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Confidence Intervals. Chantel Chang Math 480 Dr. Faber. Definitions. 1) Estimator : A random variable, , that is a function of a random sample x 1 ,x 2 ,… , x n (e.g. sample mean ).
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Confidence Intervals Chantel Chang Math 480 Dr. Faber
Definitions 1) 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, θ(e.g. true population mean, population standard deviation, or population proportion), with probability 1-α. Quantifies uncertainty of estimator. 3) Level of Significance:α, the probability of committing a Type I Error 4) Type I Error: The probability of incorrectly rejecting the null hypothesis
By the Central Limit Theorem, if we have a large enough random sample (>30), then the sample mean, , is approximately normally distributed μ=population mean σ=population s.d. n=sample size • Percent Confidence = 100(1-α)% • 95% confidence interval, α=0.05
Computation of Confidence Interval for the Population Mean 100(1-α)% Confidence Interval
Interpretation of the Confidence Interval 0.05 (error) X 25 = 1.25 For a 95% confidence interval, we should expect to see about one interval that does NOT contain the unknown parameter μ.
Example: Random Number Generator • 5 random numbers generated using FreeMat software, 10 times from a normal distribution with μ=0, andσ=1 x = randn(5,1,10);
For a 95% confidence interval, α=0.05 -Z0.025= -1.96, and by symmetry, Z0.025=1.96 =0.95 0.025= 0.025= 1.96= -1.96=
LB xave LB UB xave LB UB 0.05 (error) X 20 = 1...
Common Misconceptions/ Takeaway!! There is 0.95 probability that the confidence interval [0.0749,1.8280] contains μ. NO! μ=0, so • Before we take any samples, there is a 0.95 probability that a confidence interval will containμ. • Colloquial saying: “We are 95% confident that the interval [0.0749,1.8280] contains μ.” • If we took 100 confidence intervals, we would expect 95 of them to contain μ.
