Lesson 6 Confidence Intervals I Toward Statistical Inference Two designs for producing data are sampling and experimentation both of which should employ randomization As we have already learned one important aspect of randomization is to control bias Now we will see another positive Because chance governs our selection think of guessing whether a flip of a fair coin will produce a head or a tail we can make use of probability laws the scientific study of random behavior to draw conclusions about an entire population from which the subjects originated This is called statistical inference Statistical inference Parameter Statistic Use of probability laws the scientific study of random behavior to draw conclusions about an entire population from which the subjects originated Number that describes the population It is fixed but we rarely know it Examples include the true proportion of all American adults who support the president or the true mean of weight of all residents of New York City Number that describes the sample This value is known since it is produced by our sample data but can vary from sample to sample For example if we calculated the mean heights of a random sample of 1000 residents of New York City this mean most likely would vary from the mean calculated from another random sample of 1000 residents of New York City Examples A survey is carried out at a university to estimate the proportion of undergraduate students who drive to campus to attend classes One thousand students are randomly selected and asked whether they drive or not to campus to attend classes The population is all of the undergraduates at that university campus The sample is the group of 1000 undergraduate students surveyed The parameter is the true proportion of all undergraduate students at that university campus who drive to The statistic is the proportion of the 1000 sampled undergraduates who drive to campus to attend classes A study is conducted to estimate the true mean yearly income of all adult residents of the state of California The campus to attend classes study randomly selects 2000 adult residents of California The population consists of all adult residents of California The sample is the group of 2000 California adult residents in the study The parameter is the true mean yearly income of all adult residents of California The statistic is the mean of the 2000 sampled adult California residents Ultimately we will measure statistics and use them to draw conclusions about unknown parameters This is statistical inference assumptions being satisfied np 10 and n 1 p 10 If p is unknown then use the sample proportion II Constructing confidence intervals to estimate a population proportion NOTE the following interval calculations for the proportion confidence interval is dependent on the following The goal is to estimate p proportion with a particular trait or opinion in a population Sample statistic read as p hat proportion of observed sample with the trait or opinion we re studying Standard error of where n sample size Multiplier comes from this table Confidence Level 90 90 95 95 98 98 99 99 The value of the multiplier increases as the confidence level increases Multiplier 1 645 or 1 65 1 96 usually rounded to 2 2 33 2 58 This leads to wider intervals for higher confidence levels We are more confident of catching the population value when we use a wider interval Example In the year 2001 Youth Risk Behavior survey done by the U S Centers for Disease Control 747 out of n 1168 female 12th graders said they always use a seatbelt when driving Goal Estimate proportion always using seatbelt when driving in the population of all U S 12th grade female drivers Check assumption 1168 0 64 747 and 1168 0 36 421 both of which are at least 10 Sample statistic is 747 1168 64 Standard error 0 014 A 95 confidence interval estimate is 64 2 014 which is 612 to 668 With 95 confidence we estimate that between 612 61 2 and 668 66 8 of all 12th grade female drivers always wear their seatbelt when driving 64 2 58 014 which is 64 036 or 604 to 676 always wear their seatbelt when driving greater the confidence level the wider the interval Example Continued For the seatbelt wearing example a 99 confidence interval for the population proportion is With 99 confidence we estimate that between 604 60 4 and 676 67 6 of all 12th grade female drivers Notice that the 99 confidence interval is slightly wider than the 95 confidence interval IN the same situation the Notice also that only the value of the multiplier differed in the calculations of the 95 and 99 intervals Using Confidence Intervals to Compare Groups A somewhat informal method for comparing two or more populations is to compare confidence intervals for the value of a parameter If the confidence intervals do not overlap it is reasonable to conclude that the parameter value differs for the two populations Example In the Youth Risk Behavior survey 677 out of n 1356 12th grade males said they always wear a seatbelt To begin we ll calculate a 95 confidence interval estimate of the population proportion Check assumption 1356 0 499 677 and 1356 0 501 679 both of which are at least 10 Sample statistic is 677 1356 499 Standard error 0 136 0 499 2 0136 or 472 to 526 A 95 confidence interval estimate calculated as Sample statistic multiplier Standard Error is With 95 confidence we estimate that between 472 47 2 and 526 52 6 of all 12th male drivers always wear Comparison and Conclusion For females the 95 confidence interval estimate of the percent always wearing a their seatbelt when driving seatbelt was found to be 61 2 to 66 8 an obviously different interval than for males It s reasonable to conclude that 12th grade males and females differ with regard to frequency of wearing a seatbelt when driving confidence interval are not acceptable reasonable possibilities for the population value Using Confidence Intervals to test how parameter value compares to a specified value Values in a confidence interval are acceptable possibilities for the true population value Values not in the Example The 95 confidence interval estimate of percent of 12th grade females who always wear a seatbelt is 61 2 to This has the consequence that it s safe to say that a majority more than 50 of this population always wears their If somebody claimed that 75 of all 12th grade females always used a seatbelt we should reject that assertion The Finding sample size for estimating a population proportion When