for a 95% confidence level.
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Large-Sample Confidence Interval
for a Proportion
Once we find the critical value z*, our confidence interval for the
population proportion p is:
pˆ ± z *
pˆ (1 - pˆ )
n
One-Sample z Interval for a Population Proportion
Choose an SRS of size n from a large population that contains an
unknown proportion p of successes. An approximate level C
confidence interval for p is:
pˆ (1 - pˆ )
pˆ ± z *
n
where z* is the critical value for the standard Normal density curve
with area C between –z* and z*.
Use this interval only when the numbers of successes and failures
in the sample are both at least 15.
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Large-Sample Confidence Interval
for a Proportion
pˆ ± z *
pˆ (1 - pˆ )
n
What does the CI for p actually mean?
Here’s a picture of (Figure 6.7 on page
327) 25 confidence intervals computed
from 25 samples of the same sizenote that they vary quite a bit, but only
1 out of the 25 actually misses the
mean=p : approximately 95% of the
confidence intervals computed this
way should capture p inside…
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Example
It is claimed that 50% of the beads in a container are red. A random sample of
251 beads is selected, of which 107 are red. Calculate and interpret a 90%
confidence interval for the proportion of red beads in the container. Use your
interval to comment on the claim that ½ the beads in the container are red.
z
–1.7
0.03
0.04
0.05
 Sample proportion = 107/251 = 0.426
–1.6
0.0418 0.0409 0.0401  This is an SRS and there are 107 successes
and 144 failures. Both are greater than 15.
0.0516 0.0505 0.0495
–1.5
0.0630 0.0618 0.0606
 For a 90% confidence level, z* = 1.645
p̂(1- p̂)
p̂ ± z *
n
(0.426)(1 - 0.426)
= 0.426 ± 1.645
251
= 0.426 ± 0.051
= (0.375, 0.477)
We are 90% confident that the interval from
0.375 to 0.477 captures the actual
proportion of red beads in the container.
Since this interval gives a range of plausible
values for p and since 0.5 is not contained
in the interval, we have reason to doubt the
claim.
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Varying confidence levels
Confidence intervals contain the population proportion p in C% of
samples, in the long run. Different areas under the curve give different
confidence levels C.
Practical use of z: z*
z* is related to the chosen
confidence level C.
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C
C is the area under the standard
normal curve between −z* and z*.
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The confidence interval is thus:
p̂(1- p̂)
p̂ ± z *
n
−z*
z*
Example: For an 80% confidence
level C, 80% of the normal curve’s
area is contained in the interval.
How do we find specific z* values?
We can use a table of z (Table A) or t values (Table D). In Table D, for a
particular confidence level, C, the appropriate z* value is just above it.
Example: For a 98% confidence level, z*=2.326
We can use software. In JMP:
Create a new column, Edit Formula, and choose Normal Quantile( p ) under
Probability where p = (1-C)/2 is the area to the left of z*
Since we want the middle C probability, the probability we require is (1 - C)/2
Example: A 98% confidence level, Normal Quantile (.01) = −2.326349 (= neg. z*)
Link between confidence level and margin of error
The confidence level C determines the value of z* (in table A or D).
The margin of error m also depends on z*.
m = z * p(1- p) n
Higher confidence C implies a larger
margin of error m (thus less precision
in our estimates).
C
A lower confidence level C produces a
smaller margin of error m (thus better
precision in our estimates).
m
−z*
m
z*
Properties of Confidence Intervals
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User chooses the confidence level, C, and hence z*
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Margin of error follows from this choice as (z*)(SE of estimator)
We want
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A high level of confidence
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A small margin of error
The margin of error is smaller when
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z* (and thus the confidence level C) gets smaller
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p(1-p) is smaller
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n is larger – this is the usual way to decrease MOE –
increase the sample size!
Interpretation of Confidence Intervals
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Conditions under which an inference method is valid are never fully met in
practice. Exploratory data analysis and judgment should be used when
deciding whether or not to use a statistical procedure.
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Any individual confidence interval either will or will not contain the true
population proportion, p. It is wrong to say that the probability is 95% that
the true proportion falls in the confidence interval.
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The correct interpretation of a 95% confidence interval is that we are 95%
confident that the true proportion falls within the interval. The confidence
interval was calculated by a method that gives correct results in ~95% of all
possible samples. (See slide #13 above!)
In other words, if many such confidence intervals were constructed, ~95%
of these intervals would contain the true proportion.
HW: Read Introduction to Chapter 6 and Section 6.1 - 6.1.6; do # 6.3, 6.5-6.9
Previous HW: Read section 5.5; omit section 5.6
Do Exercises #5.85, 5.87- 5.90, 5.93-5.95, 5.99, 5.100, 5.102, 5.144