Confidence Intervals
Key Concepts:
 Form of the Confidence Interval
 Interpretation of the Confidence Interval
 Ztest intervals
General form of a confidence interval (CI)
A confidence interval estimates are intervals within which the parameter is expected to fall, with a certain degree of confidence.
The general form:
 estimate ± critical value × std.dev of the estimate
 estimate ± margin of error
For example:
 sample mean ± critical value × estimated standard error
The CIs differ based on:
 The parameter of interest, e.g., population mean, population proportion, difference in population's means, etc…
 Design of the sample: SRS, stratified, experiments
 Confidence level or a confidence coefficient, (1  α)100%, e.g., 95%, 99%, 90%, 80%, corresponding, respectively, to α values of 0.05, 0.01, 0.1, 0.2, etc…
Interpretation of a Confidence Interval
In most general terms, for a 95% CI, we say “we are 95% confident that the true population parameter is between the lower and upper calculated values”.
A 95% CI for a population parameter DOES NOT mean that the interval has a probability of 0.95 that the true value of the parameter falls in the interval.
The CI either contains the parameter or it does not contain it.
The probability is associated with the process that generated the interval. And if we repeat this process many times, 95% of all intervals should in fact contain the true value of the parameter.
What does a 99% CI say?
Would you choose a 99% or 95% CI, and why?
Tradeoffs
We want confidence coefficient to be closer to 1.
We want the sample size to be as small as possible (but not too small). This is a practical issue.
We want the CI to be as narrow as possible
 As we increase the sample estimate, the CI …?
 As we decrease st. dev, the CI …?
 As we decrease the confidence level, (1α), the CI …?
 As we increase sample size….?
zTests & Intervals
For an unknown population mean, and a known variance:
 Assumptions of the model:
 Suppose there is a normally distributed population whose standard deviation σ is known to be (say) 3 but whose mean μ may not be known. How could we estimate μ?
 Take a random sample of size n = (say) 54.
 Sample statistic
 The sample mean, \(\bar{X}\) is a good estimator of the population mean μ.
 Sampling distribution under the model assumptions:
 Via CLT is ~ N(μ, σ^{2}/n)
 We are 95% confident that μ is in the interval \(\bar{X}2\frac{\sigma}{\sqrt{n}}, \bar{X}+2\frac{\sigma}{\sqrt{n}}\).
More About Confidence Intervals
Simplified Expression for a 95% Confidence Interval
Generalizing the 95% Confidence Interval
Critical value, z /2 is a multiplier for a (1α) × 100%
For 95% CI, α = 0.5, so the Zvalue of the standard normal is at 0.025, that is z = 1.96 For any probability value (1 ) there is a number z/2 such that any normal distribution has probability (1 ) within z /2 standard deviations of the mean. Assuming that σ is known, the multiplier for a (1α) × 100% confidence interval is the (1  ½α) × 100th percentile of the standard normal distribution.
Height Example

Confidence Intervals for Proportions in Newspapers
As found in CNN in June, 2006 (here):
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The stated Margin of error: +/ 3%
Therefore, this would be the Confidence interval: 62%+/ 3%
We can be really confident that between 59% and 65% of all U.S. adults disapprove of how President Bush is handling the situation in Iraq.