Intro To Statistics
Intro To Statistics STAT 3660
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7 Chapter 11 Sample Proportions rllllll nullw P Thought Question 1 Suppose that 40 of a certain population favor the use of nuclear power for energy a If you randomly sample 10 people from this population will exactly four 40 of them be in favor of the use of nuclear power Would you be surprised if only two 20 of them are in favor How about if none of the sample are in favor sl39kl P Thought Question 2 Suppose that 40 of a certain population favor the use of nuclear power for energy b Now suppose you randomly sample 1000 people from this population Will exactly 400 40 of them be in favor of the use of nuclear power Would you be surprised if only 200 20 of them are in favor How about if none of the sample are in favor all P Thought Question 3 A 95 confidence interval for the proportion of adults in the US who have diabetes extends from 07 to 11 or 7 to 11 What does it mean to say that the interval from 07 to 11 represents a 95 confidence interval for the proportion of adults in the US who have diabetes alle P Thought Question 4 Would a 99 confidence interval for the proportion described in Question 3 be wider or narrower than the 95 interval given Explain Hint what is the difference between a 68 interval and a 95 interval all P Thought Question 5 In a May 2006 Zogby America poll of 1000 adults 70 said that past efforts to enforce immigration laws have been inadequate Based on this poll a 95 con dence interval for the proportion in the population who feel this way is about 67 to 73 If this poll had been based on 5000 adults instead would the 95 confidence interval be wider or narrower than the interval given Explain Recall from previous chapters Parameter fixed unknown numberthat describes the population Statistic known value calculated from a sample a statistic is used to estimate a parameter Sampling Variability different samples from the same population may yield different values of the sample statistic estimates from samples will be closer to the true values in the population if the samples are larger silk Recall from previous chapters Example The amount by which the proportion obtained from the sample b will differfrom the true population proportion p rarely exceeds the margin of error Sampling Distribution tells what values a statistic takes and how often it takes those values in repeated sampling Example sample proportions p s from repeated sampling would have a normal distribution with a certain mean and standard deviation x Case Study i Comparing Fingerprint Patterns Science News Jan 27 1995 p 451 rillllllrnull n Q Case Study Fingerprints o Fingerprints are a sexually dimorphic traitwhich means they are among traits that may be influenced by prenatal hormones o It is known Most people have more ridges in the fingerprints of the right hand People with more ridges in the left hand have leftward asymmetry Women are more likely than men to have leftward asymmetry 9 Compare fingerprint patterns of heterosexual and homosexual men 411 Case Study Fingerprints iQ Study Results 966 homosexual men were studied 20 30 of the homosexual men showed left asymmetry o 186 heterosexual men were also studied 26 14 of the heterosexual men showed left asymmetry II WA Case Study Fingerprints iQ A Question Assume that the proportion of all men who have leftward asymmetry is 15 Is it unusual to observe a sample of 66 men with a sample proportion 16 of 30 if the true population proportion p is 15 VG A V O I x a z x 39 Q d N r wh m s m quot 3 x 2 J Q I ll39f391 The Rule for Sample Proportions If numerous simple random samples of size n are taken from the same population the sample proportions f9 from the various samples will have an approximately normal distribution The m of the sample proportions will be p the true population proportion The standard deviation will be 191 p 71 in new 7 y w W39 1 1 a l t 1 j l l ML lt2 For rule to be valid must have ltgt Random sample ltgt Large sample size i2e n SR8 we 39 S swag Population proportmn p lt Values of f gt e39lm aiel Case Study Fingerprints g Sampling Distribution pO15 mean n66 p1 p o151 015 n 66 0044 sd it Case Study Fingerprints iQ Answer to Question oWhere should about 95 of the sample proportions lie oz mean plus or minus two standard deviations 015 20044 0062 015 20044 0238 oz 95 should fall between 0062 amp 0238 1000 Simulated Samples n66 Simulated Data p015 p015 n66 0151 o15 0044 1000 Simulated Samples n66 Simulated Data p015 approximately 95 of sample proportions fall in this interval 0062 to 0238 Is it likely we would observe a sample proportion 2 030 to N 00 L0 0 V V LO to N 00 V 0 V V LO 0 O O 39 039 Proportion of Successes 1000 Simulated Samples n30 Simulated Data p015 1000 Simulated Samples n30 Simulated Data p015 approximately 95 of sample proportions fall in this interval Is it likely we would observe a sample proportion 2 030 Con dence Interval for a Population Proportion 9 An interval of values computed from sample data that is almost sure to cover the true population proportion 9 We are highly confident that the true population proportion is contained in the calculated interval o Statistically for a 95 C in repeated samples 95 of the calculated confidence intervals should contain the true proportion Formula for a 95 Con dence Interval for the Population Proportion Empirical Rule osample proportion plus or minus two standard deviations of A p1p the sample proportion PiZWT osince we do not know the population proportion p needed to calculate the standard deviation we will use the sample proportion f in its place Formula for a 95 Con dence Interval for the Population Proportion Empirical Rule 2 19019 n pi k standard error estimated standard deviation of f7 rillll mjllm Margin of Error 2 M plus or minus part of OJ I7 051 o5 1 s 2 2 J g Formula for a Clevel Con dence Interval for the Population Proportion 131 f n where z is the critical value of the standard normal distribution for confidence level C VIIIIIFW1 rll fl iz Common Values of 2 Confidence Level Critical Value C 2 50 067 60 084 68 1 70 104 80 128 90 164 95 196 or 2 99 258 997 3 999 329 Case Study i9 Parental Discipline Brown C S 1994 To spank or not to spank USA Weekend April 2224 pp 47 What are parents attitudes and practices on discipline Case Study Survey i Parental Discipline o Nationwide random telephone survey of 1250 adults 474 respondents had children under 18 living at home results on behavior based on the smaller sample oreported margin of error 3 for the full sample 5 for the smaller sample silk Case Study Results i Parental Discipline The 1994 survey marks the first time a majority of parents reported not having physically disciplined their children in the previous year Figures over the past six years show a steady decline in physical punishment from a peak of 64 percent in 1988 The 1994 proportion who did not spank or hit was 51 Case Study Results i Parental Discipline o Disciplining methods over the past year denied privileges 79 confined child to hisher room 59 spanked or hit 49 insulted or swore at child 45 9 Margin of error 5 Which of the above appear to show a true value different from 50 slit Case Study Con dence Intervals g Parental Discipline odenied privileges 79 f 079 standard error of f V79 quot7974 20019 95 C 79 i 2o19 752 828 oconfined child to hisher room 59 f 03959 591 59 standard error of f 74 0023 95 C 59 i 2023 544 535 Case Study Con dence Intervals g Parental Discipline ospanked or hit 49 f 049 A 491 49 standard error of p 74 0023 95 C 49 i 2023 444 536 oinsulted or swore at child 45 f 045 451 45 standard error of f m 0023 95 C 45 i 2023 404 496 Case Study Results i Parental Discipline oAsked of the full sample n1250 How often do you think repeated yelling or swearing at a child leads to longterm emotional problems very often or often 74 sometimes 17 hardly ever or never 7 no response 2 oMargin of error 3 alibi s Case Study Con dence Intervals g Parental Discipline 9 hardly ever or never 7 p 007 1 standard error of f l3907 quot07250 20007 95 or 07 i 2007 056 084 9 Few people believe such behavior is harmless but almost half 45 of parents engaged in it v39l39a z Key Concepts I 9 Different samples of the same size will generally give different results oWe can specify what these results look like in the aggregate 9 Rule for Sample Proportions oCompute and interpret Confidence Intervals for population proportions based on sample proportions al31