Statistical Estimation Point Estimate – Use a Single Value from a Sample to Estimate an Unknown Parameter of a Population Є e Є ≈ e Unbiased Estimator – e is an Unbiased Estimator of Parameter Є, If E(e) = Є E(X)=µ , X is a unbiased estimator of µ , Pop Mean E(s2) = σ2 , s2 is an unbiased estimator of σ2 , Pop Variance , s is a slightly biased estimator of σ, Pop Std Dev

Interval Estimate – Bound the Parameter between a Upper and Lower Limit Lower Limit ≤ Parameter ≤ Upper Limit X - e ≤ µ ≤ X + e e – Error Bound – Maximum difference between X and µ which we expect to occur. α – Risk – Probability that the difference between X and µ will exceed e. We know from the Central Limit Theorem that the distribution for X will be Normal if we have a large enough sample size.

Sample Mean Distribution α/2 α/2 µ - e µ µ + e Standard Normal Distribution α/2 α/2 -Zα/2 Zα/2

Interval Estimate - X – e ≤ µ ≤ X + e where and n ≥ 30 so σ ≈ s Ex: n = 36, X = 20.6 yr and s = 3 yr with α = .05 Confidence Level – C = ( 1 – α )

Ex: n = 64, X = 42,000 miles, s = 10,000 miles with C = .90 & α = .10 Sample Size Calculation – Use sample size to control the error

Ex: Sample Size to Determine Ave Age for e = .5yr, σ = 3 yr, C = .95

Ex: Sample Size for Ave Cost at WSU for e = $250, σ = ?, C =.95

Small Sample Size – n < 30 t- Distribution W.S. Gosset -3 -2 -1 1 2 3 Standard Normal t (d.f. = 25) t (d.f. = 1) t (d.f. = 5)

Table of Critical Values of t df t0.100 t0.050 t0.025 t0.010 t0.005 1 3.078 6.314 12.706 31.821 63.656 2 1.886 2.920 4.303 6.965 9.925 3 1.638 2.353 3.182 4.541 5.841 4 1.533 2.132 2.776 3.747 4.604 5 1.476 2.015 2.571 3.365 4.032 23 1.319 1.714 2.069 2.500 2.807 24 1.318 1.711 2.064 2.492 2.797 25 1.316 1.708 2.060 2.485 2.787 29 1.311 1.699 2.045 2.462 2.756 30 1.310 1.697 2.042 2.457 2.750 40 1.303 1.684 2.021 2.423 2.704 60 1.296 1.671 2.000 2.390 2.660 120 1.289 1.658 1.980 2.358 2.617 1.282 1.645 1.960 2.327 2.576  t   With df = 24 and a = 0.05, ta = 1.711. 21

Confidence Interval – X - e ≤ µ ≤ X + e Ex: n =16, X = 20.6 yr, s = 3 yr, C = .95 Sample Size Calculation for Small Sample Problem?

Estimation of the Population Proportion n = sample size x = number with attribute = x/n Sample Proportion Estimator Unbiased? Standard Error for Estimator?

Confidence Interval for the Population Proportion

Ex: Percentage who plan to buy a new Car; n = 250, x = 80, C = 95 Ex: Percentage of Voters who will vote for Candidate; n = 400, x = 186, C = .90

Sample Size Calculation - Ex: Voter poll e = .01, C= .95, p = .50

Distribution of the Standard Error for Sample Proportion Ex: Sample Size to Estimate the Proportion Left-Handed

Finite Population Correction Ex: N = 5,000, e = .03, p = .30, C = .95