Statistical Hypothesis A statistical hypothesis is an assertion or conjecture concerning one or more populations.

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1 TEST OF HYPOTHESIS I a certai perspective, we ca view hypothesis testig just like a jury i a court trial. I a jury trial, the ull hypothesis is similar to the jury makig a decisio of ot- guilty, ad the alterative is the guilty verdict. Here we assume that i a jury trial that the defedat is't guilty uless the prosecutio ca show beyod a reasoable doubt that defedat is guilty. If it has bee established that there is evidece beyod a reasoable doubt ad the jury believes that there is eough evidece to refute the ull hypothesis, the jury gives a verdict i favor of the alterative hypothesis, which is a guilty verdict. I geeral, whe performig hypothesis testig, we set up the ull (H o) ad alterative (H a) hypothesis i such a way that we believe that Ho is true uless there is sufficiet evidece (iformatio from a sample; statistics) to show otherwise. Statistical Hypothesis A statistical hypothesis is a assertio or cojecture cocerig oe or more populatios. Types of statistical hypothesis: 1. Null hypothesis the hypothesis that we wish to focus our attetio o. Geerally, this is a statemet that a populatio parameter has a specified value. The hypothesis that is tested ad the oe which the researcher wishes to reject or ot to reject. Specifies a exact value of the populatio parameter. Deoted by H o.. Alterative hypothesis - The hypothesis that is accepted if the ull hypothesis is rejected. Allows for the possibility of several values. Deoted by H a or H 1. May be directioal (quatifier is < or >) or o-directioal (quatifier is ). Example State the ull ad alterative hypothesis i the followig statemets: 1. The percetage of juior high school studets who pass math subjects durig summer is 65%. Ho : 65 H a : 65. At least 5 typhoos o the average hit the coutry every year. Ho : 5 H a : 5 3. Forty-eight percet of high school graduates are computer illiterate. Ho : 48% H a : 48% 4. At most 6 out of 1 married wome i the rural areas are house wives. H : 6% H : 6% o a

2 Exercise State the ull ad alterative hypothesis. 1. At most 65% of public school childre are malourished.. O the average at least /3 of high school studets who pass their math subjects pass their physics subjects. 3. Less tha half of the ewly ursig board passers immediately get their visa i a year. 4. A ma should have at least 8 hours of sleep everyday % of elected public officials came from the same uiversity. A test of hypothesis is the method to determie whether the statistical hypothesis is true or ot. I performig statistical test of hypothesis we cosider the followig situatios: Null hypothesis TRUE FALSE Reject TYPE I Error Correct Decisio Do ot Reject Correct Decisio TYPE II Error The probability of committig a TYPE I error is also called the level of sigificace ad is deoted by a small Greek symbol alpha. Some of the commo values used for the level of sigificace are.1,.5, ad.1. For example, if =.1 for a certai test, ad the ull hypothesis is rejected, the it meas that we are 9% certai that this is the correct decisio. Importat thigs to kow before coductig a test of hypothesis: 1. Level of sigificace,. The level of sigificace,, is the probability of committig a error of rejectig the ull hypothesis whe, i fact, it is true.. Oe-tailed tests vs. Two-tailed tests Oe-tailed test of hypothesis A oe tailed test is performed whe the alterative hypothesis is cocered with values specifically below or above a exact value of the ull hypothesis. The alterative hypothesis is directioal. Two-tailed test of hypothesis A two-tailed test is performed whe the alterative hypothesis is cocered with values that are ot equal to a exact value of the ull hypothesis. The alterative hypothesis is o-directioal. 3. Test Statistic The value geerated from sample data. Test value to be compared with the critical values. 4. Critical Regio (Regio of rejectio/regio of acceptace) Depeds o the type of test to be performed.

3 If test is oe tailed, the the critical regio is cocetrated o either the left tail (for <) or the right tail of the distributio (for >). If test is two tailed, the the critical regio is distributed o each tail of the distributio. Critical values are obtaied depedig o the type of test to be performed If the test is oe tailed, the sigificace level will be the area either o the left tail or o the right tail of the distributio. If the test is two tailed, the area i each tail of the distributio will be /. Steps i hypothesis testig 1. Set up the ull ad alterative hypothesis.. Specify the level of sigificace. 3. Determie the critical regio ad the correspodig critical values. 4. Compute the value of the test statistic. 5. Make a decisio. Reject H i favor of H 1 if test statistic falls i the critical regio. Do ot reject H if it falls i the acceptace regio. 6. Draw appropriate coclusios. Remark Although most textbooks i statistics use the term accept ad reject whe iterpretig results of statistical test of hypothesis, it is very importat to uderstad that the rejectio of a hypothesis is to coclude that it is false. While the acceptace of a hypothesis merely implies that there is o sigificat evidece to say that it is false. Testig a sigle populatio mea, Suppose a radom sample of size is take from a ormal populatio with mea ad stadard deviatio. To test the claim that the populatio mea is equal to a certai value, we perform the test of hypothesis for the populatio mea,. Null ad alterative hypothesis H : = H 1: <, >, or. CASE 1: is kow or ukow but 3. Test statistic: The test statistic depeds o the case where the problem falls uder. z x

4 Critical regio/ value/s: For a oe tailed test: Reject H if z < - z for a left-tailed test z > z for a right-tailed test For a two tailed test: Reject H if z < -z/ or z > z/ for a two tailed test Note deotes the z value with a area of to the right. z z Example 1. A radom sample of 1 recorded deaths i Midaao durig the past year showed a average life spa of 71.8 years. The sample also showed a 8.9 years of stadard deviatio. Does this data idicate that the average life spa of people livig i Midaao is greater tha 7 years? Use a.5 level of sigificace. Solutio: Step 1. H : 7 ad H 1 : 7 Note that the alterative hypothesis is directioal, we are performig a oe tailed test; specifically a right tailed test. Step.. 5 (Level of sigificace) Step 3. Critical Regio z z z z Critical Regio Step 4. Test Statistic

5 Sice the populatio stadard deviatio is ukow but the sample size is large eough, that is, have x z / 1 Step 5. Sice the value of the test statistic falls uder the critical regio, we reject the ull hypothesis i favor of the alterative hypothesis. 3, we ca substitute the sample stadard deviatio for, thus we Step 6. Sice the ull hypothesis is rejected, we say that there is sufficiet evidece to say that the average life spa of people livig i Midaao is greater tha 7 years.. A salo ower believes that the average umber of their regular customers gets a haircut ad pedicure is 5. A radom sample of 5 regular customers showed that of them did a haircut ad pedicure. With =5% is the belief of the salo ower true? Assume that the stadard deviatio is 1.5. Solutio Step 1. H : 5 ad H 1 : 5 Note that the alterative hypothesis is bi-directioal, we are performig a two tailed test. Step.. 5 (Level of sigificace) Step 3. Critical Regio z z z 1 96 or z z z 1 96 z /. 5. z /. 5. Critical Regio Critical Regio Step 4. Test Statistic Sice the populatio stadard deviatio is kow we have x 5 z

6 Step 5. Sice the value of the test statistic falls uder the critical regio, we reject the ull hypothesis i favor of the alterative hypothesis. Step 6. Sice the ull hypothesis is rejected, we say that there is sufficiet evidece to refute the claim of the salo. Ad hece, we ca say that based o the sample evidece, the average umber of customers who did haircut ad pedicure is ot 5. CASE : ukow ad < 3 Test statistic x t s Critical Values/Regios For a oe tailed test: Reject H if t < - t with df = -1 for a left-tailed test t > t with df = -1 for a right-tailed test For a two tailed test: Reject H if t < -t/ or t > t/ with df - -1 for a two tailed test Example 1. I the past a study has bee made o call ceter agets o their sleepig habits. The result showed that the average umber of hours they took for sleepig is at most 8 hours. A radom sample of 5 call ceter agets where asked ad showed that the average umber of hours they took for sleepig is 6.5 with a sample variace of hours. Test whether past the study still true. Use α=.1. Solutio Step 1. H : 8 ad H 1 : 8 Note that the alterative hypothesis is directioal, we are performig a oe tailed test, specifically a right tailed test. Step.. 1 (Level of sigificace) Step 3. Critical Regio z z z z / Critical Regio Step 4. Test Statistic 1.96

7 Sice the populatio stadard deviatio is ukow but the sample size is large eough, that is, have x 5 z s 1. Step 5. Sice the value of the test statistic falls uder the critical regio, we reject the ull hypothesis i favor of the alterative hypothesis. 3, we ca substitute the sample stadard deviatio for, thus we Step 6. Sice the ull hypothesis is rejected, we say that there is sufficiet evidece to say that the average life spa of people livig i Midaao is greater tha 7 years. Exercise 1. Ritz grocery store declared that their average daily icome is P 15, with a stadard deviatio of P,. A radom sample of grocery stores of the same kid had bee asked of their daily icome ad said to have P19, o the average. If we assume that the daily icome is ormally distributed ca we coclude that Ritz grocery store daily icome declaratio is right with 95% cofidece?. Last, Cetral Luzo farmers demaded more supply o the fertilizers the govermet is providig them. They said that the supply should be kilos o the average for 1 hectares of lad. I 7, a radom sample of 35 farmers was asked the umber kilos of fertilizers a 1-hectare lad would eed. The result showed 4 kilos o the average with 3 kilos stadard deviatio. With 9% cofidece is it true that the average kilos of fertilizers eeded for a 1 hectare lad have chaged sice their last demad? Assume ormal distributio. 3. The daily average umber of major defects detected per module by the Quality Assurace team that is cosidered ormal is less tha or equal to 6. To access the Software Developmet team s quality of developed modules a radom sample of 35 modules was tested ad foud out that the mea umber of major defects per module is 8 with a stadard deviatio of major defects. Based o the result with α=5% is the daily average umber of major defects ot ormal? 4. The mayor of Las Piñas City wated to hire a batch of teachers for his ewly costructed elemetary school. Based from previous studies by his educatio committee, the average age of elemetary school teachers i Las Piñas is 4 years old with a stadard deviatio of 5. A sample of 36 ewly hired elemetary teachers was take ad the followig iformatio is obtaied: average age is 35. Does this idicate that the average age of elemetary school teachers decreased? Use a.5 level of sigificace ad assume ormality. 5. A radom sample of 8 cigarettes of a Marlboro has a average icotie cotet of 4. milligrams ad a stadard deviatio of 1.4 milligrams. Is this I lie with the maufacturer's

8 claim that the average icotie cotet does ot exceed 3.5 milligrams? Use a.1 level of sigificace ad assume the distributio of icotie cotets to be ormal. 6. A ew process for producig sythetic diamods ca be operated at a profitable level oly if the average weight of the diamods is greater tha.5 karat. To evaluate the profitability of the process, six diamods are geerated, with recorded weights:.46,.61,.5,.48,.57, ad.54 karat. a. Is there evidece to suggest that the variace of these measuremets is greater tha. karat? b. Do the six measuremets preset sufficiet evidece that the process will be profitable? Test at the.1 level. Testig a value of a sigle populatio proportio, π Suppose there are x successes i a radom sample of size draw from a ormal populatio. We wish to test whether the proportio of successes i a certai populatio is equal to some specified value. Null ad alterative hypothesis H : π = π H 1 : π π ; π > π ; or π < π Critical values/ critical regio For a oe-tailed test: Reject H if z < z α for a left tailed test. Reject H_ if z > z α for a right tailed test. For a two-tailed test: Reject H if z < zα or z > zα Test statistic pˆ z where 1 x pˆ (assumptio p 5, (1 p ) 5) Example 1. A chocolate maufacturer targets a 8 out of 1 public approval of their ew chocolate recipe to release i the market. A radom sample of 7 people where give a taste test ad resulted a 75% approval of the product. Will the compay release the product i the market with.5 level of sigificace? Solutio Step 1. H :. 8 ad H1 :. 8 Note that the alterative hypothesis is o-directioal, we are performig a two tailed test.

9 Step.. 5 (Level of sigificace) Step 3. Critical Regio z z z z z 5 z z z Critical Regio Critical Regio Step 4. Test Statistic pˆ z * Step 5. Sice the value of the test statistic falls uder the critical regio, we reject the ull hypothesis i favor of the alterative hypothesis. Step 6. Sice 1.5 is ot greater tha 1.96, we do ot reject Ho. There is o sufficiet sample evidece to refute the claim of the Maufacturer. Exercise 1. A ew papaya soap, claims that 8% of wome who used it observed ski whiteig withi two weeks of use. A kow competitor of the said product surveyed if the claim is true. The survey result said that 6 out of 1 wome observed whiteig o their ski i two weeks of use. With α= 5% is the claim of the ewly produced papaya soap true?. A evirometal o-govermet orgaizatio (NGO) declared that 7 out of 1 edagered birds i the coutry dies by hutig. The alarmig report pushed the govermet evirometal departmet to coduct their survey if such report is true to be able to put a immediate actio to it. The result showed that 67% of the edagered bird species dies i hutig. With 9% cofidece ca the govermet evirometal departmet coclude that the NGO s report is true?

10 3. The studet govermet is preparig for this year school s foudatio day. They are havig a hard time where to coduct the talet s ight to make sure there are eough seats for attedees. I the past, 7% of the total umber of studets atteds talet s ight. They coducted a survey o 1 studets ad foud out that 8 will be attedig. Test whether the umber of attedees this year will be the like the past with 95% cofidece. Testig the value of a populatio variace, Suppose we wish to test whether the populatio variace is equal to a specified value Null ad alterative hypothesis H : = H 1 : < ; > ; or. Critical values/regio For a oe-tailed test: Reject H if for a left tailed test with df = -1 1 for a right tailed test with df = -1. For a two-tailed test: Reject H if : or with df = -1. Test statistic: ( 1) s 1 Example 1. A kow cady maufacturer claims that their high-speed machies miimize the defected cadies produced. They claim that their daily productio oly cotais a variace of 1 defected cadies. A radom sample of 15 days cady productio resulted a sample variace of 165 defected cadies. Usig =5% is the claim true? Step 1. H : 1 ad H a : 1 Note that the alterative hypothesis is o-directioal, we are performig a oe tailed test; specifically, a right tailed test. Step.. 5 (Level of sigificace) Step 3. Critical Regio 5 ( 1). ( 1 1)

11 or ( 1). ( 14) Step 4. Test Statistic ( 1)s Step 5. Sice the value of the test statistic does ot fall uder the critical regio, we do ot reject the ull hypothesis i favor of the alterative hypothesis. Step 6. Sice 3.1 is ot greater tha , we do ot reject Ho. There is o sufficiet sample evidece to refute the claim of the maufacturer that the variace of defected cadies is 1.. The agecy that moitors earthquakes said itesity of earthquakes i the Philippies has a variace of. A study was coducted o 3 earthquakes that hit the coutry sice 198. A stadard deviatio of 1. resulted. With =1% is the agecy correct? Exercise 1. A bottlig compay wats to access the state of their machies i terms of defects or over spillig o their daily productio of bottled juices. If the over spillig does ot exceed a variace of bottled juices daily, the machies are said to be i good coditio. A radom sample of 6 juices resulted stadard deviatio of 5 bottles. Usig α= 5% are the machies still i good coditio?. A publishig compay cosiders a book to be error free ad ready to be out i the market if the variace error is less tha 5 per page. A radom sample of 3 pages was gathered o a book ad foud a variace of 7 errors. Usig α= 1% is the book ready to be out i the market? 3. Te ecoders from Clark Data Ceter Ic. have applied for a higher positio i the typig pool of compay ad took a typig test. The followig are the duratio of each ecoders for the test: 75, 7, 59, 6, 63, 55, 5, 7, 45, ad 85 secods. Costruct a 9% cofidece iterval for the true variace of the times required by ecoders to complete the test paragraph. 4. I a experimet with rats, a behavioral scietist used a auditory sigal hat food is available through a ope door i the cage. The scietist couted the umber of trials eeded by each rat to lear to recogize the sigal. Assumig that the populatio of umber of trials is approximately ormal, calculate a 95% cofidece iterval for the populatio variace with the give data below: Does this iterval support the claim that the variability of this data set is about?

12 Summary of the Test Statistics ad Critical Regios for Hypothesis Testig H z Value of Test Statistic x, kow or 3 x t, =-1, s ukow ad z pˆ 1 where = ( 1) s 3 x pˆ H a < > Critical Regio z z t t z z z z z z ad t t t t ad z z z z t t z z ad 1 ad z z 1 Chapter Review Choose the letter of the correct aswer. (For umbers 1-4)The average mothly icome of a gasolie boy is said to be P6,5. A radom sample was coducted to 35 gasolie boys ad foud out that the mea mothly icome is P 6, with P 3 stadard deviatio. With α=1% ca we coclude that the average mothly icome of a gasolie boy is P6,5? 1. What is the correct alterative hypothesis? a. The average mothly icome of a gasolie boy is P6,5. b. The average mothly icome of a gasolie boy is less tha P6,5. c. The average mothly icome of a gasolie boy is ot P6,5. d. The average mothly icome of a gasolie boy is at least P6,5.. Base o the alterative what type of test is it? a. oe-tailed b. two-tailed c. both a ad b d. oe 3. What is the value for Z? a b c. 1.5 d. 1.57

13 4. What is the decisio? a. Reject Ho. b. Reject H1. c. Both a ad b d. Noe (For items 5-8)A certai music club believes that 78% of musicias plays guitar. I radom sample of 5 musicias of them play guitar. With 95% cofidece is the belief of the music club right? 5. What is the value of po? a..78 b..88 c.. d What is the value of qo? a..78 b..88 c.. d What is the value of Zα/? a b c d What is the decisio? a. Reject Ho. b. Reject H1. c. Both a ad b d. Noe 9. If σ =.35, s = 3.56 ad =33 what is the value of χ? a b c d If = ad α = 1% what is the tabular value of χ if we are dealig with a two-tailed test? a b c d Commuter studets at the Uiversity of the Philippies claim that the average distace they have to commute to campus is 6 kilometers per day. A radom sample of 16 commuter studets was surveyed ad resulted to the followig data: The average distace of 31 km ad a variace of 64. The value of the test statistic for this is a. z = -.5. b. z =.394. c. t =.5. d. t = 1.5.

14 1. For a small-sample left-tailed test for the populatio mea, give the sample size was 18 ad.1. The critical (table) value for this test is a b c d Rejectio of the ull hypothesis whe it is true is called a. type I error b. type II error c. o error d. statistical error 14. The type of test is determied by a. Ho b. Ha c. α d. 1 α 15. This deotes the probability of committig a TYPE I error. a. Ho b. Ha c. α d. 1 α 16. Uder the Philippie judicial system, a accused perso is presumed iocet util prove guilty. Suppose we wish to test the hypothesis that the accused is iocet (H ) agaist the alterative that he is guilty (H 1). A type I error is committed, if ay, if the court a. covicts the accused whe, i fact, he is iocet? b. covicts the accused whe, i fact, he is guilty? c. acquits the accused whe, i fact, he is iocet? d. acquits the accused whe, i fact, he is guilty? 17. Which statemet is/are correct? I. A ull hypothesis is a claim (or statemet) about a populatio parameter that is assumed to be true util it declared false. II. A alterative hypothesis is a claim about a populatio parameter that will be true if the ull hypothesis is false. a. I b. II c. Both I ad II d. Neither I ad II For umbers 18-: Accordig to a compay s records, the average legth of all log-distace calls places through this compay i a year is 1.55 miutes. The compay s maagemet wats to check if the mea legth of the curret log-distace calls is differet from 1.55 miutes. A radom sample of 5 such calls placed through this compay produced a mea legth of miutes with a stadard deviatio of.65 miutes. Use a.5 level of sigificace. 18. What is the computed test value? a. z =.67 b. t =.67 c. z = -.67 d. t = What is the correspodig critical value? a b..575 c..633 d What is the type of test to be used? a. right tailed test b. left tailed test c. two tailed test d. caot be determied