Statistical Sciences 2244A/B Chapter Notes - Chapter 14: Confidence Interval, Interval Estimation, Statistical Parameter
22 May 2018
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Department
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Stats 2244
Chapter 14
CHAPTER 14.1
The reasoning of statistical estimation
- Example:
- The big idea is that the sampling distribution of tells us how close to μ the sample mean is
likely to be
- A confidence interval just turns that information around to say how close to the unknown
population mean μ is likely to be.
- Example:
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- Because the method has a 95% success rate, all we need is one SRS of 217 eight-year-old
American boys to compute just one interval ± 1.4
- This interval of numbers between the values − . and + 1.4 is called a 95% confidence
interval for μ.
CHAPTER 14.2
Margin of Error and Confidence Level
- Most confidence intervals take the form: estimate +/- margin of error
o The estimate is the center of the interval
o It is our unbiased guess for the value of the unknown parameter, based on the sample
data
o The margin of error shows how accurate we believe our guess is, based on the
variability of the estimate
- Confidence interval
o A level C confidence interval for a parameter has 2 parts:
▪ An interval calculated from the data, usually of the form: estimate +/- margin of
error , where the estimate is a sample statistic and the margin of error
represents the accuracy of our guess for the parameter
▪ 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
- note: the confidence interval does NOT represent the probability that a specific interval
computed from a sample contains the parameter
o it has to do with the behavior of the Cis in repeated samples
o whether or not one CI captures the parameter is unpredictable (an in practice, we will
never know if it did or not bc we dont know the value of the parameter)
o however, in the long run with a repeated samples, we expect a proportion, C, of
confidence intervals will capture the parameter
- interpreting a confidence interval:
o The confidence level is the success rate of the method that produces the interval
o We dont know whether the 9% confidence interval from a particular sample is one of
the 95% that capture μ or one of the unlucky 5% that miss.
o To say that we are 95% confident that the unknown value of μ lies between 131.1 and
133.9 cm is shorthand for We got these numbers using a method that gives correct
results 9% of the time.
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- Fig 14.2 (above) is one way to picture the idea of a 95% confidence interval
- Fig 14.3 shows the idea in a diff form
- Two things influence the size of the stimulated confidence intervals:
o The confidence level C
o Sample size n
VIDEO: confidence intervals intro part II
- We will talk about 95% confidence intervals (CI) but the same principles apply to 90% or 99%
etc
- A confidence interval is an interval estimate of a parameter that is likely to contain the true
value of the parameter
- 95% of all 9% Cis contain the true value of their population parameter
- we say we are 9% confident that our interval contains the parameter value or this is a range
of plausible values for the parameter value, at confidence level 9%
- we construct confidence intervals using sampling distributions
- recall with a large sample size, we can apply the central limit theorem (CLT) to approx. the
sampling distribution of x bar as normal
find more resources at oneclass.com
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Document Summary
The big idea is that the sampling distribution of likely to be tells us how close to the sample mean is. A confidence interval just turns that information around to say how close to population mean is likely to be. the unknown. Because the method has a 95% success rate, all we need is one srs of 217 eight-year-old. This interval of numbers between the values. Most confidence intervals take the form: estimate +/- margin of error: the estimate is the center of the interval. It is our unbiased guess for the value of the unknown parameter, based on the sample data: the margin of error shows how accurate we believe our guess is, based on the variability of the estimate. That is, the confidence level is the success rate for the method. Fig 14. 2 (above) is one way to picture the idea of a 95% confidence interval. Fig 14. 3 shows the idea in a diff form.
