Chapter 10 - Lecture 2 The independent two sample t-test and. confidence interval

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1 Assumptios Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Upooled case Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Pooled case Idepedet samples - Pooled variace - Large samples Chapter 10 - Lecture The idepedet two sample t-test ad cofidece iterval March d, 010

2 Assumptios Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Upooled case Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Pooled case Idepedet samples - Pooled variace - Large samples Assumptios Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Upooled case Details Cofidece Itervals Hypothesis Test Example Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Pooled case Itroductio Cofidece Itervals Hypothesis Test Example Idepedet samples - Pooled variace - Large samples Itroductio Cofidece Itervals Hypothesis Test

3 Assumptios Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Upooled case Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Pooled case Idepedet samples - Pooled variace - Large samples Review Let X 1,..., X m N(µ 1, σ 1 ) Let Y 1,..., Y N(µ, σ ) The two samples are idepedet. Last lecture we have see how to hadle the case of kow populatio variaces ad the case of ukow populatio variaces whe both > 30, m > 30. Today we will see what happes whe we have ukow populatio variaces with at least oe of 30, m 30.

4 Assumptios Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Upooled case Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Pooled case Idepedet samples - Pooled variace - Large samples Two cases There are two differet cases: Upooled case (o assumptios o variace) Pooled case (assumig σ1 = σ ).

5 Assumptios Details Distributio Obviously sice the populatio variaces are ukow we eed to fid the distributio of the followig radom variable T = X Ȳ (µ 1 µ ) S 1 m + S I previous lecture we have see that if both > 30, m > 30 this follows N(0, 1). If we have 30 or m 30 the this follows t ν How do we calculate the degrees of freedom ν?

6 Assumptios Details Degrees of freedom ν The degrees of freedom are foud if we roud dow to the earest iteger the followig: ν = ) ( s 1 m + s ( ) s ( 1 s ) m m The ickame of this formula is the smile face.

7 Assumptios Details (1 α)100% Cofidece Itervals A (1 α)100% CI for µ 1 µ is X Ȳ ± t ν, α s 1 m + s Remember, degrees of freedom ν is foud by usig the smile face formula.

8 Assumptios Details Hypothesis test Null Hypothesis: H 0 : µ 1 µ = 0 Test statistic: t = x ȳ 0 S 1 m + S Rejectio Regios: t ν t t ν,α if H A : µ 1 µ > 0 t tν,α if H A : µ 1 µ < 0 t tν,α/ ad t t ν,α/ if H A : µ 1 µ 0 Degrees of freedom ν is foud by usig the smile face formula

9 Assumptios Details Example Let say I wat to make a estimatio for the differece betwee male ad female studets i my Stat 319 class. I collect a sample of 15 male ad fid a average fial score of 80 ad sample variace of the scores equal to 5 ad 8 female studets ad fid a average of 78 with a sample variace of the scores of 36. Let s assume that X 1,..., X 15 N(µ 1, σ 1 ), Y 1,..., Y 8 N(µ, σ ). Fid a 95% CI for µ 1 µ.

10 Assumptios Details Example O the same example as before make a test to see if there is a differece i the scores at sigificace level 0.01.

11 Assumptios Itroductio Details I may examples, although the variaces of the two populatios are ukow, they ca be assumed to be equal. For example, the umber of credits male studets ad female studets have each semester, might have differet mea but it feels it is legitimate to assume that the two populatios have equal variace. I this case we have to use a commo value for the estimated variace. So we have T = X Ȳ (µ 1 µ ) S p m + S p = X Ȳ (µ 1 µ ) ( 1 Sp m + 1 ) How do we calculate S p? What is the distributio of the above?

12 Assumptios Itroductio Pooled variace ad distributio S p is calculated as follows: Distributio of T: S p = (m 1)S 1 + ( 1)S + m T = X Ȳ (µ 1 µ ) ( 1 Sp m + 1 ) t m+

13 Assumptios Itroductio (1 α)100% Cofidece Itervals A (1 α)100% CI for µ 1 µ is X Ȳ ± t m+, α s p ( 1 m + 1 )

14 Assumptios Itroductio Hypothesis test Null Hypothesis: H 0 : µ 1 µ = 0 Test statistic: t = x ȳ ( 0) ( 1 sp m + 1 ) t m+ Rejectio Regios: t t m+,α if H A : µ 1 µ > 0 t tm+,α if H A : µ 1 µ < 0 t tm+,α/ ad t t m+,α/ if H A : µ 1 µ 0

15 Assumptios Itroductio Example Let say I wat to make a estimatio for the differece betwee male ad female studets i my Stat 319 class. I collect a sample of 15 male ad fid a average fial score of 80 ad sample variace of the scores equal to 5 ad 8 female studets ad fid a average of 78 with a sample variace of the scores of 36. Let s assume that X 1,..., X 15 N(µ 1, σ1 ), Y 1,..., Y 8 N(µ, σ ) ad also assume equality of variaces σ = σ1 = σ. Fid a 95% CI for µ 1 µ.

16 Assumptios Itroductio Example O the same example as before make a test to see if there is a differece i the scores at sigificace level 0.01.

17 Assumptios Itroductio Questio I the previous lecture we have see what will happe if Let X 1,..., X m N(µ 1, σ1 ) Let Y1,..., Y N(µ, σ ) The two samples are idepedet. Ukow populatio variaces Both > 30, m > 30. What do you thik will happe if we add the assumptio that σ 1 = σ. Aswer: We ca use CLT to approximate with ormal.

18 Assumptios Itroductio (1 α)100% Cofidece Itervals A (1 α)100% CI for µ 1 µ is X Ȳ ± z α s p ( 1 m + 1 )

19 Assumptios Itroductio Hypothesis test Null Hypothesis: H 0 : µ 1 µ = 0 Test statistic: z = x ȳ ( 0) ( 1 sp m + 1 ) N(0, 1) Rejectio Regios: z z α if H A : µ 1 µ > 0 z zα if H A : µ 1 µ < 0 z zα/ ad t z α/ if H A : µ 1 µ 0

20 Assumptios Itroductio Example Let say I wat to make a estimatio for the differece betwee male ad female studets i my Stat 319 class. I collect a sample of 50 male ad fid a average fial score of 80 ad sample variace of the scores equal to 5 ad 35 female studets ad fid a average of 78 with a sample variace of the scores of 36. Let s assume that X 1,..., X 50 N(µ 1, σ1 ), Y 1,..., Y 35 N(µ, σ ) ad also assume equality of variaces σ = σ1 = σ. Fid a 95% CI for µ 1 µ.

21 Assumptios Itroductio Example O the same example as before make a test to see if there is a differece i the scores at sigificace level 0.01.

22 Assumptios Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Upooled case Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Pooled case Idepedet samples - Pooled variace - Large samples Sectio 10. page 493 0, 1,, 3, 4, 5, 6, 7, 8, 9, 30, 31, 3, 33, 34, 35, 36, 37, 38

Statistics for Business and Economics

Statistics for Business and Economics Statistics for Busiess ad Ecoomics Chapter 8 Estimatio: Additioal Topics Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-1 8. Differece Betwee Two Meas: Idepedet Samples Populatio meas,