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# Test 1 Review Prelim STAT 4210

UGA

GPA 3.82

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## Popular in Statistical Methods

## Popular in Statistics

This 26 page Study Guide was uploaded by Brooke Hull on Monday September 28, 2015. The Study Guide belongs to STAT 4210 at University of Georgia taught by Meghan Lutz in Fall 2015. Since its upload, it has received 139 views. For similar materials see Statistical Methods in Statistics at University of Georgia.

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Date Created: 09/28/15

1 1 1 1 1 1 1 IIIIII I nllllll IIIIIII IIIIIIl 1 1 1 1 1 1 1 533 d 52 Emory CHAPTER l0 INTRODUCTION Does the difference in sample results MEAN anything to us Do we care Is it actually significant If I drive 190 yards on average and then take golf lessons and begin to average T91 yards does this indicate a change in my average yards 39 Yes and No If I drove 191 yards on average with a sample of 30 that day I have shown significant change in my average but if I had a slightly different average over say 600 drives it isn t an indicator of practical significance Additionally it is only 1 yard it isn t enough to change my performance the way 1 point might change my grade average from an A to a B ERRORS Type I error a l reiectecl Ho but in reality Ho is a true statement For experimenters this could mean rejecting a drug that actually works Type II error B I fail to reject Ho but in reality Ho is really false For experimenters this could mean putting a drug on the market because you found there to be no significance to findings of horrible side effects but really those horrible side effects occur in 11 0 people This could also be detrimental Power the probability of reiecting Ho when Ho is false 1 3 or the probability of being right 101 says how often we are okay with rejecting the Ho when it is actually true how often are we willing to be wrong COMPARING TWO GROUPS We call this bivariate analysis Example headache duration Xi length of headache X2 type of medication used to treat the headache We get a quantitative response to a categorical question How long do headaches last on x mediciation Consider this if we give the same person two different medicines on two difference instances of headaches is this independent or dependent This is an example of dependent data The duration of the headache and response depends on the person DEPENDENT DATA DESIGN To make an experiment give us dependent data we use the following techniques if the samples have the same subjects they are automatically dependent this is called within subiects if the data design is matched pairs people matched on characteristics if the observations are taken from the same subiects at different times this is called repeated measures COMPARING TWO POPULATIONS Response categorical Predictor categorical Inference on two proportions p1 p20 or the DIFFERENCE in the population proportions of two groups is zero Saying this means they share a mean or central value on their distribution and therefore are ONE clistribution 131 is my point estimate for pl 132 is my point estimate for 192 INDEPENDENT SAMPLES In this course we only work with independent samples for population proportions It 131 and 132 are independent samples then the standard error of 131 132 is 6 231 232 Jm 231 232111232 This assumes difference variances for the number of samples or n for each group To use this formula The data must follow 1 normal distribution where 111 131 81111 1 131 along with 112 132 81112 1 132 are all at least TO There must also be d cotegoricol response simpe rdndom independent samples HYPOTHESIS DIFFERENCE TESTS FOR lgz 52 H Hypothesis H0 191 292 0 HA P1 P2 i gt lt Define here what your values represent T Test Two sample ztest for population proportions A Assumptions categorical response inclepenclent samples both groups have at least 10 successes and failures in their expected values M Mechanics Ola 005 unless otherwise specified Test statistics 131 132 0 Z0bs A A 1 1 Jp 1 pgtltn 1n 2gt p n1n2 C Conclusion We reiectfail to reiect Ho because pvalue x is less than greater than a005 Rememben Context Decision Justificaiton Example In a study of the effects of aspiring regiment on heart attack deaths Groupl People on the placebo 684 were sampled 28 died Group2 People on the aspirin regimen 676 were sampled 18 died Test if there is a difference between the aspiring and placebo treatments in heart attack death proportions H H I91 292 0 HA 291 pz 72 0 Where 19181192 are the true proportion of heart attack death rates after the placebo and aspirin treatments T Two sample ztest for proportions A independent random samples categorical response at least 10 successes and failures in each group M 010F005 Reiect the null hypothesis if the pvalue is less than Alpha 131 132 0 0041 0027 0 Z obs 1 ii 28181360 2818 1 i 1 n1 712 1360 1360 684 39 676 l43 Two tailecl test 2PrZgtl 43 or 2PrZltl 43 01528 C We fail to reject that there is a difference in heart attack death rates in placebo and aspirin treatments because the pvalue 01528 is greater than a005 BIVARIATE DATA gt Categorical explanatory variable gt Quantitative response variable If we want to know if two populations have the sample mean response we can examine this using the formula Iva 220 Efl 32is my unbiased point estimate If we have an approximately normal distribution from ACH population I know f1 f2 NlM1 M2 5 9 f1 ii If we have independent samples 2 2 S S o 0 0 0 S e x x 1 2 In thIs sItuatIon we assume 0 i G 1 2 n1 n2 Construct CI confidence interval 9 2 2 a S S x x t l 1 2 Here V is degrees freedom and is found by the following S If Smaxlt Zusev 2111 712 2 min 2 S n1 712 If maxZ Zusev 2 2 5min 1 i 1 i Til 1 111 112 1 112 this function is floored or rounded down EXAMPLE MILEAGE OF AMERICAN AND JAPANESE CARS You are interested in buying a new car and your initial criterion is fuel efficiency You want to know whether American and Japanese automobiles get the same gas mileage so you get a random sample of 249 American made cars with an average mileage of 2014 mpg and standard deviation of 641 mpg You collect another random sample of 79 Japanese automobiles with an average mileage of 3058 mpg and standard deviation of 61 1 mpg Construct a 99 Confidence interval for the true difference in gas mileage for American and Japanese automobiles smax E lt 2 gt use v 2 M 712 2 249792326 swan 111 p e icvs e a 2 52 x1 x2 i 1326 3 005 71 11 n22 2 2 2014 3058 i 2626f 611 249 79 1 2537 8343 We are 99 confident that the interval 12537 8343 contains the true difference in average gas mileage between American and Japanese automobiles Additionally we can conclude because this interval is wholly negative that Japanese automobiles have better gas mileage HYPOTHESIS TEST FOR MEAN DIFFERENCES H H0 11 2 0 HA 11 2 i 0 T Independence ttest for two samples A quantitative response independent random samples approximately normal population distributions smax smax either gt 2 01quot lt 2 for approximate degrees freedom 9an 5mm M t point estimate hyp0thesized value 321 322 O Obs standard error 2 2 51 52 711 quot2 C draw conclusions with con rex r clecision ius rificc1 rion EXAMPLE JAPANESE AND AMERICAN CARS Conduct a hypothesis test at a001 to determine whether Japanese cars are more efficient than American cars H H0 H1 2 0 HA H1 2 lt 0 Where 118 le are the true average gas mileage values for America and Japanesemade automobiles respectively T two sample ttest for difference in population means A quantitative response independent random sample approximately normal population distributions smax lt 2 gt use U n1 n2 2 249792 326 Smin 611 M aa00l reiect the null hypothesis if pvalue lt a00i 221 962 0 2014 3058 O 2 2 641 611 51 52 249 79 n1 n2 Pvalue Prt326 lt 13075 gtvalueltOOOlgt lt C We reiect the null hypothesis that the average gas mileage for American and Japanese automobiles is equal because the pvalue is less than 0001 less than a 00l We therefore conclude that there IS a difference between American and Japanese car mileage POOLED VARIANCES smax When lt 2 we assume 01 02 5mm In this case we pool our standard deviations Together 2 2 2 711 351 352 2 n2 x1x2 S and S How do pool these together You assume they have the some populdtion difference Coll their pool devidtion Sp n 1 52 71 1 52 52 52 1 1 Sp 1 12 2dndthestdnddrderrorSe p por5 p ml 1n21 n1 n2 n1 n2 APPLY AMERICAN AND JAPANESE CARS n1 1s n2 1s J249 1641279 16112 Sp n1 1n2 1 24979 2 s e 529 i i 63395i i 08186 n1 712 249 79 ii 32 204 3058 12l754 sex1 x2 08186 63395 tabs 2 When we pool tabs is i 2754 compared to on previous 13075 This is why distinguishing pooled vorionces is important RELATIVE RISK9 ASSES THE RATIO OF TWO PARAMETERS If p1 192 then 191 p2 0 so the ratio will be uni ry 1 2 Definition when the proportion is defining an undesirable outcome the above proportion is called The relative risk DEPENDENT SAMPLES This will only be seen in means for This class To get dependent samples wi rhin subjects repea recl measures ma rchecl pairs HYPOTHESIS TEST FOR DEPENDENT SAMPLES H Ho id 0 HA 61 gt lt i 0 Where id is the true difference T dependent samples ttest matched pairs ttest paired ttest A quantitative response dependent samples differences must be approximately normal STAT 4210 Test 2 Formula Sheet 191 19 13113 1901 190 n n n 171 a S 1 s 2 1 s 2 W W nl l n2 1 2 dz 2612 1702 205 171012 n n 1 n 1 perigfjvalue E1 E2 7711n2 2 191 192 771 A A A l A A l A 1 1 p1 P1n1191 P2n2p2 P1n1191 P2n2p2 A 1 2 1 2 2 n1 82 3p1n1 1712 a 1 Zc 0 5 Z0 p0lt n p0 p

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