Confidence Intervals

Chantel ChangMath 480Dr. FaberConfidence Intervals

http://upload.wikimedia.org/wikipedia/commons/8/8f/NYW-confidence-interval.svg1Definitions1) Estimator: A random variable, , that is a function of a random sample x1,x2,,xn (e.g. sample mean)

2) Confidence Interval: A range of values that include the unknown parameter, (e.g. true population mean, population standard deviation, or population proportion), with probability 1-. Quantifies uncertainty of estimator.

3) Level of Significance:, the probability of committing a Type I Error

4) Type I Error: The probability of incorrectly rejecting the null hypothesis

95% confidence interval, =0.05Percent Confidence = 100(1-)%

=population mean =population s.d. n=sample size

By the Central Limit Theorem, if we have a large enough random sample (>30), then the sample mean, , is approximately normally distributed

http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_General_CI_Experiment3Computation of Confidence Interval for the Population Mean

100(1-)% Confidence IntervalInterpretation of the Confidence Interval

0.05 (error) X 25 = 1.25For a 95% confidence interval, we should expect to see about one interval that does NOT contain the unknown parameter . Example: Random Number Generator 5 random numbers generated using FreeMat software, 10 times from a normal distribution with =0, and=1 x = randn(5,1,10);

For a 95% confidence interval, =0.05-Z0.025= -1.96, and by symmetry, Z0.025=1.96

0.025=-1.96=1.96==0.950.025=LB

xaveLBUBxaveLBUB0.05 (error) X 20 = 1...Common Misconceptions/ Takeaway!!

There is 0.95 probability that the confidence interval [0.0749,1.8280] contains . NO!Colloquial saying: We are 95% confident that the interval [0.0749,1.8280] contains .

=0, so

Before we take any samples, there is a 0.95 probability that a confidence interval will contain .

If we took 100 confidence intervals, we would expect 95 of them to contain .

Thank you!