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Hypothesis Testing

by: Emily Binakonsky

Hypothesis Testing ENGR 0020: Probability and statistics for Engineers I

Emily Binakonsky
Probability and Statistics for Engineers 1
Maryam Mofrad

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About this Document

These notes are on the sample hypothesis testing, its basic concepts, definitions, formulas, errors and approaches.
Probability and Statistics for Engineers 1
Maryam Mofrad
Class Notes
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This 3 page Class Notes was uploaded by Emily Binakonsky on Friday March 20, 2015. The Class Notes belongs to ENGR 0020: Probability and statistics for Engineers I at University of Pittsburgh taught by Maryam Mofrad in Spring2015. Since its upload, it has received 117 views. For similar materials see Probability and Statistics for Engineers 1 in Engineering and Tech at University of Pittsburgh.


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Date Created: 03/20/15
One and Two Sample Hypothesis Tests Emily Binakonsky 1 Statistical Hypotheses a General Concepts i A hypothesis is a conjecture that concerns one or more populations ii The null hypothesis is the claim that is initially assumed to be true 1 It states that there is no significant difference between the two populations concerned Any differences observed between the two specified populations are due to sampling or experimental errors 2 Denoted H0 3 It s in the form H0 9 90 null value iii The alternative hypothesis is the conjectureassertion that is contrary to thC H0 1 Denoted as HA 2 Will have either of these three forms a HA 9 gt 90 Upper tailed test b HA 9 lt 90 Lower tailed test c HA 9 i 90 Two tailed test iV You either fail to reject the H0 or you reject the Denoted H0 11 Statistical Hypothesis Testing A hypothesis test is a method for using sample data to decide whether to reject or fail to reject the null hypothesis The rejection region is the set of all test statistic values for which the null hypothesis will be rejected i Type of Error 1 Type I error a It s when you reject the null hypothesis when it is true b Denoted as a 2 Type 11 error a It s when you fail to reject the null hypothesis when it is false b Denoted as B Possible Situations for Testing a Statistical Hypothesis H0 is true H0 is false Fail to reject the Correct Decision Type 11 Ho Reject the H0 Type I Correct Decision One and Two Sample Hypothesis Tests Emily Binakonsky ii Important Properties of Hypothesis Testing 1 Type I and type II errors are related Generally speaking a decrease in one results in an increase in the probability of the other 2 The size of the rejection region can be reduced by adjusting the critical values 3 Increasing the sample size n will reduce a and 8 simultaneously 4 8 is at a maximum when the true value of a parameter approaches the hypothesized value when the null hypothesis is false iii Test Power I Is the probability of rejecting the H0 given that a specific alternative conjecture is true 111 Pvalues in Hypothesis testing decisions Is the lowest level of significance at which the observed value of the test statistic is significant The Pvalues for a Ztest 1 P value 1 cbz Upper Tailed test 2 P value M2 Lower Tailed test 3 P value Z1 q39z Two Tailed test IV Approach to Hypothesis Testing with Fixed Probability or Type I Error 1 State null and alternative hypotheses 2 Choose a fixed significance level a 3 Choose an appropriate test statistic and establish the rejection region based on a 4 Reject the H0 if the test statistic falls within the rejection region Otherwise fail to reject the null hypothesis 5 Draw scientific engineering conclusions 6 Draw the rejection region and state where the test statistic falls V Significance Testing Pvalue Approach 1 State null and alternative hypotheses 2 Choose an appropriate test statistic 3 Compute the Pvalue based on the computed value of the test statistic 4 Use your judgement based on the Pvalue and knowledge of the scientific system One and Two Sample Hypothesis Tests Emily Binakonsky VI Tests concerning a single mean i Single Sample Tests concerning a single mean 1 Assumptions a X1X2 Xn represents a random sample with mean t and variance 02 gt 0 ii Test procedure for a single Mean and variance known Null Hypothesis H0 1 10 Test statistic Z 2 X30 Alternative Hypothesis Rejection Region of Level 1 Test Upper tailed test H A t gt 0 Z 2 Z Lower tailed test HA 1 lt 0 Z S Za Two tailed test HA 1 at 0 Z S Zg 0R Z 2 Zg 2 2 VII Relationship to the Confidence Interval Estimation a The confidence interval estimation involves the computation of the bounds within which it is reasonable for the parameter in question to lie b Let s consider the following hypothesis testing at a significance level a i H 05 M 0 ii H A t 10 iii This hypothesis testing is equivalent to computing the 1001 a confidence interval on t and rejection the H0 if the 10 is outside the confidence interval


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