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# Weekly Notes 10 TMATH 110 C

University of Washington Tacoma

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This 0 page Class Notes was uploaded by Qihua Wu on Monday December 7, 2015. The Class Notes belongs to TMATH 110 C at University of Washington Tacoma taught by KENNEDY,MAUREEN C. in Fall 2015. Since its upload, it has received 8 views. For similar materials see Intro Stat Applications in Math at University of Washington Tacoma.

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Date Created: 12/07/15

Hypothesis testing testing a population claim with sample data Step 1 In the context of the study state the null hypothesis Ho the hypothesis where the population mean is equal to the claim and the alternative hypothesis Ha or H1 one that is alternative to the null hypothesis Null hypothesis is always the one with the equal sign when translated into mathematical terms Alternative hypothesis can be either two tailed or one tailed depending on the context If we are interested in only one direction Right tailed greater than the supposed mean when the age is greater than 20 Left tailed less than the supposed mean when the age is less than 20 Unless the direction is given our alternative hypothesis is two tailed Step 2 Assume the null hypothesis is true choose a test statistic whose probability distribution is known If standard deviation of the population is known use zvalues if not use t values zvalue sample meannull hypothesis mean population standard deviation sample sizequot5 t n1 sample meannull hypothesis mean sample standard deviation sample sizequot5 If the observation is close to 0 then the sample mean is very close to the null hypothesis mean and vice versa Assumptions of the statistical test The sample data results from simple random sample It is from normal distribution either the data itself is a normal distribution or with a sample size of more than 30 central limit theorem Step 3 Identify the statistical signi cance level alpha to de ne what is considered an unusual result if null hypothesis is true and identify the critical values and critical regions rejection region based on the signi cance level Compute the test statistic from the sample data determine pvalue and report decision either reject or fail to reject the claim The p value is the density in the probability distribution where the value is at or more extreme than the observed statistics A pvalue less than signi cance level would mean your observed statistic is unusual if the null hypothesis is true therefore you should reject the null hypothesis The pvalue is always less than 1 The pvalue is not the probability the null hypothesis is true given the sample data since after we had done the sample this then becomes the realization of a random variable and we do not use probability for the realization of the random variable When we use the pvalue it is more precise than the critical value because it allows us to see how close it is to the signi cance level And when we have two tail alternative we have to use pvalue times 2 since we have two sides of extreme values instead of one even if the pvalue on one side is above the signi cance level we should still times 2 since we are trying to see how far it is from the signi cance level Step 13 should be conducted before looking at the data to avoid biases Step 4 Draw conclusion in the context of the study under investigation either reject or fail to reject Types of error in hypothesis testing Type I error rejecting null hypothesis when it is true this is only affected by the signi cance level The rate this error is happening is called alpha This error happened when the observed value falls into the critical region even if the null hypothesis is true Type II error fail to reject the null hypothesis when it is false The rate this error is happening is called beta It depends under the distribution of the test statistics under the true but unknown alternative hypothesis Beta decreases when Standard deviation decreases Sample size increases Signi cance level increases Not advised because increasing the signi cance level makes it easier to reject the null hypothesis when it is true increases the rate of type I error but since it makes it easier to reject the null hypothesis then the probability of failing to reject the null hypothesis when it is false decreases Power of the test 1 beta probability the null hypothesis is rejected when it is false Correlation Linear correlation coef cient measurement of how strong is the linear relationship between the 2 variables ranges from 1 strongly negative relation to 1 strongly positive relation where 0 is when there is no relationship If there is a correlation between two variables we call the two variables covary however it does not mean one thing caused the other Assumption Data are obtained from random sampling

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