# Intro Stats II Final

 The flashcards below were created by user Jamie_Bee on FreezingBlue Flashcards. Two parts of The Statistical Inferences 1. Estimating unknown parameter(s) and constructing (1-α)100% Confidence interval for unknown parameters2. Tests of Hypothesis about the unknown parameter(s) Estimator A rule that tells us how to calculate the estimator based on the information contained in the sample. It is generally expressed as a formula which does not involve any unknown parameters in it. There are two types of estimators: Point Estimator and Interval Estimator Point Estimator An estimator given as a point or a single value Unbiased Estimator Let theta-hat be the point estimator of unknown population parameter theta [where theta could be μ or p or σ2 ] if E(theta-hat)= theta, then the point estimator theta-hat is an unbiased estimator of thetaeg. E(s)=μ Interval Estimator When an interval is constructed around the point estimate, and it is stated that this interval is likely to contain unknown population parameter(s) with a specific level. This confidence level is usually denoted by (1-α)100% where α is called the coefficient of confidence. If (1-α)100% is not given, we usually use (1-α)100%=95% Interpretation of (1-α)100% Confidence Interval In repeated sampling under identical conditions, (1-α)100% of all confidence intervals constructed in this manner will enclose the unknown mean μ 3 quantities to decrease the width of the Confidence Interval 1. Confidence level (1-α)100% or zα/2 (not ideal, lowers the probability that our confidence interval contains the unknown mean μ) 2. Population variance σ2 or σ (not ideal, as recalculating variance is time consuming and costly, as we have to go through the entire population)3. Sample size n (ideal) Margin of Error (for the estimate of unknown mean μ) Denoted by E and defined as the quantity that is subtracted and added to the sample mean to obtain (1-α)100% confidence interval. Also called the "bound on the error of estimation" or "the maximum error" or "the estimation is within.." Interpretation of E We can say with probability (1-α)100% that the maximum error is within ±E when estimating μ by x-bar The most conservative estimate of n When we have no prior information about p or q, we use p=.5 and therefore q=.5 so that the variance of p-hat, v(p-hat) is maximized. The sample size n obtained using p=.5 and q=.5 is called the most conservative estimate of n. Statistical Hypothesis A conjecture about the unknown population parameter(s). The conjecture may or may not be true. There are two types of statistical hypothesis for each situation, called the Null Hypothesis and the Alternative Hypothesis Null Hypothesis Denoted by H0, and states that the unknown population parameter is equal to a specific value. The Null Hypothesis always has an equal sign in it, and this is the hypothesis that is actually tested. Alternate Hypothesis Denoted by HA , and defined as the complement or negation or opposite of the Null Hypothesis (H0) Type I error denoted by α and represents the probability of rejecting H0 given H0 is true. The value of alpha is also called the significance level of the test. Type II error denoted by β and represents to probability of accenting H0 given H0 is is false. Note: β≠(1-α)! The Power of the Test 1-β, where β is Type II error. Both β and α cannot be reduced simultaneously for fixed sample size n (one goes up when the other goes down). Increasing n maximizes the power of the test, as it lowers both β and α. The Classical or Critical Value Approach to testing Hypothesis I. Formulate H0 and HA II. Select an appropriate test statistic (zcalculated )III. Fix the level of significance (α) and formulate the decision ruleIV. Write your conclusion in words The Decision Rule Aka the Critical Region or Rejection Region, depends on HA and α. If HA is two-sided, we use zα/2 and -zα/2 or t(n-1, α/2) and -t(n-1, α/2). Otherwise we use zα or -zα in the same direction of HA The mean and standard deviation always have... The same units! the P-value An alternate method to test H0 , the P-value is the probabillity, assuming H0 is true that the statistic zc would take an extreme or mre extreme value than the actually observed value. In fact, the p-value is the smallest calculated α or Type I error assuming H0 is true. Thus, we reject H0 if α>p-value. Three methods for Testing Hypothesis a) The Classical or Critical Value Approachb) the p-value Approachc) If HA is two-sided, (1-α)100% confidence interval for μ (ie, Reject H0 if μ=μ0 does not lie in the (1-α)100% confidence interval) Assumptions for using (1- )100% Confidence Interval for two populations when σ is known 1) Two samples are random & independent2) Both samples came from two independent, normal populations3) σ12 (σ1) and σ22(σ2) are known Assumptions for using (1- )100% Confidence Interval for t-distribution 1) Two samples are random & independent2) Both samples came from normal populations3) σ12 (σ1) and σ22(σ2) are unknown but equal The point estimator for the unknown common variance σ2 is sp2 To test the hypothesis about unknown p1 & p2 we combine the information given in both samples to compute estimated variance of p1 & p2 To construct a (1- )100% confidence interval for p1 and p2 we do NOT combine the information contained in both samples to compute the estimated variance A goodness of fit test Tests the Null Hypothesis that the observed frequencies follow a pattern or theoretical distribution. The test is goodness-of-fit because the hypothesis tested is how good the observed frequencies fit a given pattern The -squared goodness of fit test used to test whether of not the sampled multinomial data is in agreement with the hypothesized distribution. OR Testing 3 or more unknown population proportions. In a goodness of fit test, when is the Null Hypothesis rejected? A good agreement between the observed and expected frequencies results in a small value of . A perfect agreement would result in =0. Thus the Null Hypothesis is rejected if is large [upper tail test] For tests of Independence between Criterion A and B... Ho: The two criteria A&B are INDEPENDENT or not related (HoI) For tests of Independence between Criterion A and B... HA: The two criteria A&B are DEPENDENT or related (HAD) For tests of independence, is  For tests of Independence, Eij= For a 2x2 Contingency table, testing for independence for two criteria is equivalent to testing H0: P1=P2 vs HA:P1≠P2 Test of Homogeneity A test of homogeneity involves testing the H0: the proportions of elements with certain characteristics in two or more different populations are the samevsHA: the proportions are not the same Analysis of Variance (ANOVA) A procedure used to test the Null Hypothesis that the means of three of more populations are equal. The grand mean for all k-samples is SST, SSB AND SSW must always be... positive, since it's they're the sum of squares The point estimator for the unknown common variance σk2 is Mean Square within (MSW) Assumptions for ANOVA a. k-samples are random and independentb.Each of the k-samples came from a normal populationc. σ21, σ22, σ23, ,σ2k are unknown but equal The Statistical Linear Regression model is SSxx or SSyy must always be... positive Sxy can be... positive or negative To Construct a (1- )100% Confidence Interval for unknown B, use , where When solving for b in the estimated regression model... write the equation out in general form first. If there it is "a-bx" that means "a+(-b)x" which implies b is negative The population correlation denoted by and defined as the strength of the relationship between two variables, x&y. The sample correlation denoted by r. Since the population correlation is usually unknown, the point estimator of the population correlation is the sample correlation r. The range for is -180% or .8 AuthorJamie_Bee ID279922 Card SetIntro Stats II Final DescriptionStudy help for the Stats II final Updated2014-07-30T15:35:50Z Show Answers