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# 324 Class Note for STAT 100 at PSU

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This 7 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 28 views.

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Date Created: 02/06/15

Apr 8 Statistic for the day In the United States in 2001 Units of blood donated 15 million Number of donors 8 million Units transfused 14 million Number of patients 49 million Assignment Prepare for midterm 3 Speaking of college basketball In a Gallup poll conducted Dec 58 2004 people were asked For each of the following say Whetheryou are a fan of that sport or not College basketball 41 answered yes or somewhat The ne print from gallup com 3nn margin oferror sample size1003 Formula for estimating the standard deviation of a sample proportion don t need histogram 41 x 1741 016 1003 lfwe happen to know the true population proportion we use it instead ofthe sample proportion What to expect from sample proportions Facts ngerprints may be in uenced by prenatal hormones Most people have more ridges on right hand than left People who have more on the left hand are said to have leftward asymmetry Women are more likely to have this trait than men The proportion of all men who have this trait is about l5 n in a study of 186 heterosexual and 66 homosexual men 26 14 heterosexual men showed the trait and 20 30 homosexual men showed the trait Reference Haiku A Y and KimuraD Dermatoglyphic Asymmetry and Sexual Orientation in Men Behavioral Neuroscience Vol 108 No 6 00371206 Dec 94 is it unusual to observe a sample of66 men and observe a sample proportion of307 We now know what the distribution of sample proportions based on a sample of 66 should look like We will suppose that the true proportion in the population of men is 15 Standard 3915X 1 deviation 66 Thus a sample proportion of 30 would be 3015I044 341 standard deviations above the true mean assuming that the sample is a representative sample from the population Histogram ur prupumuns Wm Normal Curve n as true pmpumun is standard dEviauun n44 is a in 5 New 4 a an mm at m n 235 n harriasemal men a gamma dwatians The sample proportion tor homosexual men 30 is too large to so me from the expected distribution of sample proportions Sample means measurement variables Suppose we want to estimate the mean weight at PSU Hist urarn urvveigm with Normal Curve Fraquarrm Wm Data from stat t00 survev spring 2004 Sample size 237 Mean value is t52 5 pounds Standard deviation is about 240 r t004 35 Hypothetical result using a quotpopulationquot that resembles our sample Histogram urinnn rnearrswith rrurrnal curve based on samples ur STE 237 Extremely interesting The histogram of means is bellshaped even though the original population was skewed Standard deviation is about t57 r t484 94 2 25 Formula for estimating the standard deviation of the sample mean don t need histogram Just like in the case otproportions we would like to have a simple formula to nd the standard deviation otthe mean Without having to resample a lotottimes Suppose we have the standard deviation of the original sample Then the standard deviation otthe sample mean is standard deviation of the data alsample size Example SAT math scores Suppose nationallv we know that the SAT math test has a mean ott00 points and a standard deviation of t00 points Draw by hand a picture otwhat you expect the distribution otsample means based on samples of size t00 to look like Sample means have a normal distribution mean 500 standard deviation t00t0 t0 So draw a bell shaped curve centered at 500 with 95 otthe bell between 500 720 480 and 500 20 520 m rmz rmi rum Normal curve ofSAT means sample size1oo A sample of t00 SAT math scores with a mean ot540 would be very unusual A sample ott00 with a mean ot5t0 would not be unusual Score True proportion known lfyou play 400 games ofcraps where Wlll the proportion of games you Win lie 95 ofthe time True proportion mean ofsample proportions 493 Standard deviation ofsample proportions 050 Answer Between 4932O50 and 4932050 or between 393 and 593 Other confidence coefficients An example Suppose we want a 74 con dence interval instead of 95 How many standard deviations span the middle 74 of the normal curve 74 confidence interval see p 457 512mm nnrrml curve Since 74 is in the middle there is 43 in eitherend n So ndzfor 43andz 74A for 87 13a 13 Wegetz1113 90con dence interval sample proportion i113Std Dev Table of diminishing returns New SDOld SD Sample sizeincreased by 42 4 times 43 9 times 44 46 times 45 25 times 46 36 times 400 times Imagine that someone in your family has just handed a baby to you Imagine yourself holding the baby With which arm are you holding the baby Research question Do mothers tend to hold their babies on the left side over their hearts Rationale the sound ofthe human heart is soothing to the new born baby and the mother instinctually knows this Can we nd support for this hypothesis in data Lee Salk in a 4973 Scienti c American article describes a series of observations and experiments to investigate this hypothesis See pages 475 ofthe textbook Study 1 monkeys Satk observed 42 rhesus monkeys In Bronx Zoo hotdmg babres 40 hetd the baby on tett Suppose thrs Is a sampte otRhesus monkeys Frnd a 90 con dence Interyat tor the poputatron proportron of monkey mothers who hotd baby on te sampte proportron 4042 95 sampte srze 42 standard deyratron otsampte proportron 034 number otstandard deyratrons tor90 t 64 90 con dence Interyat 95 1464gtlt034 95 i 055 895 to t 005 895 to 1 01pr4 Study 2 mothers both right and Iett handed Ot255 rrght handed mothers 83 hetd baby on tett They sard It was more naturat smce It ees the rrght hand for domg thmgs Ot32 te handed mothers 78 hetd baby on te They sard It was better to hotd baby In dommant arm Rrghthanded 98 con dence Interyat t sampte proportron 83 2 sampte srze 255 3 standard deyratron of the sampte proportron 024 4 number of standard deyratrons for 98 2 33 5 98 con dence Interyat 83 i 2 33gtlt 024 83 i 056 774 to 886 Left handed mothers 90 con dence Interyat t sampte proportron 78 sampte srze 32 standard deyratron 073 number of standard deyratrons for 90 464 90 con dence Interyat 78 i t 64gtlt073 78 i 12 66 to 90 01pr Study 3 shoppers Researchers tortered around a supermarket parkrng tot and recorded In Whrch arm the shoppers carrred therr grocery bags of 438 shoppers 50carrred bags on tett 95 con dence Interyat t sampte proportron 50 Study 4 paintings and sculpture Ot466 pamtmgs and scutpture of the Madonna and chrtd 80 hetd baby on te 98 con dence Interyat t sampte proportron 80 sampte srze 466 standard deyratron otsampte proportron 049 number otstandard deyratron tor98 2 33 98 con dence Interyat 80 i 2 33gtlt 049 80 i 044 756 to 844 npr 2 sampte srze 483 3 standard deyratron otsampte proportron 024 4 number of standard deyratrons tor95 2 5 95 con dence Interyat 50 i 2gtlt 024 50 i 048 452 to 548 Summary canmence Imeryats Im Propaman hatmng nerrr arr tett at E El 95 as E N 95 I I I I I an m handed monkeys quothanded shDPPErs rr ass rr 2 rr 42 rr 255 rr on What makes for Wrder Intervats Smatter samptes targer con dence coef crents r requencv Example Estimate mean of pairs of jeans owned by a student at PSU Husnnum m Jams Mean 78 pairs St Dev 58 pairs Sample size 222 Give a 98 con dence interval Example Estimate mean of pairs of jeans owned by a student at PSU ofSEMs for 98 con dence 233 58 Mean 78 pairs SEM 1222 0394 St Dev 58 pairs Sample size 222 98 con dence interval 78 1 233X04 Give a 98 78 i 09 or 69 to 87 de e quotlaw39 lnterpretation We estimate that the population of Penn State students owns 78 pairs ofjeans on average 98 con dence interval is 69 to 87 pairs a reasonable range of values for the true population mean Fibonacci Sequence Guess the next numbers in the sequence 1 1 2 3 5 8 13 21 34 Called a Fibonacci sequence Ratios of pairs after a while equal approximately 618 eg 813 615 1321 619 2134 618 wtdth length N width I 18 length then the rectangle is called a golden rectangle Width to Length ratios for rectangles appearing on beaded baskets otthe Shoshoni 0693 0662 0690 0606 0570 0749 0652 0628 0609 0844 0654 0615 0668 0601 0576 0670 0606 06110553 0633 0625 0610 0600 0633 0595 Wanna 12mm mam mm mm bums m 2 D75 2 01 m 2 Balderr mum m ass 2 the sarne wtthout the heartbeat for a new group of newborns Babtes were dtvtded tnto three groups 39ltght btrth wetght 2510 r 3000 grarns 39rnedturn btrth wetght 3010 73500 grarns and 39heavy 3510 grarns and above The wetghtchange from day 1 to day 4 was recorded We want to know tfthe populatton rneans for treatrnent heartbeat and control no heartbeat are close or not Light birth weight llean Std Dev sarnp stze SEM Treatment heartbeat 65 50 35 8 45 Control no heartbeat 720 60 28 11 33 Treatment J8452 11332 1413 SEM 845 Control SEM 1133 Light birth weight Difference in sample means 85 9 Standard error ofthe difference 1413 g 95 con dence interval for the true difference 85 2X1413 which is 567to 11326 Since zero is not in the Cl zero is not a plausible value Thus we have found a statistically signi cant difference Perfect pitch and the brain The researchers found 39rnustctans wtth perfect pttch rnean dPT r 57 39rnustctans wtthout perfect pttch rnean dPT r 23 Questton Are the dPT rneans close or not is there a dtfference between rnustctans wtth and wtthout perfect pltCh7 Equtvalently we ask is the dtfference tn rneans 577 r 23 7 34 close to 07 musicians musicians perf pitch no perfpitch means 7 57 r 23 sample tt t9 size SD Qt 17 SEM 019 039 SEN 019Z 0392 043 Diffin means 7 57 r r 23 r 34 95 Cl 7 Elli 2x 043 which is r 43 to r 26 Conclusion They are not close There is a difference General conclusions There is a signi cant difference between the asymmetry ofthe PT for musicians with perfect pitch and both musicians without perfect pitch and nonrmusicians This strongly suggests that there is a relationship between the physical structure of the PT in the brain and perfect pitch ability Categorical ordinal categorical nominal quantitative discrete or quantitative continuous CN l Eye color QC l Weight QD I Number ofsiblings CN l Gender QC l Time in 100meter dash QD I Number of cigarettes smoked in a day CN l Building where your rst class occurs CO I Year in school frl soljrl sr Consider a clock that s 5 minutes fast I Valid orinvalid l Reliable or unreliable l Biased or unbiased Answer valid reliable and biased Consider a scale that is sometimes several pounds too low sometimes several pounds too high I Valid or invalid l Reliable or unreliable l Biased or unbiased Answer valid unreliable and unbiased

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