Estimating)the)Population)Proportion)with)Confidence)Intervals))Lecture)16))Unknown)p)• In)real)life,)knowing)the)value)of)p)is)unusual))• Sample)surveys)were)developed)to)discover)p)because)we)simply)never)know)it))• Your)text)goes)through)an)example)(on)page)317),)the)key)ideas)are)what)we)already)know) )(sample)proportion))and)n(sample)size))and)can)come)up)with)an)estimated)standard)error)!"!"#= ))Real)Survey))• The)national)Retailer)Federation)just)completed)a)Mother’s)Day)survey)• It)is)revealed)that)over)half)(54.3%))of)the)8,724)US)adults)surveyed)will)take)mom)out)for)a)meal.)This)is)a)sample)statistic)from)a)very)large)sample)• But)what)percentage)of)all)US)adults)will)take)mom)out)Sunday?)Is)it)higher,)lower,)or)the)same?)Is)it)more)than)50%?))What)do)we)know))• P)is)unknown)but) )is)54.3%)and)known)• It’s)a)random)sample)so)it)is)most)probably)unbiased,)though) )may)not)be)exactly)equal)to)p))• We)can)use)it)to)calculate)standard)error))• The)standard)error)is)about).5%)and)this)means)the)population)parameter)is)very)near)the)sample)statistic.)A)small)SE)means)the)sample)is)precise.))• The)sample)size)is)large)(8724))so)the)sampling)distribution)of) )is)near)normal)and)centered)around)the)true)p.))• Therefore,)there)is)a)68%)chance)that)the) )is)within)one)standard)error)(SE))of)p,)a)95%)chance)that)is)within)2SE)of)p)and)a)99.7%)chance)that)it )is)within)3SE)of)p.)Even)at)3SE)away)it)is)almost)certain)that)true)population)parameter)is)within)3*).5%)of)1.5%)of)54.3%)(the)sample)percentage).)))Confidence)• Therefore,)we)can)be)quite)confident)that)the)population)parameter)was)captured)in)the)interval)54.3%)±!1.5%)• The)±!1.5%)is)called)the)“margin)of)error”)it)indicates)how)far)from)the)population)parameter)our)estimate)can)be.)))Information)from)a)confidence)Interval))• A)confidence)interval)gives)us)a)range)of)possible)values)for)the)parameter)Unknown pIn real life, knowing the value of p is unusual.Sample surveys were developed to discover p because we simply never know it.Your text goes through an example (p. 317), the key ideas are that we know (sample proportion) and n (sample size) and can come up with an estimated standard error Unknown pIn real life, knowing the value of p is unusual.Sample surveys were developed to discover p because we simply never know it.Your text goes through an example (p. 317), the key ideas are that we know (sample proportion) and n (sample size) and can come up with an estimated standard error Unknown pIn real life, knowing the value of p is unusual.Sample surveys were developed to discover p because we simply never know it.Your text goes through an example (p. 317), the key ideas are that we know (sample proportion) and n (sample size) and can come up with an estimated standard error Unknown pIn real life, knowing the value of p is unusual.Sample surveys were developed to discover p because we simply never know it.Your text goes through an example (p. 317), the key ideas are that we know (sample proportion) and n (sample size) and can come up with an estimated standard error What do we know?P is unknown but is 54.3% and known.It’s a random sample so it is most probably unbiased, though may not be exactly equal to pWe can use it to calculate a standard errorThe standard error is about .5% and this means the population parameter is very near the sample statistic. A small SE means the sample is precise.Unknown pIn real life, knowing the value of p is unusual.Sample surveys were developed to discover p because we simply never know it.Your text goes through an example (p. 317), the key ideas are that we know (sample proportion) and n (sample size) and can come up with an estimated standard error Unknown pIn real life, knowing the value of p is unusual.Sample surveys were developed to discover p because we simply never know it.Your text goes through an example (p. 317), the key ideas are that we know (sample proportion) and n (sample size) and can come up with an estimated standard error Sample)Proportions)• A)confidence)interval)states)our)level)of)confidence)in)the)correctness)of)this)interval.)This)confidence)interval)is)3SE)which)gives)us)a)confidence)level)of)99.7%.)Which)means)we)can)be)confident)that)the)percentage)of)all)US)adults)is)very)near)to)54.3%.)))WHY)ARE)WE)CONFIDENT)OF)THE)CORRECTNESS?))Why)are)we)confident?)• When)parameter)p)is)unknown)but)the)random)sample)that)generates)estimates)(pchat) ))meets)the)CLT)conditions)we)know)they)should)be)nearby)p.))• Except)we)do)not)know)if)the)sample)estimate) )is)lower)than)p)or)higher)than)p.))• So)the)constructed)confidence)interval)“reaches)out”)in)both)directions)(higher)and)lower))than)pchat.)))• The)width)of)the)interval)id)determined)by)how)confident)we)wish)to)be.))Settling)the)Confidence)Level)• The)confidence)level)tells)us)how)often)we)can)expect)to)capture)the)true)parameter)p)WHY ARE WE CONFIDENT IN THE CORRECTNESS?RECALL THE SAMPLING DISTRIBUTION IS NORMALUnknown pIn real life, knowing the value of p is unusual.Sample surveys were developed to discover p because we simply never know it.Your text goes through an example (p. 317), the key ideas are that we know (sample proportion) and n (sample size) and can come up with an estimated standard error Unknown pIn real life, knowing the value of p is unusual.Sample surveys were developed to discover p because we simply never know it.Your text goes through an example (p. 317), the key ideas are that we know (sample proportion) and n (sample size) and can come up with an estimated standard error Why are we confident?When parameter p is unknown but the random sample that generate estimates (p-hat ) meets the CLT conditions we know they should be nearby p.Except we do not know if the sample estimate is lower than p or higher than pSo the constructed confidence interval “reaches out” in both directions (higher and lower) than p-hat.The width of the interval is determined by how confident we wish to be.• It)is)a)measure)of)our)success)rate,)it)doesn’t)tell)us)how)successful)a)single)confidence)interval)will)be,)only)what)we)can)expect)form)the)method))•