Essay About Tests Of Significance And True Value Of The Parameterany

Confidence Intervals / Tests of Significance & Power
Confidence Intervals / Tests of Significance & PowerConfidence intervalsPurpose – to estimate an unknown parameter with an indication of how accurate the estimate is and of how confident we are that the result is correct68-95-99.7 rule says that the probability is about 0.95 that x bar will be within two standard deviations of the population mean score Estimate +/- margin of error – form for confidence intervalConfidence interval for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameterAny normal distribution has probability about 0.95 within +/- 2 standard deviations of its meanTo construct a level C confidence interval:Find the number z* such that any normal distribution has probability C within +/- z* standard deviations of its meanIf two tail, you find the z* using table A on the left side and on the right side, the area in the middle = C = our confidence interval Probability C that x bar lies between (u – z*(std dev/sqr rt of n)) and (u + z*(std dev/sqr rt of n)): = M = (u +/- z*(std dev/sqr rt of n))Choosing Sample SizeTo obtain a desired margin of error m, plug in the value of std. dev. And the value of z* for your desired confidence level and solve for the sample size nn = Tests of Significance Purpose: assess the evidence provided by the data in favor of some claim about the population parameters; compare observed data with a hypothesis whose truth we want to assessExample – need to conclude whether or not the true means of the two random samples are different Key steps:Started with a question about the difference between two sample means and are trying to determine whether or not the data are compatible with no differenceCompare the means of the data and fine the difference between them Turn the results into probabilities If the probability is very small – we have observed something that is very unusual or the assumption (that there is no difference in the means) isn’t trueNull hypothesis – the statement being tested in a test of significanceAlternative hypothesis – the statement we hope/suspect to the true; is true when we reject the nullTest StatisticMeasures compatibility between the null hypothesis and the data Purpose: use it for the probability calculation that we need for our test of significanceZ = (estimate – hypothesized value)/(standard deviation of estimate)P-ValuesProbability, assuming null hypothesis is ture, that the test statistic would take a value as extreme or more extreme than that actually observedSmaller the P-value, the stronger the evidence that the null hypothesis is not true To solve:Find Z = (estimate – hypothesized value)/(standard deviation of estimate)Check Table A for equivalent z score and find the percent/probability Need to times the probability by two since it is on both sides of the curve and then you get the percentage of observing a difference as outside of the main area of the curve Significance LevelIf the p-value is as small or smaller than alpha, we say that the data are statistically significant at level alphaSteps for Tests of SignificanceState null hypothesis and determine alternative hypothesisCalculate the value of the test statistic on which the test will be based – to measure how far the data are from the nullFind the p value for the observed data – this is the probability that the test statistic will weigh against the nullState a conclusion – conclude whether you can reject null or notTests for a Population Mean SummaryzTest for Pop Mean = z = (Xbar – mu0)/(std. dev./sq. rt. n)Two-Sided Significance Tests / Confidence Intervals?Two-sided tests rejects null hypothesis exactly when the value mu0 falls outside a level 1-mu confidence interval for the mu (mean)P-values versus fixed alphaP-value is the smallest level of alpha at which the data are significant P-value gives us a better sense of how strong the evidence is Power & Inference as a DecisionPower = probability that a fixed alpha significance test will reject the null when a particular alternative value of the parameter is true Use when:“is N subjects a large enough sample for this project?”Calculate power in 3 steps:State null and alternative hypotheses, as well as the significance level, alphaFind values of Xbar that will lead us to reject the nullUse z test → z = (Xbar – mu0)/(std. dev./sq. rt. n)Plug in the z score for z from Table A for whatever probability of test significance you choose Solve for Xbar → as the unknown variableCalculate probability of observing these values of Xbar when the alternative is trueUse same formula as in part 2, but solve for P instead of XbarHigh power is desirable Ways to increase power:Increase alphaIncrease sample sizeDecrease standard deviationConsider a new alternative hypothesis further away from mu0Types of ErrorType I – reject null and accept alternative when null is trueType II – accept null when alternative is trueInference for Mean of a Population / Comparing Two MeansT distributionUse when standard deviation is not knownWhen standard deviation is not known,         we estimate it with the sample standard deviation (s/sq. rt. n), which is equal to the standard error of the statisticOne-sample T statistic:T =(Xbar – mu)/( s/sq rt n)Degrees of freedom?One-Sample T Confidence IntervalC = Xbar +/- t* (s/sq rt n)Margin of error = t*(s/sq rt n)One-Sample T TestSame as with z testt = (Xbar – mu0)/(s/sq. rt. n)Calculate power of T testWe assume a fixed level of significance, often alpha = 0.05Step 1 – decide on a standard deviation, significance level, whether the test is one-sided or two-sided and an alternative value of mu to detectWrite the event that the test rejects the null in terms of XbarFind probability of this even when population mean has this alternative valueComparing Two MeansUse when you are comparing responses from two groups and the responses in each group are independent of those in the other groupTwo Sample Z StatisticZ = Two Sample T Significance Test T = Two Sample T Confidence Interval Questions:When do you use t vs. s? Is it just when you don’t know the standard deviation? How do you choose S in the power equation?Comparing Two Proportions & Chi-Squared TestsTwo-way tables: organize data by two factors (e.g., group by age, then by education)Marginal distributions – summarize each variable independentlyTwo-way table describes the relationship between variables Chi-square test:Purpose: to determine if the differences in sample proportions are likely to have occurred by just chance because of the random samplingNull hypothesis in this test = no relationship between the variablesCompare actual counts with expected counts to test thisExpected count = (row total x column total) / nFormula:[pic 1]Large values for x2 → evidence against H0When to use?All individual expected counts are 1 or more (≥1) No more than 20% of expected counts are less than 5 (< 5)