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# 507 Note for STAT 50300 at Purdue

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Date Created: 02/06/15

Hypothesis Testing Statistics 503 Simonsen Required Elements State the scienti c question to be answered De ne the parameters of interest Let H1 be State the null hypothesis H0 both mathematically in terms of parameters and in words Decide based on the problem statement whether the test is directional or not and state the alternative hypothesis HAboth mathematically and in words Name the type of statistical test to be used and the distribution the test statistic follows under H0 State the signi cance level at to be used and the corresponding critical value of the test statistic De ne the rejection region Calculate the test statistic from the data Compare the test statistic to the rejection region or give the Pvalue and compare it to ac Make a decision about the null hypothesis a If the test statistic is in the rejection region state quotreject H0quot b If not state quotdo not reject H0quot 10 Form a scienti c conclusion based on that decision If 9a then start with quotThis study provides evidence quot if 9b then start with quotThis study does not provide evidencequot followed by quotat the 7 signi cance level thatquot followed by the verbal statement of HA These steps are to be followed every time you do a hypothesis test Remarks Use a complete sentence with no symbols This can be fairly general u or H1 and m or 11 or 111 and 112 or Usually an equality Occasionally e g Ftest WMW it may be too cumbersome to express HA mathematically Include the df or other parameters for that distribution if appropriate The critical value and rejection region depend on 0c the df and distribution from 5 and whether HA is directional Show all calculations Ie test statistic lt or gt critical value You MAY NOT use the word quotacceptquot anywhere in this step Ifusing Pvalue reject H0 when P lt 0c Use a complete sentence with no symbols Use similar wording to step 4 You MUST NOT af rm H0 See examples Don39t get too creative Null Alternative Hypothesis Hypothesis nondirectional directional T39teStS H0 H A rejection HA rejection df ts 13900 CI region region One HH0 H H0 tslteth HltH0 tsltetu 111 iHo far Sample tS gt tuz H gt 0 ts gt ta SE7 y i tuZSE TWO H1 H2 H1 3 H2 ts lt itaZ H1 lt H2 is lt tu J71 72 for Alin Independent tS gt taz SE Y1 7 Y2 i Samples H1 gt H2 ts gt ta yl yl totZSE a Paired 111 112 111 at u tS lt itaz 111 lt Hz tS lt ita ndil J71 J72 for uluz S l amp es tS gt tuz H1 gt 2 ts gt ta SEQ 7 y i tustdlff df SE12 SE222 lt n n 7 2 SEquot SE 1 2 quot1 1 quot2 1 Action reject H0 do not reject H0 H0 is true type I error correct nonrejection Truth I false pos1t1Ve true negative H0 1s false correct rejection type 11 error true positive false negative Example quottwoindependentsample nondirectional ttestquot In a study of the periodical cicada Magicicada septendecim researchers measured the hind tibia lengths of the shed skins of 00 individuals Results for males and females are shown in the accompanying table Assuming that hind tibia lengths follow a normal distribution compare the mean hind tibia lengths for male and female cicades usin an appropriate hypothesis test Tibia Length micrometre units Group n mean SD Males 54 7842 287 Females 48 8044 352 Solution notes in square brackets are for your information and are not part of the solution Do male and female cicadas have the same mean tibia length Let H1 be the mean tibia length for male cicadas and u be the mean tibia length for female cicadas H0 H1 u Male and female cicadas have the same mean tibia length HA H1 at u Male and female cicadas have different mean tibia lengths Use a twoindependentsample nondirectional ttest SE2 SE2 2 M has atdistribution with df SE SEquot SE quot1 1 quotz 1 On a quiz or exam use df n1 n 7 2 Test at significance level at 005 critical value is tozs 1986 Use at 005 unless a different level is specified in the problem Ifts lt 71986 or tS gt 1986 then will reject H0 tS 91 under H0 s2 s2 2872 3522 USE 1 2 L 2 54 06408 quot1 quot2 71 72 7842 8044 7 SE 06408 tS lt 71986 so reject H0 at level 005 If tS were gt 1986 would also reject H0 If1986 lt tS lt 1986 would not reject H0 This study provides evidence at the 05 significance level that male and female cicadas have different mean tibia lengths If Hg was not rejected conclude quotThis study does not provide evidence at the 05 significance level that male and female cicadas have different mean tibia lengthsquot 3152 ts If you wish you may also include the Pvalue in the concluding sentence Place it in parentheses Pi immediately before the word quotatquot ie quotThis study provides evidence P00021 at the 05 significance level that male and female cicadas have different mean tibia lengthsquot or in the case of nonrejection quotThis study does not provide evidence P 021 at the 05 significance level that male and female cicadas have different mean tibia lengths quot Example quotonesample nondirectional ttestquot In a study of the effect of aluminum intake on the mental development of infants a group of 92 infants who had been born prematurely were given a special aluminum depleted intravenous feeding solution At age 18 months the neurologic development of the infants was measured using the Bayley Mental Development Index which is designed so that 100 is the average score in the general population How do premature infants fed this solution compare in neurologic development to the general population BMDI score n mean s 92 97 95 21 73 Solution notes in square brackets are for your information and are not part of the solution Do premature infants fed a special aluminumdepleted intravenousfeeding solution have different neurological development than the general infant population Let u be the mean BMDI score for premature infants fed the solution Let no 100 the mean BMDI score for the general infant population H0 u no premature infants fed the solution have the same mean BMDI score as the general population HA H1 at u premature infants fed the solution have a different mean BMDI score than the general population Use a onesample nondirectional ttest tS has a tdistribution with df n7 1 92 7 1 91 under H0 Test at significance level at 005 critical value is tozs 1986 Use at 005 unless a different level is specified in the problem Ifts lt 71986 or tS gt 1986 then will reject H0 SE i 2173 n M ts y 100 2 9795 100 41905 SE 2265 71986 lt tS lt 1986 so do not reject H0 at level 005 If tS were lt itozs or gt tozs then would reject H0 This study does not provide evidence at the 05 significance level that premature infants fed the solution have a different mean BMDI score than the general population If Hg was not rejected conclude quotThis study provides evidence at the 05 significance level that premature infants fed the solution have a different mean BMDI score than the general populationquot 2265 If you wish you may also include the Pvalue in the concluding sentence Place it in parentheses Pi immediately before the word quota quot ie quotThis study does not provide evidence P037 at the 05 significance level that premature infants fed the solution have a different mean BMDI score than the general populationquot or quotThis study provides evidence P 0037 at the 05 significance level that premature infants fed the solution have a different mean BMDI score than the general population quot Example quottwoindependentsample directional ttestquot A pain killing drug was tested for efficacy in 50 women who were experiencing uterine cramping pain following childbirth Twenty five of the women were randomly allocated to receive the drug and the remaining 25 received a placebo inert substance Capsules of drug or placebo were given before breakfast and again at noon A pain relief score based on hourly questioning throughout the day was computed for each woman The possible pain relief scores ranged from 0 no relie to 56 complete relief for 8 hours Summary results are shown in the table Assuming that pain relief scores follow approximately a normal distribution test for e icacy of the drug at reducing uterine cramping pain PAIN RELIEF SCORE Treatment n Mean SD Placebo 25 3196 1205 Drug 25 2532 13 78 Solution notes in square brackets are for your information and are not part of the solution Is the drug more effective than the placebo at reducing pain Let H1 be the mean pain relief score of women who take the placebo Let u be the mean pain relief score of women who take the drug H0 H1 u The drug and the placebo are equivalent at reducing uterine cramping pain or quot The mean pain relief score of women who take the drug is the same as that for women who take the placeboquot H A M lt u The drug is more effective than the placebo at reducing uterine cramping pain or quotThe mean pain relief score of women who take the placebo is less than that of women who take the drugquot Use atwosample directional ttest SE12 SE 2 ts y1 yz SE has atdistribution with df SE4 SE 47 under H0 1 quot1 1 quotz 1 Test at signi cance level or 005 critical value is t05 1678 Use or 005 unless a different level is specified in the problem Ifts lt 71678 then will reject H0 Note if HA were quotgtquot then this statement would be quotIfts gt 1678 then will reject H0quot 2 2 2 2 USE 1 Z is2 1205 1378 366 n1 n2 25 25 71 72 3196 2532 SE 366 tS gt 71678 so do not reject H0 at level 005 If tS lt 71678 then would reject H0 This study does not provide evidence at the 005 significance level that the drug is more effective than the placebo at reducing uterine cramping pain If Hg was rejected conclusion would be quotThis study provides evidence at the 005 significance level that the drug is more effective than the placebo at reducing uterine cramping painquot The Pvalue for this example is P0962 ts 1813 Example onesample directional ttest In a study of a viral disease researchers wish to know ifthe viral infection elevates body temperature We know that the mean healthy body temperature in the general population is 3 7 0 C A group of 25 patients su ering from this virus have their temperatures taken We observe that their mean temperature is 375 C and their SD is 10 C Assuming that body temperatures follow approximately a normal distribution use these data to determine whether the viral infection elevates body temperature Solution notes in square brackets are for your information and are not part of the solution Do people with this viral infection have elevated body temperature Let u be the mean body temperature of all people infected with this virus Let no 370 the mean healthy body temperature in the general population H0 u uo people infected with this virus have normal mean body temperature HA u gt no people infected with this virus have elevated mean body temperature Use a onesample directional ttest y H0 tS has a tdistribution with n71 24 df Test at signi cance level at 005 critical value is t05 1711 Use at 005 unless a different level is speci ed in the problem Ifts gt 1711 then will reject H0 Note if HA was quotltquot then this statement would be quotIfts lt 1711 then will reject H0quot s 10 SE 020 J J25 ts y 37537002 25 SE tS gt 1711 so reject H0 at level 005 If tS lt 1711 then would not reject H0 This study provides evidence at the 005 signi cance level that people infected with this virus have elevated mean body temperature If Hg was not rejected conclusion would be quotThis study does not provide evidence at the 005 signi cance level that people infected with this virus have elevated mean body temperaturequot The Pvalue for this example is P00098 Example nondirectional Wilcoxon Mann Whitney WMW test A plant physiologist conducted an experiment to determine whether mechanical stress a ects the growth of soybean plants Young plants were randomly allocated to two groups of 13 plants each Plants in one group were mechanically agitated by shaking for 20 minutes daily while plants in the other group were not agitated After 16 days of growth the total stem length cm of each plant was measured with the results given in the accompanying table Assuming that plant stem lengths do not follow a normal distribution compare the treatments to determine whether mechanical stress changes the distribution of soybean stem length Does mechanical stress change the growth of soybean plants H0 the distribution of soybean plant stem lengths is the same whether mechanically stressed or not HA the distribution of soybean plant stem lengths is different when mechanically Control Stress 252 247 295 257 301 265 301 270 302 271 302 272 303 273 306 277 311 287 312 289 314 297 335 300 343 306 stressed than when not stressed Use a nondirectional WMW test Us maXK1 K2 follows the WMW null distribution with n 13 n 13 Test at level at 005 critical value is 124 ist Z 124 then will reject H0 K111012121212121213131313131485 K2 0111111111227205 check K1 K2 1485205 169nln2 1313169 okay Us maXK1 K2 1485 Us 1485 2 124 so reject H0 This study provides evidence at the 005 signi cance level that the distribution of soybean plant stem lengths is different when mechanically stressed than when not stressed Pvalue lt 0001 WMW tests can also be done with directional alternative hypotheses in which case only K1 or K2 depending on direction would be used We did not cover directional WMW tests Example quotpairedsample nondirectional ttestquot A study was conducted to compare the sodium content in the plasma of female southern fur seals with the level in the milk of these seals The following observations of the sodium content in mmoVL were obtained in 10 randomly selected seals Assuming that sodium content in milk and plasma follows a normal distribution conduct an appropriate hypothesis test with 0 004 Solution notes in square brackets are for your information and are not part of the solution Do milk and plasma differ in their sodium content in female southern fur seals Let H1 be the mean sodium content in milk Let u be the mean sodium content in plasma Seal Milk Plasma Difference 1 93 147 54 2 104 157 53 3 95 142 47 4 815 141 595 5 95 142 47 6 95 147 52 7 765 148 715 8 805 144 635 9 795 144 645 10 87 146 59 mean 887 1458 571 s 899 461 795 H0 H1 uz milk and plasma have the same mean sodium content in female southern fur seals H A ul 3 H2 milk and plasma have different mean sodium content in female southern fur seals Use a paired sample nondirectional ttest ts yd W has a tdistribution with df 10 7 1 9 under H0 diff Test at level at 004 critical value is toz 2398 If tS lt 2398 or if tS gt 2398 then will reject H0 SE sdff795251 I ym 71 y2 571 SE W W Jfo S SE M diff tS lt 72398 so reject H0 This study provides evidence at the 004 signi cance level that milk and plasma have different mean sodium content in female southern fur seals The Pvalue is 296XlO399 for this example Example quotpairedsample directional ttestquot A random sample of 9 middle aged men who did not exercise regularly ub 39ect was selected to participate in a study to test the effect of regular exercise on resting heart rate RHR First the RHR of each subject was measured Then each man was required to exercise on a treadmill for 30 minutes 4 times a week for 6 weeks Afterwards the RHR of each man was measured again The results are shown in the table The standard deviation of the di erences is 5 388 Assuming that RHR has a normal distribution use these data to test at the 0 0 significance level whether regular exercise decreases mean RHR mean 849 Note that the Diff column is not given in the question but is part of SD 6 75 your solution Solution notes in square brackets are for your information and are not part of the solution Does regular exercise decrease mean RHR Let H1 be the mean RHR of men before exercising regularly Let u be the mean RHR of men after exercising regularly H0 H1 u regular exercise has no effect on mean RHR HA H1 gt u regular exercise decreases mean RHR Use a paired sample directional ttest ts yd W has a tdistribution with df n 7 1 9 7 1 8 under H0 diff Test at level at 001 Critical value is t01 2896 If tS gt 2896 then will reject Ho If tS lt 2896 then will not reject H0 SW 5388 WWW tS lt 2896 so do not reject H0 E E 849 803 46 2537 SE 1796 ydx m 1796 S SE0 7 SE 1796 803 46 572 5388 A This study does not provide evidence at the 001 significance level that regular exercise decreases mean RHR The Pvalue is 00174 for this example 1 l 2 2 l l l 2 2 2 Example nondirectional sign test A study was conducted to compare the sodium content in the plasma of female Seal Milk Plasma Diff 7 southern fur seals with the level in the milk of these seals The following 1 93 147 39393939 quot observations of the sodium content in mmoVL were obtained in 10 randomly 2 104 157 7 selected seals Assuming that sodium content in milk and plasma do not follow 3 95 142 7 a normal distribution conduct an appropriate hypothesis test with 0 0 02 4 81 5 141 7 Note the assumption of nonnormality may be implicit in the problem rather 5 95 142 7 than stated explicitly as above The Diff column is not calculated for you 6 95 147 7 but is part of your solution 7 76 5 148 7 Solution notes in square brackets are for your information and are not part of 8 80395 144 T the solution 1 T Do milk and plasma differ in their sodium content in female southern fur seals Let it be the probability for any seal that the milk has higher sodium content than does plasma H0 11 12 For any seal it is equally likely that milk or plasma will have higher sodium content OR milk and plasma have the same sodium content H A 11 12 For any seal it is not equally likely that milk or plasma will have higher sodium OR milk and plasma have different sodium content Use a nondirectional sign test BS maXNN7 has a folded binomial10 05 distribution under H0 Test at level or 002 critical value is B02 10 from Table 7 will reject H0 if BS 2 10 N7 0 N7 10 so BS N7 10 BS 2 10 so reject H0 This study provides evidence at the 002 signi cance level that milk and plasma have different sodium content Pvalue calculation do if requested Where Y is Binomialn10 11 05 Pvalue 2PrY z 10 2PrY 10 2 13051 05 0001953 Pvalue justification this is why it works don t actually do this in your problem notice that N 7 N7 n and we can think of these as the number of successes and failures Pvalue 13an B 235 13an N7 ZBSPrHU N7 235 PrHU N7 233an n N 235 PrN 235 1t05PrN Sn BS n052PrN 235 05y 05H 2 quot 305 y3 11 05 by symmetry since 11 05 Example quotdirectional sign testquot A random sample of 9 middle aged men who did not exercise regularly ub at was selected to participate in a study to test the effect of regular exercise on resting heart rate RHR First the RHR of each subject was measured Then each man was required to exercise on a treadmill for 30 minutes 4 times a week for 6 weeks Afterwards the RHR of each man was measured again The results are shown in the table Assuming that RHR does not have a normal distribution use these data to test at the 001 significance level whether regular exercise decreases mean RHR 0 9 Note the assumption of nonnormality may be implicit in the problem rather than stated explicitly as above The Diff column is not calculated for you but is part of your solution Solution notes in square brackets are for your information and are not part of the solution Does regular exercise decrease RHR Let it be the probability for any man that he has higher RHR before exercising regularly H0 11 05 regular exercise has no effect on RHR OR men are equally likely to have higher RHR before or after exercising regularly HA 11 gt 05 regular exercise decreases RHR OR men are more likely to have higher RHR before exercising regularly than after exercising regularly Use a directional sign test BS N has a binomial8 05 distribution under H0 We completely discard subject 3 because ofa tie so n is reduced by 1 If HA were 11 lt 05 then we would use BS N instead Test at level or 001 Critical value is chml 8 Table 7 If BS 2 8 then will reject H0 BS N 7 Since Bs lt 8 we do not reject H0 This study does not provide evidence at the 001 significance level that regular exercise decreases mean RHR Pvalue calculation do if reguested Where Y is Binomialn 8 11 05 Pvalue PrY z 7 PrY 7PrY 8 057 051 058 05 81058 00352 Pvalue justification this is why it works don t actually do this in your problem If HA were 11 lt 05 then would do this with N7 Pvalue Pan B 2 BS Pan N 05y 05H y 3 23 PrN 235 1105 n 05 y l l 2 2 l l l 2 2 2

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