Working with Two Populations. Comparing Two Means

Transcription

1 Workig with Two Populatios Comparig Two Meas

2 Coditios for Two-Sample Iferece The data are from two radom samples from two distict idepedet populatios. Normality. Two sample t procedures are more robust tha oe sample. Whe applyig the Cetral Limit theorem, we use the sum of the sample sizes to determie the accuracy. Each populatios eeds to be at least 0 times their respective sample size.

3 Cofidece Itervals Draw two idepedet radom samples of size ad of size from ormal populatios. The cofidece iterval for is: Oe-sample mea x t * s Two-sample mea x x t * s s

4 Test of Sigificace To test the hypothesis H o : = Oe-sample mea t xs o Two-sample mea t x x s s

5 Degrees of freedom We ever wat to over estimate the accuracy of a t procedure. Choose of the smaller sample size if a calculator or computer program is uavailable to determie the degrees of freedom.

6 The beauty of the graphig calculator The distributio of the two-sample t statistic is close to the t distributio with degrees of freedom df give by: s s s s df

7 Example A dieticia has developed a diet that is low i fats, carbohydrates, ad cholesterol. Although the diet was iitially iteded to be used by people with heart disease, the dietitia wishes to examie the effect this diet has o the weights of obese people. Two radom samples s of 00 obese people each are selected, ad oe group of 00 is placed o the low-fat diet. The other 00 are placed o a diet that cotais approximately the same quatity of food but is ot as low i fats, carbohydrates ad cholesterol. Each perso, the amout of weight lost (or gaied) i a 3-week period is recorded. The results are below. Low-Fat Diet Other Diet Mea weight loss 9.3 lbs 7.4 lbs Sample variace.4 6.3

8 Form ad iterpret a 95% cofidece iterval for the differece betwee the populatio mea weight losses for the two diets. sample t-iterval mea weight loss for low-fat mea weight loss for other Give two idepedet radom samples. Both samples are very large, so Cetral Limit Theorem guaratees approximately ormal distributios. Assume both populatios at least 000 people. x x t * s s df = ,3.7 We are 95% cofidet that the true mea differece lies betwee.673 ad 3.7 pouds more for the low-fat diet tha the other diet.

9 Test to see if there is a differece i weight loss for the two diets. sample t-test t mea weight loss for low-fat mea weight loss for other H o : H a : (same coditios as sample t-iterval from part a) x x s s df = 93 p-value=.006 =.05 We reject H o. Sice the p-value <, there is eough evidece to believe the low-fat diet weight loss is differet tha the other diet.

10 Example A radom sample of studets from a high school were chose to determie if their sittig pulse rate was lower tha their stadig pulse rate. Each studet s pulse rate was measured i both positios. Ca we coclude the sittig pulse rate is lower?

11 Ca we coclude the sittig pulse rate is lower? sample t-test true mea pulse rate sittig true mea pulse rate stadig H o : H a : Give two idepedet radom samples

12 sample special case Northwest Austi tha the air pollutio i cetral Austi. Air pollutio measuremets are take from both areas ad are icluded i the table below. Air pollutio is measured i parts per millio. Does there seem to be a sigificat differece i the air quality of Northwest Austi ad cetral Austi? Cetral Northwest

13 sample special case Northwest Austi tha the air pollutio i cetral Austi. Air pollutio measuremets are take from both areas ad are icluded i the table below. Air pollutio is measured i parts per millio. Does there seem to be a sigificat differece i the air quality of Northwest Austi ad cetral Austi? Date 5/3 5/6 5/9 7/ 7/5 7/8 8/ 8/6 Cetral Northwest

14 Matched pair t-test A radom sample of studets from a high school were chose to determie if their sittig pulse rate was lower tha their stadig pulse rate. Each studet s pulse rate was measured i both positios. Ca we coclude the sittig pulse rate is lower?

15 We fail to reject H o sice the p-value is great tha =.05, there is ot eough evidece to believe the true mea differece is greater tha 0 idicatig sittig pulse is ot lower. Matched pair t-test true mea differece (stadig d H : 0 H : 0 0 Collectio Give radom sample Boxplot is roughly symmetrical, data is approximately ormal. Safe to ifer populatio at least 40 studets. t d a d sittig) differece df = 3 p-value.409 Box Plot

16 Two proportios

17 Cofidece Iterval: Differece of proportios ˆ ˆ pˆ z* p p ˆ ˆ pˆ z* p p a f p p z * pˆ pˆ pˆ pˆ

18 Cofidece Iterval: Differece of proportios pˆ pˆ pˆ pˆ pˆ pˆ pˆ pˆ pˆ pˆ pˆ pˆ

19 Cofidece Iterval: Differece of proportios Draw a sample of size from a populatio p ad draw a idepedet radom sample of size from aother populatio p, the cofidece iterval for the differece of proportios is a f p p z * p a p f p a p f

20 Test of Sigificace To test the hypothesis H o : p p z p p ca hf F ca hf c c c c a f H G I K J p p p p p c p c where p c x x

21 I the late 70 s there were itesive atismokig campaigs sposored by both federal ad private agecies. Suppose that the America Cacer Society radomly sampled,500 adults i 979 ad the sampled,000 adults i 98 to determie whether there was evidece that the percetage of smokers had decreased. Give a 90% proportio z-iteval p proportio who smoked i 979 p proportio who smoked i 98 Give two radom samples take from two idepedet populatios. Assume that both populatios are at least 0,000 people.

22 I the late 70 s there were itesive atismokig campaigs sposored by both federal ad private agecies. Suppose that the America Cacer Society radomly sampled,500 adults i 979 ad the sampled,000 adults i 98 to determie whether there was evidece that the percetage of smokers had decreased. Give a 90% p ( p) p ( p ) Both samples sizes are large eough for a ormal distributio.

23 p p z * a f a f p p p p a f a f., a f We are 90% cofidet that the true differece i proportios is betwee.0309 ad.0849 higher for those who smoked i 979 tha those who smoked i 98.

24 Do these data idicate that the proportio of smokers decreased over this -year period?.0 proportio z-test H0 : p p Ha : p p p p z F a f p p HG I K J a ff H500 z 356. p - value =.0009 p We reject H o. Sice p-value<, there is eough evidece to believe the proportio who smoked i 979 is higher tha the proportio i 98. I K