Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: χ = npq

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1 8.6. Chi-Square ( χ ) Test [MATH] From the DeMoivre -- Laplae Theorem, when ( 1 p) >> 1 1 m np b( n, p, m) ϕ npq npq np, where m is the observed number of suesses in n trials, the probability of suess is p, and q = 1 p. So, the random variable m np χ = npq an be onsidered as the normalized random variable. Squaring both sides of the equation gives χ = = = ( m np) npq ( p + q)( m np) npq ( ) + ( (1 )) ( m np) = + np ( n m nq) ( m E[ X ]) ( n m E[ X ]) p m np q m n q npq nq = + E[X] E[ X ] Therefore, Pearson's umulative test statisti, whih asymptotially approahes a χ distribution, is: ( observed eted ) exp χ = exp eted In Ti-84, > = Pr( X χ ) χ df ( lower, upper, df ) 17

2 Chi-Square tests are used in three ways: The goodness-of-fit test for univariate. Categorial homogeneity with bivariate ategorize first, and then ollet sample data. Categorial independene with bivariate ollet sample data first, then ategorize. Mathematially, the last two tests are the same. Notie that the χ distribution is not symmetrial. The tests are one-sided and right-tailed. The expeted ounts an be alulated as Expeted Count = row olumn table Observed Count (OC) / Expeted Count (EC) C1 C C3 C4 Total R1 R R3 R4 Total 18

3 The test statisti χ is ( observed eted ) exp χ = exp eted The degree of freedom is Provided the df = ( row 1)( olumn 1) row and/or olumn are greater than one. The P-value is ( ) P = Pr X > χ value The sample size must be large enough so that the following onditions are met: 1.) No expeted ounts are less than one..) All of expeted ounts should be no less than five and if they are not, 3.) No more than % of the expeted ounts are less than 5. The null hypothesis assumes that there is no assoiation between the two studied variables, and the alternative hypothesis assumes that there is assoiation between the two studied variables. Example A lub at THS sells donuts for fundraising. It sells plain, strawberry, blueberry, and innamon donuts. The members of the lub wonder if there is a preferene for one of these types of donuts or if eah type is preferred by the same proportion of the students. Suppose a sample of 6 sells is given below. The table entries are observed frequenies or ounts. Assume that the signifiant level is.5. ( The goodness-of-fit test for univariate) Plain Strawberry Blueberry Cinnamon Observed Count

4 Step Statement Question 1 1.) State number of olumn n and number of rows n r. Calulate the degree of df = ( n 1) n 1 freedom: ( ).) State the signifiant level: α 1.) Test: χ Test: The goodness-of-fit test for univariate. Categorial homogeneity with bivariate ategorize first, and then ollet samples. Categorial independene with bivariate ollet samples first, then ategorize..) Null Hypothesis : H 3.) Alternative Hypothesis: H a r Expeted Counts: 3 row olumn table Conditions: 1.) No expeted ounts are less than one..) All of expeted ounts should be no less than five and if they are not, 3.) no more than % of the expeted ounts are less than 5. Test Statisti : 13

5 ( observed eted ) exp χ = expeted 4 ( χ ) P = Pr X >, df value [Ti-84] 1.). Stats ->TESTS-> C: χ - Test 3.) Calulate Rejet H, if P value < α Solution: Step Statement Question 1 1.) State number of olumn n and number of rows n r. Calulate the degree of df = ( n 1) n 1 freedom: ( ).) State the signifiant level: α 1.) Test: χ Test: The goodness-of-fit test for univariate. Categorial homogeneity with bivariate ategorize first, and then ollet samples. Categorial independene with bivariate ollet samples first, then ategorize..) Null Hypothesis : H 3.) Alternative Hypothesis: H a r n = 4, nr = 1 df = 4 1 = 3 α =.5 A univariate goodness-of-fit hi-square test. 1 H : p = pp = ps = pb = p = =.5 4 H : Not all proportions equal to.5 a Expeted Counts: Expeted Counts 131

6 3 4 row olumn table Conditions: 1.) No expeted ounts are less than one..) All of expeted ounts should be no less than five and if they are not, 3.) no more than % of the expeted ounts are less than 5. Test Statisti : ( observed eted ) exp χ = expeted ( χ ) P = Pr X >, df value [Ti-84] 1.). Stats ->TESTS-> C: χ - Test 3.) Calulate Rejet H, if P value < α np p = nps = npb = np = 6(.5) = 15 Eah olumn has more than 5 entries, so, the onditions are satisfied. Let EC be expeted ounts and OC be the observed ounts. OC/EC Plain Straw B-berry Cinnamo n OC 13/15 1/15 16/15 19/15 χ = ( observed exp eted ) expeted ( 13 15) ( 1 15) ( 16 15) ( 19 15) = = pvalue Sine = χ df (,1,3).574 > α. p value > α, there is an insuffiient evidene to rejet the null hypothesis. That is, there is no onvining evidene that the four types of donuts are not equally preferred. 13

7 Example The data of SRS of 9 THS students was olleted to ategorize themselves as liberal, moderate or onservative. A two-way table with resulting data is given below: Liberal Moderate Conservative Total Males Females Total Do these data provide evidene of an assoiation between politial philosophy and gender at THS? Assume the signifiant level is.1 and the sample was olleted SRS without ategorized first by gender. (The test for Independene) Step Statement Question 1 1.) State number of olumn n and number of rows n r. Calulate the degree of df = ( n 1) n 1 freedom: ( ).) State the signifiant level: α 1.) Test: χ Test: The goodness-of-fit test for univariate. Categorial homogeneity with bivariate ategorize first, and then ollet samples. Categorial independene with r 133

8 bivariate ollet samples first, then ategorize..) Null Hypothesis : H 3.) Alternative Hypothesis: H a Expeted Counts: 3 row olumn table Conditions: 1.) No expeted ounts are less than one..) All of expeted ounts should be no less than five and if they are not, 3.) no more than % of the expeted ounts are less than 5. Test Statisti : ( observed eted ) exp χ = expeted 4 ( χ ) P = Pr X >, df value [Ti-84] 1.). Stats ->TESTS-> C: χ - Test 3.) Calulate Rejet H, if P value < α Solution: Step Statement Question 1.) State number of olumn n n = 3, nr =, df = (3 1)( 1) = 134

9 1 3 and number of rows n r. Calulate the degree of df = ( n 1) n 1 freedom: ( ).) State the signifiant level: α 1.) Test: χ Test: The goodness-of-fit test for univariate. Categorial homogeneity with bivariate ategorize first, and then ollet samples. Categorial independene with bivariate ollet samples first, then ategorize..) Null Hypothesis : H 3.) Alternative Hypothesis: H a Expeted Counts: row olumn table Conditions: 1.) No expeted ounts are less than one..) All of expeted ounts should be no less than five and if they are not, 3.) no more than % of the expeted ounts are less than 5. Test Statisti : ( observed eted ) exp χ = expeted r α =.1 Sine there is more than one row and the data were olleted before ategorized, so the test is for independene. H : Gender and politial affiliation are independent. H a : Gender and politial affiliation are dependent. O / E Lib Mod Con Tot M 16/1.8 1/11.4 6/7.8 3 F /3. /.6 16/ Tot Example for ell alulation: M-Lib ell: 36(3) / 9 = 1.8 All data are SRS and the expeted ounts are greater than five, so the onditions are satisfied. 135

10 ( χ ) P Pr X, df value = > ( observed exp eted ) χ = expeted ( ) ( ) ( 6 7.8) = ( 3.) (.6) ( ) =.15 4 [Ti-84] 1.). Stats ->TESTS-> C: χ - Test 3.) Calulate Rejet H, if P value < α pvalue = χ df (.15,1, ) =.341 > Sine the pvalue is greater than the signifiant level, there is not enough evidene to rejet the null hypothesis. That is, no relationship between gender and affiliation an be onluded. α Example [MC11] The following two-way table resulted from lassifying eah individual in a random sample of residents of a small ity aording to level of eduation (with ategories "earned at least a high shool diploma" and "did not earn a high shool diploma") and employment status (with ategories "employed full time" and "not employed full time"). Earned at least a high shool diploma Did not earn a high shool diploma Employed full Not employed Total time full time Total If the null hypothesis of no assoiation between level of eduation and employment status is true, whih of the following expressions gives the expeted number who earned at least a high shool diploma and who are employed full time? (A) (B) (C) (D) (E)

11 Solution: The answer is B. From row olumn 9(8) Expeted ount = =. table 157 Example [FRQ 134] The Behavioral Risk Fator Surveillane System is an ongoing health survey system that traks health onditions and risk behaviors in the United States. In one of their studies, a random sample of 8,866 adults answered the question Do you onsume five or more servings of fruits and vegetables per day? The data are summarized by response and by age-group in the frequeny table below. Do the data provide onvining statistial evidene that there is an assoiation between age-group and whether or not a person onsumes five or more servings of fruits and vegetables per day for adults in the United States? Solution: Average Yes No (97) = or older

12 χ = ( observed exp eted ) exp eted ( 31 4.) ( ) ( ) ( ) ( ) ( ) = = df = ( 1)(3 1) =, p = 1 F χ (8.983, ) =.11 < α. The null hypothesis is value rejeted. That is, the sample data provide strong evidene that there is an assoiation between age group and onsumption of fruits and vegetables for adults in the United States. 138