Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CHAPTER 7 SUMMARY CENTRAL LIMIT THEOREMS FOR x and p s : Main point of Central Limit Theorems: Any statistic that comes from a random sample that is large enough so that the “law of large numbers” applies will possess three characteristics: 1. Consistency: The statistic will have equal probability of understating or overstating the actual value of the population parameter it is estimating. 2. Unbiasedness: On average, and with highest probability, the statistic will assume the correct value of the population parameter it is estimating. 3. Efficiency: The larger the sample used to compute the statistic estimating the relevant population parameter, the smaller the error dispersion (standard error) of the statistic. Thus we can state three Central Limit Theorem versions (A, B, and C)for the statistics x and ps : Statistic x ps Consistency Case A: Normal distribution, when is known Case B: t distribution with (n-1) degrees of freedom, when is unknown and we must use s as an estimate/proxy. Case C: Normal Distribution Unbiasedness x x x x p p ps s Efficiency x n x p s s n p(1 p) n p s (1 p s ) n CHAPTER 8: Interval Estimation SUMMARY OBJECTIVES: 1. To construct interval estimates for population means, , and population proportions, p, using randomly selected sample information, along with sampling distributions. 2. To determine proper sample sizes, n, for statistical inference about populations from samples. Parameter Best Point Estimate p x ps Interval estimate x –E<< x +E ps – E < p < ps + E Margin of Error for n large enough E = z· x ; ( known) z from table E.2 Case A E = z· p s (n p s >5 and n 1 p s >5) z from table E.2 Case C Margin of Error for n not large enough E = t· x ( unkown) t from table E.3 Case B Procedure cannot be applied to p s when sample is not large enough Proper sample size for inference n=(z/E)2 n=pq(z/E)2 Four (4) steps to calculate Confidence Intervals: 1. Estimate sample statistics from sample and sampling distributions (n, x or p s , s, and x or p s ). 2. Use best point estimate, x or p s , as interval midpoint. 3. Calculate the number of standard deviations you will deviate from midpoint, using confidence level (z) and sampling distribution information ( x or p s ); i.e., calculate the "maximum error", E = z· x or t· x , in the case of means. Or E = z· p s , in the case of proportions. TO CALCULATE z: when sample size is "large enough" (table E.2) Divide the complement of the confidence level by 2. Find the above area in table E.2 and read it “inside-out”. This identifies the (negative) number of standard deviations, z-score, associated with the desired confidence level. TO CALCULATE t: when sample size is not "large enough" (table E.3) Divide the complement of the confidence level by 2. This determines the table E.3 column to use. Count degrees of freedom (df = n - 1). This determines the table E.3 row to use. Read the t score "outside-in". 4. To create interval estimates we add and subtract the error allowance, E, from the best point estimate, x or p s .