Student Academic Learning Services Page 1 of 7. Statistics: The Null and Alternate Hypotheses. A Student Academic Learning Services Guide

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1 Student Academic Learning Services Page 1 f 7 Statistics: The Null and Alternate Hyptheses A Student Academic Learning Services Guide Student Services Building (SSB), Rm 204 This dcument last updated: 7/27/2011

2 Student Academic Learning Services Page 2 f 7 The Null and Alternate Hyptheses: befre we begin The Null and Alternate Hypthesis statements are imprtant parts f the analytical methds cllectively knwn as inferential statistics Inferential statistics are methds used t determine smething abut a ppulatin, based n the bservatin f a sample 1 Infrmatin abut a ppulatin will be presented in ne f tw frms, as a mean (µ) r as a prprtin (p) Use the ppulatin mean (µ) in the hypthesis statements when the questin gives yu infrmatin abut the ppulatin in the frm f an average e.g. the average travel time was 40 minutes, µ = 40 minutes Use the ppulatin prprtin (p) in the hypthesis statements when the questin gives yu infrmatin abut the ppulatin in the frm f a fractin, percentage, r decimal e.g. 4 ut f 5 dentists agree, p = ⅘ r p = 80% r p =.80 The Null Hypthesis: H0 Stating the Null Hypthesis is the starting pint f any hypthesis testing questin slutin When slving a prblem, it is written as H 0 : The Null Hypthesis is the stated r assumed value f a ppulatin parameter (the mean r prprtin that is being analyzed) What the questin says the ppulatin is ding The current r reprted cnditin The necessary infrmatin tends t be in the first sentence f the prblem When trying t identify the ppulatin parameter needed fr yur slutin, lk fr the fllwing phrases: It is knwn that Previus research shws The cmpany claims that A survey shwed that When writing the Null Hypthesis, make sure it includes an = symbl. It may lk like ne f the fllwing: e.g. H 0: µ = 40 minutes e.g. H 0: µ 40 minutes e.g. H 0: µ 40 minutes 1 Basic statistics fr business & ecnmics, Duglas A. Lind [et al.]. 3 rd Canadian Ed., McGraw-Hill Ryersn, Trnt. Student Services Building (SSB), Rm 204 This dcument last updated: 7/27/2011

3 Student Academic Learning Services Page 3 f 7 The Alternate Hypthesis: H1 The Alternate Hypthesis accmpanies the Null Hypthesis as the starting pint t answering hypthesis testing questins When slving a prblem, it is written as H 1 : The Alternate Hypthesis is the stated r assumed value f a ppulatin parameter if the Null Hypthesis (H 0 ) is rejected (thrugh testing) The necessary infrmatin tends t be fund in the last sentence f the prblem (r the sentence ending in a? ) When trying t identify the infrmatin needed fr yur Alternate Hypthesis statement, lk fr the fllwing phrases: Is it reasnable t cnclude Is there enugh evidence t substantiate Des the evidence suggest Has there been a significant There are three pssible symbls t use in the Alternate Hyptheses, depending n the wrding f the questin Use when the questin uses wrds/phrases such as: is there a difference...? is there a change...? Use < when the questin uses wrds/phrases such as: is there a decrease? is there less? are there fewer? Use > when the questin uses wrds/phrases such as: is there a increase? is there mre? When writing the Alternate Hypthesis, make sure it never includes an = symbl. It shuld lk similar t ne f the fllwing: e.g. H 1: µ < 40 minutes e.g. H 1: µ > 40 minutes e.g. H 1: µ 40 minutes Student Services Building (SSB), Rm 204 This dcument last updated: 7/27/2011

4 Student Academic Learning Services Page 4 f 7 Reading the Questin Here is an example prblem t demnstrate the prcess f creating Null and Alternate Hypthesis statements. Example A recent survey f cllege campuses acrss Ontari claims that students spend an average f 2.7 hurs a day using their cell phnes. A randm sample f 35 Durham Cllege students shwed an average use f 2.9 hurs a day, with a standard deviatin f 0.4 hurs. D Durham Cllege students use their cell phnes mre than the typical Ontari cllege student? Step 1: Find the ppulatin infrmatin Read the questin carefully and try and find infrmatin that is being presented as, r claims t be, fact. In the first sentence we see the phrases A recent survey and claims that (bth are gd indicatrs that the infrmatin we need is in that sentence) Next, determine if yu are wrking with a ppulatin average (µ) r ppulatin prprtin (p) The infrmatin is given t us in the frm f an average (2.7 hurs) s we knw we will use µ in the Null and Alternate Hypthesis statements S far the Null and Alternate Hypthesis statements lk like this: H 0 : µ H 1 : µ 2.7 hurs 2.7 hurs Step 2: Determine the peratrs (math symbls) Read the questin carefully and find the sentence that ends in?. It is ften (but nt always) the last sentence f the prblem Examine the wrding f the questin sentence, lking fr wrds/phrases that indicate which peratr t use The example questin asks, D Durham Cllege students use their cell phnes mre than the typical Ontari cllege student? Because the phrase mre than is used in the questin, we will use the greater than symbl (>) The Null and Alternate Hypthesis statements nw lk like this: H 0 : µ 2.7 hurs H 1 : µ > 2.7 hurs Student Services Building (SSB), Rm 204 This dcument last updated: 7/27/2011

5 Student Academic Learning Services Page 5 f 7 The Null and Alternate Hypthesis statements must ppse each ther S if the Alternate Hypthesis uses the greater than symbl (>), the Null Hypthesis statement must use the less than symbl (<). Hwever, we knw that the Null Hypthesis statement must als include an equals symbl (=) Cmbining the less than symbl and the equals symbl we have the less than r equals t symbl ( ) The Null and Alternate Hypthesis statements nw lk like this: H 0 : µ 2.7 hurs H 1 : µ > 2.7 hurs Prblem Types During yur curse yu will be asked t analyze and slve a variety f different hypthesis testing questins. Listed belw are sme f the prblem types yu may encunter, and what the Null and Alternate Hypthesis statements might lk like. One Sample A cmparisn f sample data t the stated ppulatin infrmatin t determine if the stated ppulatin infrmatin is still true Use the ppulatin symbls in yur hypthesis statements (µ and p) Tw Samples e.g. H 0: µ = 40 hurs H 1: µ 40 hurs e.g. H 0: p =.20 H 1: p.20 A cmparisn f data frm ne sample t data frm a different sample t determine if the tw ppulatins they came frm are the same Use the ppulatin symbls in yur hypthesis statements (µ and p) When cmparing multiple samples, the Null and Alternate Hypthesis statements can be written tw ways. Be sure t check with yur instructr as t which is preferred e.g. H 0: µ A = µ B r H 0: µ A - µ B = 0 H 1: µ A µ B H 1: µ A - µ B 0 e.g. H 0: p A = p B r H 0: p A - p B = 0 H 1: p A p B H 1: p A - p B 0 Student Services Building (SSB), Rm 204 This dcument last updated: 7/27/2011

6 Student Academic Learning Services Page 6 f 7 Mre than Tw Samples (ANOVA) A cmparisn f sample data acrss mre than tw samples r treatments t determine if the ppulatins are the same When perfrming an ANOVA, yu may be asked t cmment n the variatin/variance f the samples r the means f the samples. Be sure t lk fr the fllwing statements t determine what symbls t use in yur hypthesis statements When Is there (mre/less/difference) variatin Use the ppulatin variance symbl (σ 2 ) in the hypthesis statements e.g. H 0: σ 2 A = σ 2 B r H 0: σ 2 A - σ 2 B = 0 H 1: σ 2 A σ 2 B r H 1: σ 2 A - σ 2 B 0 When Is there difference in the mean/average Use the ppulatin mean symbl (µ) in the hypthesis statements e.g. H 0: µ A = µ B = µ C H 1: the means are nt equal Linear Regressin An analysis f the relatinship between tw variables within a sample t determine the affect changing ne f them (the independent variable) has n the ther (dependent variable) The Null and Alternate Hypthesis statements use the ppulatin crrelatin cefficient (ρ) instead f the ppulatin mean, prprtin, r variance Nte that this symbl is called rh (sunds like rw ). Althugh it lks like the letter p it is nt, and has a very different meaning e.g. H 0: ρ = 0 H 1: ρ 0 Student Services Building (SSB), Rm 204 This dcument last updated: 7/27/2011

7 Student Academic Learning Services Page 7 f 7 Multiple Regressin An analysis f the relatinship between multiple variables within a sample t determine the relatinship (strength and nature) f thse variables. In multiple regressin there is a single dependent variable, but multiple independent variables There are usually mre than ne set f hypthesis statements needed t cmplete the prblem when perfrming a multiple regressin analysis The first set is used when perfrming a glbal test t see if there is a relatinship between any f the independent variables and the dependent variable. The Null and Alternate Hypthesis statements use the symbl β t represent the net regressin cefficients in the ppulatin. There is a β fr each independent variable in the prblem e.g. H 0: β 1 = β 2 = β 3 = 0 H 1: nt all β are 0 Later in a prblem yu may be asked t test each independent variable s regressin cefficient n its wn. In this case, yu will create Null and Alternate Hypthesis statements fr each independent variable e.g. H 0: β 1 = 0 H 0: β 2 = 0 H 0: β 3 = 0 H 1: β 1 0 H 1: β 2 0 H 1: β 3 0 Chi-Squared (χ 2 ) A cmparisn t bserved data t expected data Symbls are nt used in the Null and Alternate Hypthesis statements e.g. H 0: there is NO difference between the (bserved frequency) and the (expected frequency) H 1: there IS a difference between the (bserved frequency) and the (expected frequency) Student Services Building (SSB), Rm 204 This dcument last updated: 7/27/2011

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