If Y is normally Distributed, then and 2 Y Y 10. σ σ

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1 ull Hypothei Significance Teting V. APS 50 Lecture ote. B. Dudek. ot for General Ditribution. Cla Member Uage Only. Chi-Square and F-Ditribution, and Diperion Tet Recall from Chapter 4 material on: ( ) ( μ) E μ E Z μ If i normally Ditributed, then and ( ) EZ μ 0 Z ( i μ) EZ ( ) EZ ( ) E E i μ μz Z. 0 If the population ditribution of i normally ditributed, then Z will alo be normally ditributed, and i called the tandard ormal Z. ow conider the ditribution of quared Z : Z i ( i μ ) Imagine what the hape of the ditribution o Z would look like. Think of quaring each alue under a tandard normal ditribution. Thi produce an outcome which eliminate all negatie alue, and yield a probability ditribution which appear to peak at zero, and how declining probabilitie a Z increae. otice where it appear to be located (i.e., the expected alue). Thi quared Z ditribution i called a Chi-Squared Ditribution. It expected alue i.0, and it ariance i.0. It i thu the ditribution of one quared Z. ow conider umming quared Z, where each i randomly, and independently, ampled. Thu we produce a new Random ariable, where the alue i the um of quared Z. Z + Z, Z + Z + Z, and in general, Z i i 3 3 It i alo the cae that + + (Alo can take diff where df are alo diff)

2 Simulation of Std ormal Z and Chi-Square Run with 4,997 and graphed in SPSS a a Hitogram. Thu the interal repreent grouping of the X alue into equal interal for purpoe of production of thee hitogram. The actual ditribution are continuou. Each Chi-quare ditribution i titled CHISQ(df) where the number repreent degree of freedom. ote the aymmetry of Z een with thi large - imulation i not perfect. ote on all ditribution, E ( ), where df, and. Thu each of thee fie ditribution i centered on their expected alue.

3 Algebraic Bai relating Chi-Squared Variable, Population and Sample Variance, and the One-Sample Diperion Tet Starting point i re-examination of, in light of aboe: Thu, ( μ) i ( ) and i ( μ) ( ) ( μ ) i Z Recall that - df Therefore, for a ampling ituation where data point are ampled randomly, and a ariance calculated: ( ) alo, from a two group pooled ituation: ( + ) Pooled + ( ) Thu rearranging the aboe formula gie. Thu and more generally, and et Thee algebraic ubtitution enable the creation of a tet about population diperion hypothee. 3

4 Goal: () We want to tet the hypothei that a population ariance (unknown) equal a pecified alue. () Ue the tandard approach of HST (3) Create a tet tatitic which reflect the bet critical tet of the hypothei. 0 H 0 :, where i ome pecified alue (e.g., H 0 : ) Conider a ampling ituation where we draw a random ample of ize, and calculate, the ample ariance on thoe data. Do our data gie u reaon to reject the null hypothei? H : o if two-tailed, non-directional < o if one-tailed (ue left tail of tet ditribution) if two-tailed, (ue right tail of tet ditribution) > o Form the tet tatitic from the aboe algebra: ( ) Example: One-tailed, alpha.05 H 0 : and 5 > 5 0 H : 350 (thu 8.7) [(5)350]/ Uing Appendix, Table IV, we ee that the Critical Value for Thu our oberation fall in the region of rejection and we reject H 0. The ample data are deiant enough o that (with thee df) we can reject the null. Ue table 4 for df up to 00. Aboe that, ue the large ample approximation in Hay ection 9.6, epecially equation Alo examine ection 9.5 to ee that confidence interal for and are aailable, and baed on the ue of the chi-quared ditribution. Aumption: ormality of the random ariable population ditribution i more important here than for the location tet conidered earlier (ection 9.7 in Hay). 4

5 F-Ditribution The F-ditribution i defined a the ratio of two chi-quared ariable, each diided by their own repectie df. Thu each F ha both numerator ( ) and denominator ( ) aociated with it. A i the cae for t and Chi-quared, F i a family of ditribution. Thi i why table of critical alue are extenie combination of numerator and denominator df. ow conider the uage of F to tet two-ample diperion hypothee. The Two-Sample Diperion Problem and H 0 : where repreent population ariance of two population H A :, yielding a non-directional, two-tailed tet here in thi illutration Our tak, once again i to find an appropriate tet tatitic. Recall that and more generally, So, then and when ample ariance are calculated on two different ample, drawn at random. If the null i true, then the following ratio i ditributed a F, (when the random ampling and normality aumption are met) i ditributed a F (,) 5

6 E(F, ) /( -) Thu the expected alue of an F i cloe to.0, but not exactly. A denominator df rie, the expected alue will be cloer to one. ow, if we draw two ample, at random, from the ame population, or from two population which hae the ame ariance, then i ditributed a F with -, and - Thi approximation hold well if the normality aumption i met. Conider an illutration: Sample I 400 (d0) II 44 (d) 5 F(0,4) 400/44.78 It i typical in thi tet to place the larger of the two alue in the numerator and ue only the upper tail of the F ditribution. Een though it i technically a two-tailed tet, thi arrangement allow u to place the whole alpha in the upper tail and implify the proce. (See Hay ection 9.9 for ue of the F-table and how to find lower tail probabilitie if eer neceary) From Table V, we ee that the C.V. for an F (0,4).60 when α.05 Concluion: reject H 0 a a -tailed tet with α.05 We conclude that the firt ample wa deried from a population which did not hae the ame ariance a the population from which the econd ample wa deried. otice two thing:. The ratio of the d doe not hae to exceed / in order for rejection of ome null hypothee.. Recognize how thi might help u in deciding the appropriatene of the homogeneity of ariance aumption for the two-ample location tet we conidered earlier. From Table V, note that C.V. F (,5) 4.54 and qrt why intereting? 6

7 Relationhip among the Ditribution. ormal i parent for each and limiting form of t and chi-quare.. t approximate td normal Z when df are large. 3. Chi-quare i a um of quared Z 4. F i the ratio of two chi-quared, each diided by their own df 5. t F (, V ) Proof of point #5: t μ from the approximation permitted in the one-ample location tet t t ( μ) Diide numerator and denominator by pop.d. ( μ) ( μ) ( μ) / M ( ) Multiply both numerator and denominator by, and rearrange numerator to place the term in the denominator of the numerator ratio. ow quare both ide: t ( μ) F (, ) M Z M From definition hown aboe, both the numerator and denominator are now chi-quared ariable, each diided by their repectie df. It i alo poible to derie thi proof for the two-ample t, but we won t here. 7

8 Conidering thee four primary probability denity function, and familie of function, it would help to ummarize each of their repectie Expected Value and Variance (not hown for F) and recognize their hape a or df become large. Rely, in part, on ection 9. in the textbook for thi. 8