Statistics Lecture 13 Inference for a Population Mean

Transcription

1 The govermet claims that studets ear a average of $4500 durig their summer break from studies. A radom sample of studets gave a sample average of $3975 ad a 95% CI was foud to be ($355,$445). This iterval is iterpreted to mea that:. If the study were to repeated may times, there is a 95% probability that the true populatio mea summer earigs is ot $4500 as the govermet claims.. Because our specific CI does ot cotai the value $4500 there is a 95% probability that the true populatio mea summer earigs is ot $ If we were to repeat our survey may times, the about 95% of all the CI will cotai the value $ If we repeat our survey may times, the about 95% of our cofidece itervals will cotai the true populatio value of the mea earigs of studets. Statistics - Lecture 3 Iferece for a Populatio Mea Two-sample Tests for differece i meas

2 Comparig Two Samples Up to ow, we have looked at iferece for oe sample Our et focus i this course is comparig the data from two differet samples For ow, we will assume that these two differet samples are idepedet of each other ad come from two distict populatios Populatio :, Populatio :, Sample :, s Sample :, s Blackout Baby Boom Revisited Nie moths (Moday, August 8th) after Nov 965 blackout, NY Times claimed a icreased birth rate Already looked at sigle two-week sample: foud o sigificat differece from usual rate (430 births/day) What if we istead look at differece betwee weekeds ad weekdays? Su Mo Tue Wed Thu Fri Sat Weekdays Weekeds

3 Two-Sample Z test We wat to test the ull hypothesis that the two populatios have differet meas H 0 : = or equivaletly, - = 0 Two-sided alterative hypothesis: - 0 If we assume our populatio SDs ad are kow, we ca calculate a two-sample Z statistic: We ca the calculate a p-value from this Z statistic usig the stadard ormal distributio Two-Sample Z test for Blackout Data To use Z test, we eed to assume that our pop. SDs are kow: =.7 ad = 4.5 We ca the calculate a two-sided p-value for Z=7.5 usig the stadard ormal distributio From ormal table, P(Z > 7.5) is less tha 0.000, so our p-value = P(Z > 7.5) is less tha We reject the ull hypothesis at -level of 0.05 ad coclude there is a sigificat differece betwee birth rates o weekeds ad weekdays 3

4 Two-Sample t test with ukow variaces We still wat to test the ull hypothesis that the two populatios have equal meas (H 0 : - = 0) If ad are ukow, the we eed to use the sample SDs s ad s istead, which gives us the two-sample T statistic: The p-value is calculated usig the t distributio, but what degrees of freedom do we use? df ca be complicated ad ofte is calculated by software Simpler ad more coservative: set degrees of freedom equal to the smaller of ( -) or ( -) Two-Sample t test for Blackout Data To use t test, we eed to use our sample stadard deviatios s =.7 ad s = 4.5 We eed to look up the tail probabilities usig the t distributio Degrees of freedom is the smaller of - = or - = 7 4

5 Two-Sample t test for Blackout Data From t-table with df = 7, we see that P(T > 7.5) < If our alterative hypothesis is two-sided, the we kow that our p-value < = 0.00 We reject the ull hypothesis at -level of 0.05 ad coclude there is a sigificat differece betwee birth rates o weekeds ad weekdays Same result as Z-test, but we are a little more coservative 5

6 Two-Sample Cofidece Itervals I additio to two sample t-tests, we ca also use the t distributio to costruct cofidece itervals for the mea differece Whe ad are ukow, we ca form the followig 00 C% cofidece iterval for the mea differece - The critical value t k* is calculated from a t distributio with degrees of freedom k k is equal to the smaller of ( -) ad ( -) Cofidece Iterval for Blackout Data We ca calculate a 95% cofidece iterval for the mea differece betwee birth rates o weekdays ad weekeds: We get our critical value t k* =.365 is calculated from a t distributio with 7 degrees of freedom, so our 95% cofidece iterval is: Sice zero is ot cotaied i this iterval, we kow the differece is statistically sigificat! 6

7 Cofidece Iterval for Blackout Data We ca calculate a 95% cofidece iterval for the mea differece betwee birth rates o weekdays ad weekeds: We get our critical value t k* =.365 is calculated from a t distributio with 7 degrees of freedom, so our 95% cofidece iterval is: Sice zero is ot cotaied i this iterval, we kow the differece is statistically sigificat! Two-Sample t test with ukow variaces Oe more alterative: Suppose we are comparig two populatios that have differet meas but the same stadard deviatios. We wat to ifer about the differece betwee the meas whe the stadard deviatio is ukow. We are assumig that both populatios have the same stadard deviatio but we have two estimates S ad S (the two samples stadard deviatios). The best way to combie theses two estimates to give a more iformative estimator. The pooled estimator of the variace is s p ( ) s ( ) s 7

8 8 Two-Sample t test with ukow variaces The test statistics that should be used i this situatio is Calculate the P-value by usig the t distributio with ( ) degrees of freedom ad the compare it to the appropriate sigificace level Alteratively if we are testig a two-sided hypothesis we ca costruct the appropriate CI: 0 ) ( s T p * * ),( ) ( s t s t p p Matched Pairs Sometimes the two samples that are beig compared are matched pairs (ot idepedet) Eample: Seteces for crack versus powder cocaie We could test for the mea differece betwee X = crack seteces ad X = powder seteces However, we realize that these data are paired: each row of seteces have a matchig quatity of cocaie Our t-test for two idepedet samples igores this relatioship

9 Matched Pairs Test First, calculate the differece d = X - X for each pair The, calculate the mea ad SD of the differeces d Quatity Crack X Seteces Powder X Differece d = X - X Matched Pairs Test Istead of a two-sample test for the differece betwee X ad X, we do a oe-sample test o the differece d Null hypothesis: mea differece betwee the two samples is equal to zero H 0 : d = 0 versus H a : d 0 Usual test statistic whe populatio SD is ukow: p-value calculated from t-distributio with df = 8 P(T > 5.4) < so p-value < 0.00 Differece betwee crack ad powder seteces is statistically sigificat at -level of

10 Matched Pairs Cofidece Iterval We ca also costruct a cofidece iterval for the mea differece d of matched pairs We ca just use the cofidece itervals we leared for the oesample, ukow case Eample: 95% cofidece iterval for mea differece betwee crack ad powder seteces: Summary of Two-Sample Tests Two idepedet samples with kow ad We use two-sample Z-test with p-values calculated usig the stadard ormal distributio Two idepedet samples with ukow ad We use two-sample t-test with p-values calculated usig the t distributio with degrees of freedom equal to the smaller of - ad - Two idepedet samples with ukow ad ad assume they are equal We use two-sample t-test with pooled variace estimator. The p- values is calculated usig the t distributio with + - degrees of freedom Two samples that are matched pairs We first calculate the differeces for each pair, ad the use our usual oe-sample t-test o these differeces 0

11 Summary of Two-Sample Tests JMP! How to make a oe sample t-test like we have leared How to make two sample t-tests