inference: confidence intervals chapter 8. 8.1 what are point and interval estimates of population...
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INFERENCE: CONFIDENCE INTERVALS
Chapter 8
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8.1 What are Point and Interval Estimates of Population Parameters?
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Point Estimate and Interval Estimate
A point estimate is a single number that is our “best guess” for the parameter
An interval estimate is an interval of numbers within which the parameter value is believed to fall.
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Point Estimate vs. Interval Estimate
A point estimate doesn’t tell us how close the estimate is likely to be to the parameter
An interval estimate is more useful It incorporates a
margin of error which helps us to gauge the accuracy of the point estimate
aiaccess.net
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Properties of Point Estimators Property 1: Good
estimator has sampling distribution centered at the parameter An estimator with
this property is unbiasedSample mean
[proportion] is an unbiased estimator of population mean [proportion]
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Properties of Point Estimators
Property 2: Good estimator has smaller standard error than other estimators Falls closer than
other estimates to parameterSample mean has
a smaller standard error than sample medianmontenasoft.com
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Confidence Interval
An interval containing the most believable values for a parameter
The probability that the interval contains the parameter is the confidence level Close to 1, most
commonly 0.95 comfsm.fm
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Logic of Confidence Intervals
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Logic of Confidence Intervals
gmatclub.com
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Margin of Error
Measures accuracy of point estimate in estimating a parameter
Multiple of standard error
1.96 standard errors for a 95% confidence interval of p
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CI for a Proportion
19% of 1823 respondents agreed, “It is more important for a wife to help her husband’s career than to have one herself.” Assuming standard error is 0.01, calculate a 95% CI for p who agreed.
2.bp.blogspot.com
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8.2 How Can We Construct a Confidence Interval to Estimate a Population Proportion?
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Finding the 95% CI for p
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Protecting the Environment
“Are you willing to pay much higher prices in order to protect the environment?”1154 respondents, 518 willing Find and interpret a 95% confidence interval
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Needed Sample Size for CI of P
eaststaffsbc.gov.uk
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Confidence Levels Other than 95%
95% confidence means a 95% chance that
contains p With probability
0.05, CI misses p To increase chance
of a correct inference, use larger confidence level, such as 0.99
(se)*1.96 p̂
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We must compromise between desired margin of error and desired confidence of a correct inference
Confidence Levels Other than 95%
thefreemarketeers.files.wordpress.com
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CI for a Proportion
Of 598 respondents, 366 said yes, “If the wife in a family wants children, but the husband does not, is it all right for the husband to refuse to have children?” Assuming standard error is 0.01, calculate a 99% CI for p who said yes.
vitaminsandnutrition.files.wordpress.com
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Error Probability for CI Method?
(se)*z ˆ p
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Summary
pirate.shu.edu
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Effects of Confidence Level and Sample Size on Margin of Error
The margin of error for a confidence interval:Increases as confidence level increasesDecreases as sample size increases
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Interpretation of Confidence Level
For 95% confidence intervals, in the long run about 95% of those intervals would give correct results, containing the population proportion, p
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8.3 How Can We Construct a Confidence Interval to Estimate a Population Mean?
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Construct CI for μ
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Properties of the t Distribution Bell-shaped and
symmetric about 0
Probabilities depend on degrees of freedom, df = n-1
Has thicker tails (more spread out) than standard normal distribution
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t-Distribution
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95% Confidence Interval for μ
boost.org
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t-CI for Other Confidence Levels?
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Table B
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If the Population is Not Normal, is the Method Robust?
Basic assumption is normal population distribution
Many distributions far from normal
Abnormal distributions are okay for large n because of the CLT
CIs using t-scores usually work well: except for extreme outliers, the method is robust
urbanbacon.com
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Standard Normal Distribution is t-Distribution with df = ∞
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8.4 How Do We Choose the Sample Size for a Study?
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Sample Size for Estimating p
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Sample Size For Exit Poll
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Sample Size for Estimating μ
statcan.gc.ca
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Education of Black South Africans
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Factors Affecting Choice of Sample Size?
1. Desired precision, as measured by the margin of error, m
2. Confidence level3. Variability in data - If
subjects have little variation (σis small), we need fewer data than with substantial variation
4. Financial
schoolloans.org
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Using a Small n
The t methods for μ are valid for any n, but look for extreme outliers or great departures from normal
For a population proportion, p, the method works poorly for small samples because the CLT no longer holds
stat.psu.edu
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CI for p with Small Samples
businessandlearning.com