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Extends to F-test for variance equality, independent samples comparisons, and notes on Behrens–Fisher and Cochran–Cox weighted t approaches for unequal variances.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/exercise-7-studentss-t-test-paired-and-independent-t-test/472929/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/exercise-7-studentss-t-test-paired-and-independent-t-test/472929.png","ImageObject",300,407,{"name":42,"@type":43},"Marglet","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-08","2026-09-30",true,{"@type":52,"interactionType":53,"userInteractionCount":30},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What are the steps to test a single population mean when n \u003C 30 using t test?","Question",{"text":62,"@type":63},"State the null hypothesis µ = µ0 and the alternate hypothesis (two-tailed or one-tailed forms). Choose the level of significance (typically 5% or 1%), compute the test statistic, compare tcal with the critical table value ttab, then accept H0 if tcal \u003C ttab or reject H0 if tcal > ttab.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How is inference made in a t test using the table value?",{"text":67,"@type":63},"After computing the test statistic tcal, compare it with the table value ttab at the chosen level of significance and degrees of freedom. If tcal \u003C ttab, the procedure accepts H0; if tcal > ttab, it rejects H0 and concludes a significant difference between the sample mean and the population mean.",{"name":69,"@type":60,"acceptedAnswer":70},"Why is the F-test used before comparing two means from independent samples?",{"text":71,"@type":63},"Before applying the t test for equality of two means, the exercise uses the F-test to check whether the variances are equal. If Fcal \u003C Ftab, variances are treated as equal; if Fcal > Ftab, variances are unequal, affecting how the t test is applied.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},472929,1791425785,{"code":4,"msg":81,"data":82},"success",[83,87,91,94,99,104,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":92,"slug":93},70,"exam",{"id":95,"doc_module":4,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},5,"Comic",60,"comic",{"id":100,"doc_module":4,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},6,"Technology",50,"technology",{"id":105,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},7,"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":95,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":30,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145},2336477982702,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Exercise.7  \nStudents’s t test – paired and independent t test  \nTest for single Mean (n\u003C30)  \n1. Form the null hypothesis Ho: µ= µo  \n(i.e) There is no significance difference between the sample mean and the population mean ie., µ= µo  \n2. Form the Alternate hypothesis H 1 : µ≠µo (or µ>µo or µ\u003Cµo)  \nie., There is significance difference between the sample mean and the population mean  \n3. Level of Significance  \nThe level may be fixed at either 5% or 1%  \n4. Test statistic  \nWhere  ~~ ~~~~ ~~  \n| |\n| --- |\n| |\n\nWhere s =  \n5. Find the table value ttab=t(0 .05,(n-1) d.f  \n1. Inference If tcal\u003Cttab  \n(i) We accept the null hypothesis H0  \n(ii) There is no significant difference  \n(or) if tcal> ttab  \n(i)We reject the null hypothesis H0 (ie) we accept the alternative hypothesis  \n(ii)There is significant difference between the sample mean and the population mean.  \nExample 1  \nBased on field experiments, a new variety green gram is expected to given an yield of 12.0quintals per hectare. The variety was tested on 10 randomly selected farmers fields. The yield ( quintals/hectare) were recorded as 14.3,12 .6,13 .7,10 .9,13 .7,12 .0,11 .4,12 .0,12 .6,13 .1. Do the results conform the expectation?  \nSolution  \nNull hypothesis H0 : µ=12.0  \n(i.e) the average yield of the new variety of green gram is 12.0 quintals/hectare. Alternative Hypothesis: H 1 :µ≠ 12.0  \n(i.e) the average yield is not 12.0 quintals/hectare  \nLevel of significance: 5 % Test statistic  \nFrom the given data  \ns =  \n=  \n=  \n= 1.0853  \nNow   ~~ ~~    \n=    \nTable value  \nt(0 .05,9)=2.262 (two tailed test)  \nInference tcal\u003Cttab  \nWe accept the null hypothesis H0  \nWe conclude that the new variety of green gram will give an average yield of 12 quintals/hectare.  \nNote  \nF-test is used to test the equality of two means  \nif   \nwhere  is the variance of the first sample whose size is n 1.  \n is the variance of the second sample whose size is n2.  \nOtherwise  \nif   \nInference Fcal\u003CFtab  \nWe accept the null hypothesis H0.(i.e) the variances are equal.  \nTest for equality of two means (Independent Samples)  \nGiven two sets of sample observation x 11,x 12,x 13…x 1n . Similarly x21,x22,x23…x2n of sizes n1 and n2 from the normal population.  \n1. Using F-Test , test their variances  \n(i)Variances are Equal: Ho:., µ 1= µ2  \nH 1 µ 1≠µ2 (or µ 1\u003Cµ2 or µ1> µ2)  \nTest statistic  \nWhere  \nVariances are equal  \n(a) When the samples have unequal variances and equal number of observations(n1=n2), the test statistic is  \n(b) When the samples have unequal variances and unequal number of observations (n1≠n2), the test statistic is  \nThis statistic follows neither t nor normal distribution but it follows Behrens-Fisher test. The Behrens – Fisher test is laborious one. An alternative simple method has been suggested by Cochran & Cox. In this method the critical value of t is altered as tw (i.e) weighted t test  \nWhere t 1=t(n1-1) d.ft2=t(n2-1) d.f  \nExample 2  \nA group of 5 patients treated with medicine. A is of weight 42,39,38,60 &41 kgs. Second group of 7 patients from the same hospital treated with medicine B is of weight 38, 42, 56, 64, 68, 69, & 62 kgs. Find whether there is any difference between medicines?  \nSolution  \nHo:., µ 1= µ2 (i.e) there is no significant difference between the medicines A and B as regards on increase in weight.  \nH 1 µ 1≠µ2 (i.e) there is a significant difference between the medicines A and B Level of significance = 5%  \nBefore we go to test the means first we have to test their variability using F-test. F-test  \nHo:., σ 12=σ22  \nH1:., σ 12≠σ22  \nFtab(6,4) d.f=6.16  \n⇒ Fcal\u003CFtab  \nWe accept the null hypothesis H0.(i.e) the variances are equal. Test statistic  \nWhere  \n=  \nTable value  \nttab[(5+7-2)=10]d.f at 5% l.o.s = 2.228 Inference:  \ntcal\u003Cttab  \nWe accept the null hypothesis H0  \nWe conclude that the medicines A and also B do not differ significantly.  \nExample 3  \nThe summary of the results of an yield trial on onion with two methods of propagation is giv","cbCaiskcUoBpRE5a","https://ap.wps.com/l/cbCaiskcUoBpRE5a","pdf",820453,14,"English","# Students’s t test – paired and independent t test\n## Test for single mean (n\u003C30)\n## Test for equality of two means (independent samples)\n## F-test for equality of variances\n## Worked examples (green gram, medicines, onion yield, rubber plants)","[{\"question\":\"What are the steps to test a single population mean when n \\u003c 30 using t test?\",\"answer\":\"State the null hypothesis µ = µ0 and the alternate hypothesis (two-tailed or one-tailed forms). Choose the level of significance (typically 5% or 1%), compute the test statistic, compare tcal with the critical table value ttab, then accept H0 if tcal \\u003c ttab or reject H0 if tcal \\u003e ttab.\"},{\"question\":\"How is inference made in a t test using the table value?\",\"answer\":\"After computing the test statistic tcal, compare it with the table value ttab at the chosen level of significance and degrees of freedom. If tcal \\u003c ttab, the procedure accepts H0; if tcal \\u003e ttab, it rejects H0 and concludes a significant difference between the sample mean and the population mean.\"},{\"question\":\"Why is the F-test used before comparing two means from independent samples?\",\"answer\":\"Before applying the t test for equality of two means, the exercise uses the F-test to check whether the variances are equal. If Fcal \\u003c Ftab, variances are treated as equal; if Fcal \\u003e Ftab, variances are unequal, affecting how the t test is applied.\"}]","Exercise 7 - Students’s t test - paired and independent t test | PDF",1790781644,35]