10.1 estimating with confidence - ds...
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10.1 Estimating with
Confidence
Chapter 10
Introduction to Inference
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Statistical Inference Statistical inference provides methods for
drawing conclusions about a population from
sample data.
Two most common types of statistical inference
are confidence intervals and significance tests.
Both report probabilities that state what would
happen if we used the inference method many
times.
When you use statistical inference, you are
acting as if the data are a random sample or
come from a randomized experiment.
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IQ and Admissions
How would the sample mean vary if we took many samples of 50 freshmen from this same population?
To estimate the mean of our population, we use the mean of our sample, x.
Although our sample mean, x is an unbiased estimator of our population mean μ, it will rarely be exactly equal to it, so our estimate has some error.
The sampling distribution of x tells us how big the error is likely to be when we use x to estimate μ.
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Interpreting a confidence interval
Our sample of 50 freshmen gave a sample
mean of 112. And a standard deviation of
σ = 15.
The resulting interval is 112 +/- 4.2, which
can be written as (107.8, 116.2).
We say that we are 95% confident that the
unknown mean IQ score for all Big City
University freshmen is between 107.8 and
116.2.
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Confidence interval and confidence
level
A level C confidence interval for a
parameter has two parts: A confidence interval calculated from the data,
usually of the form estimate +/- margin of error
A confidence level C, which gives the probability
that the interval will capture the true parameter
value in repeated samples. That is, the confidence
level is the success rate for the method.
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Caution!
Probabilities are long-run relative frequencies, and the idea simply doesn’t apply to a found interval.
An already constructed interval either does or does not contain the population value.
It is correct to give the meaning of “confidence” in terms of probability (e.g. “the probability that my method of constructing intervals will capture the true population value is .95”)
It is never correct to interpret a found interval using the language of probability.
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Assignment
P. 624 exercises 10.1 – 10.3, 10.5
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Conditions for constructing a
Confidence Interval
SRS
Normality
Independence
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Critical Values
The number z* with probability p lying to
the right under the standard Normal curve
is called the upper p critical value of the
standard Normal distribution.
z* can be found using either Table A or
Table C.
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*x zn
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Inference Toolbox
Step 1: Parameter
Step 2: Conditions
Step 3: Calculations
Step 4: Interpretation
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Assignment
P. 632 exercises 10.7 – 10.12
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How Confidence Intervals Behave:
We would like high confidence and a small
margin of error
High confidence says that our method
almost always gives correct answers
A small margin of error says that we have
pinned down the parameter quite precisely
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The margin of error gets smaller
when…
z* gets ______
σ gets ______
n gets ______
smaller smaller
larger
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Example 10.6
Changing the confidence level
What effect does changing the confidence
level have?
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Choosing the sample size
You can arrange to have both high confidence and a small margin of error by taking enough observations.
To obtain a desired margin of error, substitute the value of z* for your desired confidence level, set the expression for m less than or equal to the specified margin of error, and solve the inequality for n.
Example 10.7
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The required sample size may not be
practical due to the time and money
involved.
The size of the sample determines the
margin of error…the size of the population
does not influence the sample size
needed.
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Some Cautions on using the
formula for confidence intervals
pg 636
The data must be from an SRS from the population.
This formula does not work for more complex probability sampling designs.
Fancy formulas cannot rescue badly produced data.
Beware of outliers.
Be careful of small samples and non-normal populations.
You must know the standard deviation of the population.
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The most important caution concerning
confidence intervals is that the margin of
error in a confidence interval covers only
random sampling errors.
Undercoverage and nonresponse in a
sample survey can cause additional errors
that may be larger than the random
sampling error.
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Assignment
P. 637 exercises 10.13 – 10.18
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What statistical confidence does
and does not say: A 95% confidence interval means that we are 95% confident that the mean score for the population lies in the interval.
These numbers were calculated by a method that gives correct results in 95% of all possible samples.
The probability is 95% that the true mean falls in that interval.
No randomness remains after we draw one particular sample and get from it one particular interval. The true mean either is or is not in the interval.
The probability calculations of standard statistical inference describe how often the method gives correct answers.
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Exercises
Section 10.1 exercises on page 640,
10.19- 10.26
Make sure you do 10.26!