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Quantitative Methods

QMS 202

Jason Chin- Tiong Chan

Winter

Description

Central limit theorem The Standard Normal Probability Distribution The standard normal distribution has mean of 0 and For a large sample size the sampling distribution ofis approximately normal irrespective of the xstandard deviation of 1 It is also called the z shape of the population distribution The mean and standard deviation of the sampling distribution of distributionareQQ and The sample size is usually considered to be large if 30un xWA zvalue or standard normal value is the distance xW xnbetween a selected value designed x and theMean of the Sample Meansdivided by the population population mean Q The mean of the distribution of the sample mean will be exactly equal to the population mean if we are QxWThe formula is standard deviationzQQable to select all possible samples of a particular size from a given population W xArea Under the Normal CurveStandard Error of the Mean Approximately 68 percent of the area under theIf the standard deviation of the population is W the standard deviation of the distribution of the normal curve is within one standard deviation of thesample mean is mean WQs Confidence Interval for a Population meanknown Properties of the student t distribution W It is like the normal distribution a continuous distributionThe level of confidence is symbolized by 1001vEIt is like the normal distribution bellshaped and symmetricalis the proportion in the tails of the distribution that where E There is not one t distribution but rather a family of t distributions is outside the confidence interval The proportion in the upperAll t distributions have a mean of 0 but their standard deviations differaccording to thesample size tail of the distribution is 2E and the proportion in the The t distribution is more spread out and flatter at the centre than the standard normal2E lower tail of the distribution is distribution As the sample size increases the t distribution approaches the normal 1 sample z interval distribution because the errors in usingto estimatedecrease with larger samplessW With 90 confidence we could say that the population mean Confidence Interval for a Population meanunknown W daily sales is between 379220 and 400700The Confidence Interval for a Population mean 1 sample t interval When C inc Range increases as wellDegree of freedomn 1What is a HypothesisConfidence Interval Estimation for the Proportion 1A hypothesis is a statement about a population is the sampleThe point estimate for the population proportion T2 In statistical analysis we make a claim that is state a hypothesis collect data andxproportionwhereis the sample size andis the nxthen use the data to test the claim p n3 In most cases the population is so large that it is not feasible to study all the itemsnumber of items in the sample having the characteristics of interest objects or persons in the population For example it would not be possible to contact population proportionTevery accountant in Canada to find out his or her annual income4 An alternative to measuring or interviewing the entire population is to take a sample z critical value from the standardized normal distributionfrom the population and test a statement to determine whether the sample does or samplesize n does not support the statement concerning the population and are greater than 5assuming both xxn What is Hypothesis Testing Use 1 prop Z Interval 1 A procedure based on sample evidence and probability theory which determinewhether the hypothesis is a reasonable statementDetermining Sample Size 2 The procedure for testing a hypothesisSample size determination for the meanStep1Define the parameters of interest1 There are 3 factors that determine the size of a sample noneStep2State null and alternative hypothesesof which has any direct relationship to the size of the populationStep3Identify a level of significance2They areStep4Identify the test statisticithe degree of confidence selectedStep5Compute the value of test statistic and pvalue iithe maximum allowable error e Step6State the statistical decision and business conclusioniii the variation in the population W 1Null HypothesisA statement about the value of a population parameter H3The sample size determination for the mean o 22zWAlternative Hypothesis H A statement that is accepted if the sample datan A2e provide evidence that the null hypothesis is false 2 Level of significance The probability of rejecting the null hypothesis when it is true Sample size determination for the proportion 3Test Statistics A value determined from sample information used to determine1 There are 3 factors that determine the sample size whether or not to reject the null hypothesis2 They are 4pvaluethe probability of observing a sample value is as extreme as or more ithe degree of confidence selected extreme than the value observed when the null hypothesis is true5ii the maximum allowable error eType I error Rejecting the null hypothesis when it is trueiii the population proportion T6 Type II error Do not reject the null hypothesis when it is false3The sample size determination for the proportion 4 Conduct a hypothesis test for a population mean with known population standard2deviation Z testone sample mean z test1zTT n5 Conduct a hypothesis test for a population mean with unknown population 2estandard deviation T testmean sample mean ttestFor npie p o56 Conduct a hypothesis test for a population proportion1 prop z testpie sign One tailed test lower and upper two tailed test when Ho and HA doesnt P valuesigma do not reject HoP value sigma reject HoWhen Hodivide sigma by 2 to calculate t critical anddont compare it with p value HaT statP valueSig 2 Ha t stat p value Sig2 HaT statP value1sig2Ha t stat p value1sig2

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