Module 8: Interval Estimation

θ is fixed while θ hat_n is a random variable which provides the single best value to estimate θ 

 θ hat is unbiased when bias = E(θ hat_n) - θ = 0 

 θ hat is consistent when θ hat_n -> θ 

 Mean squared error MSE = E(θ hat_n - θ) 2 = bias(θ hat_n ) + V(θ hat_n ) 

 If bias -> and se -> as n -> infinity then θ hat_n is consistent 

 Probability is stronger than samples, probability standard error eventually converges to 0 as n approaches infinity but samples converge to a normal distribution which is not necessarily the same as the population distribution. 

 We the estimator variability (se) to provide an interval of parameter values that are "supported" by the sample. 

 

 A 1 - α confidence interval for a parameter θ is an interval C n = (a; b) where a = a(X 1 , ...X n ) and b = b(X 1 , ...X n ) are functions of the data such that:  P(θ ∈ C n ) >= 1 - α

 ; Where θ is the actual population mean. C n is random and θ is fixed. 

 The confidence interval (a; b) capture the true mean with confidence 1 - α. We commonly use 95% confidence intervals which corresponds to α = .05. This does NOT mean there is 1- α chance/probability the parameter falls in the interval. The correct interpretation: If we repeatedly take samples of size n from a fixed and stable population and build a 95% confidence intervals, 95% of these intervals would contain the true unknown parameter. 

 CI For Mean of a Normal Distribution 

 If σ 2 is known:  X bar +/- Z α /2 *α x 

 If σ 2 is unknown:  X bar +/- t ( α /2,n-1) *S/sqrt(n); Where S 2 = 1/(n-1) * sum(x i -x bar ) 2 

 Using S in place of SD causes more uncertainty, thus increasing the size of the CI. 

 We can similarly find the confidence interval of a proportion in a similar manner: 

 

 Chi-Square DIstribution 

 

 The above represents the chi-squared distribution with n degrees of freedom. E(Q) = n and V(Q) = 2n. 

 The distribution of X 2 n-1 is not symmetrical, so instead of centering our CI (a,b) on the mean, we look for symmetry so that the bounds P(θ < a) = α/2 and P(θ > b) = α/2. 

 

 We derive the variance of a distribution through Fisher's theorem (not shown). The CI comes out to: 

 

 Pearson's product Moment Correlation Coefficient is between -1 and 1 and represents the correlation between 2 variables. 

 

 Although rarely used, you could find a confidence interval for this value.