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Statistical Methods and Computing

by: Vance Bode Sr.

Statistical Methods and Computing 22S 105

Marketplace > University of Iowa > Natural Sciences and Mathematics > 22S 105 > Statistical Methods and Computing
Vance Bode Sr.
GPA 3.72

Mary Cowles

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Mary Cowles
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This 6 page Class Notes was uploaded by Vance Bode Sr. on Friday October 23, 2015. The Class Notes belongs to 22S 105 at University of Iowa taught by Mary Cowles in Fall. Since its upload, it has received 32 views. For similar materials see /class/228075/22s-105-university-of-iowa in Natural Sciences and Mathematics at University of Iowa.

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Date Created: 10/23/15
2281105 Statistical Methods and Computing Introduction to Hypothesis Testing Lecture 14 Mar 10 and 147 2008 Kate Cowles 374 SH7 33570727 kcowles statuiowaedu Example I claim that my husbands resting pulse rate is 45 beats per minute This is very low and would be typical of either a highly trained athlete or a sick individual To test my claim7 you wish to measure his restr ing heart rate on 5 different occasions Here7 the population of interest is all possir ble measurements of my husbands resting pulse rate My claim may be interpreted as saying that the mean M of this population of values is 45 beats per minute 2 Introduction to Hypothesis Testing Recall that statistical inference is using data contained in a sample to draw conclusions or make decisions about the entire population from which the sample is taken Two main goals of statistical inference o estimation of unknown population parame ters otesting speci c hypotheses about unknown population parameters The purpose of hypothesis testing is to assess the evidence provided by data about some claim concerning a populationMk Moore D s The 3am 13mm gfsmmms 4 Suppose the measurements you get are 42 52 43 48 47 The sample mean i 464 Does this provide evidence against my claim We will consider this question by asking what would happen if my claim were true and we repeated the sample of 5 measurements many times 5 Suppose rst that we knew that the standard deviation of measurements of my husbands restr ing heart rate was a 4 beats per minute 0 lf the claim that M 45 is true7 the same pling distribution of i from 5 measurements is normal with mean M 45 and standard deviation 0 4 179 a B 0 We can judge whether any observed i is sure prising by nding it on this distribution 7 The alternative hypothesisis the claim forwhich we are trying to nd evidence 0 symbolized H 1 ln the example about my husbands heart rate7 your alternative hypothesis probably was Hazngt45 The p value of the test is the probability7 come puted assuming that H0 is true7 that the ob served outcome would take a value as extreme as or more extreme than7 what we actually ob served 0 Small prvalues are evidence against the null hypothesis 6 Terminology of hypothesis tests The null hypothesis is the statement being tested 0 The test is intended to assess the strength of evidence against the null hypothesis 0 Usually is a statement of no e ect no diff ference777 nothing going on o The null hypothesis is commonly symbolized as Hot 0 H0 is a statement about an unknown popur lation parameteri oExample H0 M 45 8 The result of a hypothesis test is a decision The possible outcomes are called 0 Rejecting the null hypothesis 0 Not rejecting the null hypothesis Before we carry out the test7 we must decide how strong we will require the evidence to be in order for us to reject Hot We specify this in terms of a signi cance level 0 The signi cance level is how small we will require the prvalue to be in order to reject Hot 0 symbol is a 0 conventional choices are a 05 and a 01 Example my husbands resting heart rate We will choose a 05 as the signi cance level at which to carry out the test To nd the prvalue of our results7 we will stanr dardize i so we can use the normal table 0 Remember the prvalue is computed assume ing H0 is true7 so the value of u to use is the value stated in H0 fin 0N5 464745 179 z 078 11 Onesided and twosided tests of hy potheses The hypothesis test we just conducted was one sided test We were interested only in showing that the value of the unknown parameter diff fered from that given in H0 in one direction H0u45 Haugt45 We might also have stated the hypotheses this way H0u 45 Haugt45 10 According to Table A7 the probability of a value this large or larger is 0218 We would say that for this test result p 0218 Since this is larger than a 057 we cannot reject the null hypothesis That is7 we have de cided that the evidence was not suf cient to re ject my claiml 12 In specifying null and alternative hypotheses 0 There must be no overlap in the range of values included in the two hypotheses o All possible values of the unknown popula tion parameter must be covered in one or the other of the two hypotheses Twosided hypothesis tests Example We wish to compare fasting serum cholesterol levels in persons over 21 living in a group of islands in the South Paci c with typical levels found in the US We know that levels in adults over 21 in the US are approximately normally distributed with 0 mean 190 mgdl 0 standard deviation 40 mg dll We have no idea what the relative levels of serum cholesterol are on the islands as compared with the US The hypotheses for our two sided test are H0 u 190 Ha 11 7f 190 Before we look at our data7 we will decide on the signi cance level a for our test Let us choose a 051 We then perform blood tests on 100 adults from the islands and nd that the sample mean level i 1815 mgdll To carry out our hypothesis test7 we note that7 if H0 is true7 the sampling distribution of i is normal with M 190 Uj 4 14 We will assume that the levels on the islands are normally distributed with 0 unknown mean u 0 known standard deviation 40 mgdl 16 We will standardize the value of i that we ob served to nd out how likely we would have been to get a value as extreme as what we got7 or more extreme7 if H0 were true 9 M0 0W 1815 7 190 4 72125 We must nd out what area under the standard normal curve lies 0 to the left of 721125 0 and to the right of 21125 The answer is 1017 1017 1034 17 This is the p 7 value for the test Since p lt 05 we reject the null hypothesis and conclude that serum cholesterol levels are different among adult residents of the Paci c lslands than among adults in the US 1815 7 190 381 72231 18 One sample ttests If we dont know the population standard devir ation7 then we 0 estimate it with the sample standard devia tion 5 o compute a t statistic rather than a z statistic 0 compare to a t distribution with the appro priate degrees of freedom Example If we do not assume that we know a for serum cholesterol levels among residents of the Paci c lslands From the sample of 100 adults7 we compute s 381 mgdl We then compute i M0 t sv 20 We try to use Table C to nd the area to the left of 721231 and to the right of 2231 under a t curve with 99 degrees of freedomi The closest we can come is that under a t curve with 100 degrees of freedom7 the area in one tail would be between 01 and 102 Thus we conclude that the prvalueis somewhere between 02 and 04 SAS can do a much better job for usl It would provide a prvalue of 102791 Thus7 if we had chosen a 057 we would reject the null hypothesis Types of error in hypothesis testing a probability of making Type I error H 0 M M0 probability of making Type ll error Ha I M 74 0 True state of the world H0 is false H0 is true Reject H0 Correctl Type I error Do not reject H0 Type ll error Correctl reject H0 l H0 is true a P Pfail to reject H0 l H0 is false or7 put another way 23 24 power 1 i probability of The corresponding value for the sampling disr correctly rejecting tribution of i if H0 is true is H0 when it is i M ZU false depends on our de nition of 45 1645 179 H a Return to the example of my husbands resting 479 heart rate 0 What value of i would have been required in order to reject H0 M 45 in favor of Ha M gt 45 if a 05 For a standard norrnal7 z 1645 cuts off the upper 05 area


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