certain How are we?
Lecture 8 Confidence intervals for the mean
Sociology 549, Paul von Hippel 1
Review of central limit theorem (CLT) Turning the CLT around to get confidence intervals, e.g.
Confidence interval for a mean:
We are 95% confident that new sociology BAs (population) have an average starting salary (parameter) between $27,369 and $30,299 (interval)
Overview
Sociology 549, Paul von Hippel
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Central limit theorem for the mean: Review
The sample mean is usually within a couple standard errors of the population mean. In 95% of all samples,
Y is in Y 1.96 Y
where Y = Y / N
In 99% of all samples,
Y is in Y 2.58 Y
where Y = Y / N
Sociology 549, Paul von Hippel
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Central limit theorem for the mean: Using abbreviated normal table
The sample mean is usually within a couple standard errors of the population mean. In Confidence% of all samples,
Y is in Y Z Y
where Y = Y / N
where Z comes from the "z (standard normal)...table"
Confidence z ... ... 95% 1.96 ... ...
Sociology 549, Paul von Hippel
Area from -Z to +Z - Z +Z
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Central limit theorem for the mean: So what?
If you know the population mean, the central limit theorem helps you guess the sample mean. But in practice you know the sample mean, and want to guess the population mean.
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Central limit theorem: The sample mean is usually within a couple standard errors of the population mean. In Confidence% of all samples,
Turning around the central limit theorem: Normal confidence interval
Y is in Y Z Y
where Y = Y / N
Confidence interval: The population mean is usually within a couple standard errors hopefully close of the sample mean. We are Confidence% confident that
Y is in Y ZSY
where SY = SY / N
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Sociology 549, Paul von Hippel
Normal confidence interval: Example #1 data
National Association of Colleges and Employers Sample of N=92 Sociology BAs, graduating 2000 01
not a Random Sample but we'll pretend it is
Variable: starting salary (Y) in thousands
Cases: 38.0, 28.0, 28.0, 24.6, ...
Y = 28.834 SY = 7.095
Sociology 549, Paul von Hippel
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Normal confidence interval: Example #1 calculation If we want 95% confidence, then Z=1.96.
Confidence z ... ... 95% 1.96 ... ... Y is in Y ZSY where SY = SY / N
Y is in 28.834 1.96SY
where SY = 7.095 / 92 = .7397
Y is in 28.834 1.96(.7397) Y is in 28.834 1.450 Y is between 27.384 and 30.284
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Normal confidence interval: Example #1 interpretation
We are 95% sure that the average salary in the population of new soc BAs, 200001, is between $27.384K and $30.284K Just an average. Doesn't mean 95% of individual salaries are in the interval.
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Point estimate
Definitions
best single-number guess at a parameter rarely right, often close $28.834K is a point estimate for the average salary of new sociology BAs
Y is a point estimate of Y
Confidence interval
range (interval) of guesses at a parameter width reflects uncertainty: vague but usually right ($27.384K, $30.284K) is a 95% confidence interval for the average salary of new sociology BAs
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Y ZSY is a confidence interval for Y
The confidence interval covers the population mean unless you've got a weird sample.
What is confidence?
"95% confidence" = In 95% of all samples, the confidence interval includes the population mean. In 5% of all samples, the confidence interval fails to cover the mean. Do you know if you've got a weird sample? No! So 5% of the time you're wrong. typical samples 95%
weird samples
Sociology 549, Paul von Hippel
weird samples
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What is confidence?
Is a 99% confidence interval narrower or wider than a 95% confidence interval?
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Problem with the normal confidence interval
Central limit theorem: The sample mean is usually within a couple standard errors of the population mean. In Confidence% of all samples,
Y is in Y Z Y
where Y = Y / N
Confidence interval: The population mean is usually within a couple standard errors not identical of the sample mean. We are Confidence% confident that
Y is in Y ZSY
where SY = SY / N
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Sociology 549, Paul von Hippel
The normal confidence interval assumes
sY = Y But sY is just an estimate of Y Uncertainty (sampling error) in this estimate is not reflected in normal confidence interval
Sampling error in sY
Normal confidence interval is
too certain, too confident too narrow We say it's right in 95% of all samples
But it's actually right in < 95%
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The t distribution
Normal distribution : How many true (population) standard errors Y = Y / N separate Y from Y ? t distribution : How many estimated (sample) standard errors SY = SY / N separate Y from Y ?
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t distribution
The t distribution
reflects uncertainty sY about as an estimate of Y If N is large,
then sY is close to Y, so t distribution is close to standard normal distribution (black)
Instead of N, though, we use degrees of freedom (df)
df=N-1
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t confidence intervals for the mean
We are Confidence% confident that
Y is in Y tsY
where sY = sY / N
t comes from your "z (standard normal) and t table" t Degrees of freedom df=N-1. Degrees of freedom (df)
Confidence 90% 91% 92% 93% 94% 95% 96% 97% 98% 99% 99.9% z 1.64 1.70 1.75 1.81 1.88 1.96 2.05 2.17 2.33 2.58 3.29 100 1.66 1.71 1.77 1.83 1.90 1.98 2.08 2.20 2.36 2.63 3.39 50 1.68 1.73 1.79 1.85 1.92 2.01 2.11 2.23 2.40 2.68 3.50 40 1.68 1.74 1.80 1.86 1.94 2.02 2.12 2.25 2.42 2.70 3.55 30 1.70 1.75 1.81 1.88 1.95 2.04 2.15 2.28 2.46 2.75 3.65 25 1.71 1.76 1.82 1.89 1.97 2.06 2.17 2.30 2.49 2.79 3.73
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Normal confidence intervals:
Normal vs. t confidence intervals for the mean: Comparing the formulas
Y is in Y Z Y
t confidence intervals:
where Y = Y / N where sY = sY / N
Y is in Y tsY
Differences: 1. Z vs. t (Appendix B vs. Appendix C) 1. vs. s (population vs. sample) 1. t uses degrees of freedom df=N-1. Z doesn't. t intervals are wider. Confidence level is more accurate.
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National Association of Colleges and Employers
Confidence interval for the mean: Example #2 data
Sample of N=92 Sociology BAs, graduating 2000 01 Variable: starting salary in thousands
Cases: 38.0, 28.0, 28.0, 24.6, ...
Y = 28.834 SY = 7.095
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If we want 95% confidence, then since df=N1=91 (closest to 100), t=1.98.
Confidence z 95% 1.96 t Degrees of freedom (df) 100 50 40 30 25 1.98 2.01 2.02 2.04 2.06
Confidence interval for the mean: Example #2 calculation
Y is in Y tSY Y is in 28.834 1.98SY
where SY = SY / N where SY = 7.095 / 92
Y is in 28.834 1.98(.740) Y is in 28.834 1.465 Y is between 27.369 and 30.299
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Confidence interval for the mean: Example #2 interpretation
We are 95% sure that the average salary in the population of new soc BAs, 200001, is between $27,369 and $30,299 Just an average. Doesn't mean 95% of individual salaries are in the interval.
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Normal vs. t confidence intervals for the mean: Comparing the results, Example #2
95% normal confidence interval: ($27,384, $30,284) 95% t confidence interval: ($27,369, $30,299) The t interval is wider. But just $30 wider. df is large, so t is close to Z
Here df=N-1=91. In student GSS, df=N-1=1427, so t even closer to Z.
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Sample of N=15 US adults from student GSS Variable: CHILDS ("How many children have you ever had?")
Cases: 3,6,2,2, ...
Confidence interval for the mean: Example #3 data
small sample. Maybe t will make a difference.
Y = 2.38 SY = 1.502
Sociology 549, Paul von Hippel ...