AP Stats Test Quick Review: Confidence Intervals

00:33:54
https://www.youtube.com/watch?v=vc43CRPgrG8

Ringkasan

TLDREl vídeo tracta sobre la revisió dels intervals de confiança en estadístiques AP. S'explica com construir un interval de confiança per a mitjans de mostres i proporcions, com interpretar-los, i la importància del nivell de confiança. Es descriu el procés de calcular el marge d'error i les condicions que cal verificar per garantir la precisió de l'interval. També es discuteix la influència del valor crític en la grandària de l'interval, i es subratlla que el nivell de confiança no és una probabilitat, sinó una afirmació sobre la fiabilitat dels intervals construïts a partir de totes les mostres possibles.

Takeaways

  • 📊 Importància dels intervals de confiança per a la inferència estadística
  • 📝 Convenció d'interpretar intervals: '95% confiat que...'
  • ✅ Condicions per a la construcció d'un interval: aleatorietat, mida de la mostra
  • 🔍 Distingir entre Z star i T star segons la situació
  • ⚖️ La interpretació del nivell de confiança: no una probabilitat
  • 🔄 El marge d'error resulta de la desviació estàndard i el valor crític
  • 📈 Construcció de l'interval per mitjà de mostres i estadístiques
  • 🚀 El procés de càlcul en quatre passos: identificació, verificació, construcció, interpretació
  • ❓ Efecte de la grandària de la mostra en la precisió de l'interval
  • 🧐 Com ajudar a determinar significativitat amb intervals de confiança

Garis waktu

  • 00:00:00 - 00:05:00

    En aquest vídeo, es revisen els intervals de confiança per a l'examen d'AP Statistics. Es destaca la importància de saber com construir un interval de confiança a partir d'una estadística de mostra, ja sigui per a mitjanes o proporcions, i com interpretar aquests intervals. També es discuteix el significat del nivell de confiança, aclarint que no és una probabilitat, sinó una afirmació sobre la precisió dels intervals construïts a partir de múltiples mostres.

  • 00:05:00 - 00:10:00

    S'explica que un interval de confiança per a un paràmetre poblacional es basa en una estadística de mostra, com la mitjana o la proporció. Es destaca la importància de les distribucions de mostres i com es pot calcular un interval de confiança mitjançant una fórmula que inclou la mitjana de la mostra i el marge d'error, que depèn del valor crític i la desviació estàndard de l'estadística.

  • 00:10:00 - 00:15:00

    Es presenten les condicions necessàries per construir un interval de confiança, incloent la necessitat que la mostra sigui aleatòria, que sigui menor del 10% de la població i que la mostra sigui suficientment gran. Es discuteixen les condicions específiques per a mostres de proporcions i mitjanes, incloent el teorema del límit central.

  • 00:15:00 - 00:20:00

    Es detallen els passos per calcular un interval de confiança, incloent la identificació del que es vol estimar, la verificació de les condicions, la construcció de l'interval i la seva interpretació. Es fa èmfasi en la importància de comunicar el que representa l'interval de confiança en el context del problema.

  • 00:20:00 - 00:25:00

    S'explica com determinar el valor crític (Z* o T*) necessari per construir un interval de confiança, depenent del nivell de confiança desitjat. Es presenten exemples de càlcul de Z* i T* utilitzant calculadores i taules, així com la importància de conèixer els graus de llibertat en el cas de T*.

  • 00:25:00 - 00:33:54

    Finalment, es discuteix com els intervals de confiança poden ajudar a entendre les proves de significança. Es presenta un exemple d'un interval de confiança per a la diferència entre dues proporcions, destacant que si l'interval conté zero, no hi ha evidència suficient per rebutjar la hipòtesi nul·la, indicant que no hi ha una diferència significativa entre les dues proporcions.

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Peta Pikiran

Video Tanya Jawab

  • Què és un interval de confiança?

    Un interval de confiança és un rang de valors que s'utilitza per estimar un paràmetre poblacional, basant-se en una estadística de mostra.

  • Com es calcula un interval de confiança?

    Es calcula a partir d'una estadística de mostra, afegint i restant un marge d'error que depèn del valor crític i la desviació estàndard.

  • Quina és la importància de la confiança del 95%?

    El 95% de confiança significa que, si es repetís l'experiment nombroses vegades, aproximadament el 95% dels intervals de confiança obtinguts contindrien el valor veritable.

  • Què representa el marge d'error?

    El marge d'error és la quantitat que s'afegix i resta a la estadística de mostra per construir l'interval de confiança.

  • Quines condicions s'han de verificar per construir un interval de confiança?

    Les condicions inclouen la col·lecció aleatòria de la mostra, que la mostra sigui menor del 10% de la població i que la mostra sigui prou gran.

  • Com es determina el valor crític Z o T?

    El valor crític Z o T es determina segons el nivell de confiança que es vulgui assignar al interval.

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Gulir Otomatis:
  • 00:00:00
    got in this video for the AP statistics
  • 00:00:02
    test we're going to start reviewing
  • 00:00:04
    confidence intervals so what exactly do
  • 00:00:06
    you need to know about confidence
  • 00:00:08
    intervals for the AP stats test you
  • 00:00:10
    definitely need to know how to construct
  • 00:00:12
    one based on a sample statistic that can
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    either be for a sample mean or a sample
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    proportion you also need to know how to
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    construct a confidence interval based on
  • 00:00:21
    the difference whether that difference
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    be B whether that difference be between
  • 00:00:25
    two sample proportions or two sample
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    means you definitely know how need to
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    know how to interpret a confidence
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    interval that's really important that
  • 00:00:36
    comes up a lot on multiple choice you
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    also need to know how to explain the
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    level of confidence when we talk about
  • 00:00:42
    being 95% confident what does that
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    actually mean a lot of kids accidentally
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    think it's a probability it is not a
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    probability and also we need to make
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    sure we understand how we could actually
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    use confidence intervals to draw
  • 00:00:56
    conclusions about significance all right
  • 00:01:00
    so let's jump right into it so let's
  • 00:01:03
    make sure we truly understand what a
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    confidence interval is for a confidence
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    interval for a population parameter
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    based on a sample statistic is very easy
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    to understand first we want to estimate
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    a population mean mu or a population
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    proportion P so always think about the
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    fact that we're using a kompis interval
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    to try to find the true population
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    parameter which is a mean mu or a
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    proportion P now to do this the first
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    thing we need is to take a sample that's
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    either a sample mean x-bar or a sample
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    proportion P hat then we simply have to
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    calculate how many standard deviations
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    we are willing to reach up and down of
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    hopes of capturing the true population
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    mean or proportion so confidence
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    intervals are directly based on sampling
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    distributions which there was a video
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    about how they recommend watching that
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    video first actually so what we do is we
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    think about the fact that if we think
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    about all possible samples that's what a
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    sampling
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    distribution shows the results of all
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    possible samples and we certainly expect
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    the truth whether it be a mean or a
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    proportion to be in the center and we
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    understand that some samples can be a
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    little higher and some samples can be a
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    little bit lower so if we were to grab
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    one of those random samples like right
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    here maybe this is our X bar or our P
  • 00:02:33
    hat but notice because of variability
  • 00:02:37
    that is not matching up exactly with
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    what's true but if we cast an interval
  • 00:02:44
    around that sample by going up a little
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    bit I'm sorry that would be down down a
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    little bit and up a little bit we
  • 00:02:51
    created interval from here to here and
  • 00:02:55
    notice that the truth did get caught in
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    that interval and as long as we get one
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    of the many samples that are very close
  • 00:03:04
    to the truth
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    when we build that interval around it we
  • 00:03:08
    should contain the truth so how do you
  • 00:03:11
    actually calculate a confidence interval
  • 00:03:13
    well the first thing is you start off
  • 00:03:15
    with this very generic formula it is a
  • 00:03:18
    sample a sample statistic whether that
  • 00:03:20
    be x-bar or p hat and from that sample
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    statistic you go up and down plus and
  • 00:03:29
    minus what we call the margin of error
  • 00:03:31
    now this entire back part is the margin
  • 00:03:34
    of error it is the critical value
  • 00:03:36
    multiplied by the standard deviation of
  • 00:03:39
    the statistic the critical value is AZ
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    star or 80 star based on how confident
  • 00:03:47
    the question asks us to be we'll talk
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    more about that in a second the standard
  • 00:03:53
    deviation of the statistic is what we
  • 00:03:55
    learned back with sampling distributions
  • 00:03:58
    if you're working with proportions the
  • 00:04:00
    standard deviation of the statistic is P
  • 00:04:02
    times 1 minus B divided by n with a
  • 00:04:06
    giant square root around it if you're
  • 00:04:08
    working with means it is the standard
  • 00:04:09
    deviation of the population divided by
  • 00:04:12
    the square root of your sample size n
  • 00:04:14
    now remember those are those two
  • 00:04:17
    formulas allow us to understand what I
  • 00:04:19
    was trying to show you up in this
  • 00:04:20
    picture here that samples naturally are
  • 00:04:23
    allowed to very little
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    so we're using that natural variation to
  • 00:04:27
    create this interval because we're
  • 00:04:29
    saying hey listen our sample statistic
  • 00:04:32
    isn't a hundred percent accurate but if
  • 00:04:35
    we go up a little bit or doubt a little
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    bit because of natural variation we
  • 00:04:39
    should capture the truth just keep in
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    mind it's a four-step process this is
  • 00:04:45
    how I teach it other teachers may teach
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    it differently but I teach four steps
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    step one is to always identify what it
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    is you're trying to find I'm trying to
  • 00:04:54
    estimate the true population proportion
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    of boys who wear glasses I'm trying to
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    estimate the true mean amount of time a
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    high school a high schooler takes to get
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    to school in the morning always identify
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    what you try to find second step is to
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    check the conditions now the first two
  • 00:05:11
    conditions are always the same the
  • 00:05:12
    sample must have been collected randomly
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    to avoid bias the second condition is
  • 00:05:17
    that your sample must be under 10% of
  • 00:05:20
    the population to assume independence
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    because typically when we sample we do
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    not replace so when I go and grab a
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    person out of the population to sample I
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    don't put them back right I already
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    selected them I don't want them to
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    selectively be selected again so as long
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    as our sample size is under ten percent
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    of population any small change that not
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    replacing could cause is negligible the
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    third condition is what changes the
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    third condition in general is that the
  • 00:05:48
    sample must be big enough now if you're
  • 00:05:51
    working with proportions big enough
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    means that your sample must contain
  • 00:05:55
    within it ten successes or more and ten
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    or more failures if you're working with
  • 00:06:01
    means the big enough condition actually
  • 00:06:04
    could take on three different options
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    option number one if your population is
  • 00:06:10
    already known to be normal then your
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    sampling distribution is guaranteed to
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    be normal so any sample size even small
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    samples of size four or three are big
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    enough if your pot if your sample is
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    thirty or larger than the central limit
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    theorem can help you because even if the
  • 00:06:28
    population is unknown or non normal the
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    central limit theorem says that the
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    sampling distribution will still be
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    normal as long as your sample is thirty
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    or larger
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    the final scenario is if your sample is
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    under 30 and you do not know that your
  • 00:06:44
    population is normal here you simply
  • 00:06:47
    need to take a quick look at your data
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    to make sure that there's no major
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    skewness or no major outliers in it
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    after checking all those conditions the
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    third step to a confidence interval is
  • 00:06:57
    all the work them about to show you it's
  • 00:06:59
    actually building the interval using
  • 00:07:01
    this formula that I just went through
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    after that you need to interpret your
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    interval you need to explain what that
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    interval represents and it's a very
  • 00:07:10
    simple to do that you're simply saying
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    I'm 95% confident that the true
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    population blank filled in with the
  • 00:07:16
    context of the problem
  • 00:07:17
    is between this value in this value okay
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    and we'll go over that as well all right
  • 00:07:23
    let's move on to talking about the
  • 00:07:25
    critical value the Z star or T star
  • 00:07:28
    obviously this value is needed anytime
  • 00:07:30
    you're building a confidence interval it
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    is entirely based on how confident you
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    want to be if you want to be 95%
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    confident or 90% confident or 99 percent
  • 00:07:41
    confident this value will change well
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    all you have to do is think about how
  • 00:07:45
    far you're willing to reach obviously if
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    I want to be 99 percent confident I need
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    to reach out a little bit wider so my
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    interval is going to be bigger if I only
  • 00:07:54
    want to be 90 percent confident then my
  • 00:07:56
    interval can actually be a little bit
  • 00:07:58
    smaller because I don't have to reach
  • 00:07:59
    out as one so first off anytime you're
  • 00:08:03
    working with proportions you need to use
  • 00:08:04
    Z star so how do you find Z star how do
  • 00:08:07
    you find your critical value for a
  • 00:08:09
    specific level of confidence well the
  • 00:08:12
    first option is using your calculator
  • 00:08:14
    2nd VARs and pull up invert norm invert
  • 00:08:18
    norm will help you find your critical Z
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    star but you have to be very careful
  • 00:08:23
    what to enter in first you asked for the
  • 00:08:26
    area this is the area at the very very
  • 00:08:29
    bottom so you have to process this if I
  • 00:08:32
    want to be 95% confident that means that
  • 00:08:35
    there's 95% in the middle that means
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    that 5% is being left out but because of
  • 00:08:42
    symmetry that 5% gets split evenly
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    two-and-a-half percent at the bottom and
  • 00:08:46
    two and a half percent the top and that
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    is what the area wants the bottom so
  • 00:08:52
    I'm going to type in 0.025 because for
  • 00:08:55
    95% confident that would put two and a
  • 00:08:57
    half percent at the bottom kind of a
  • 00:09:00
    little bit tricky that this is how
  • 00:09:01
    invert norm works but sorry that's how
  • 00:09:03
    it works so your Z star is 1.96 it does
  • 00:09:07
    say negative because we looked at the
  • 00:09:09
    bottom but remember you're going up and
  • 00:09:11
    down so technically your Z star is
  • 00:09:13
    positive or negative so to repeat this
  • 00:09:17
    process for 90% confident 90% confident
  • 00:09:21
    puts 5% at the bottom and it also puts
  • 00:09:24
    5% of the top because 90% in the middle
  • 00:09:27
    10% left out 5% on each end and invert
  • 00:09:31
    norm only once the bottom end so one
  • 00:09:33
    point six four four nine one more time
  • 00:09:37
    for 99% confident you have to put in
  • 00:09:41
    point zero zero five because of your 99%
  • 00:09:45
    confident that means 1% is left out and
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    that is split evenly a half a percent on
  • 00:09:50
    the bottom and a half a percent at the
  • 00:09:53
    top and the half a percent the Bob is
  • 00:09:55
    what we want so we get a Z star two
  • 00:09:58
    point five seven five now these are the
  • 00:10:01
    three most common levels of confidence
  • 00:10:02
    if you're asked about a weirder level
  • 00:10:05
    like 96 or 98% then you just have to
  • 00:10:07
    process it through the way I just
  • 00:10:09
    explained all right how do you get a
  • 00:10:11
    t-star anytime you're working with means
  • 00:10:14
    you need a t-star so anytime a promise
  • 00:10:17
    concerning with population means you
  • 00:10:19
    need a t-star
  • 00:10:20
    so how do you get a t-star well once
  • 00:10:23
    again right underneath invert norm is
  • 00:10:25
    invert T this will help you calculate T
  • 00:10:28
    stars it also asks for the area at the
  • 00:10:30
    bottom so if you're 95% confident once
  • 00:10:33
    again that will put 0.025 or
  • 00:10:36
    two-and-a-half percent at the bottom now
  • 00:10:38
    the other thing it asks for is your
  • 00:10:39
    degrees of freedom remember the T models
  • 00:10:41
    are all based on how many degrees of
  • 00:10:42
    freedom you have kind of a weird name
  • 00:10:44
    but simply degrees of freedom is your
  • 00:10:46
    sample size minus one so you are
  • 00:10:48
    required to know how big your sample is
  • 00:10:50
    so let's just say we have a sample of 40
  • 00:10:52
    well then that would give us 39 degrees
  • 00:10:55
    of freedom and our T star for 95%
  • 00:10:58
    confident 40 degrees I'm sorry 39
  • 00:11:01
    degrees afford 39 degrees of freedom
  • 00:11:04
    from a sample
  • 00:11:04
    forty would give us a ZT star of 2.0 two
  • 00:11:08
    to seven now keep in mind it does say
  • 00:11:11
    negative but remember it's actually
  • 00:11:12
    positive and negative now another way to
  • 00:11:15
    get your Z stars or your T stars is to
  • 00:11:18
    use a t-chart a t-chart is provided to
  • 00:11:21
    you on the back of the APS it's a
  • 00:11:23
    success or in the is basically it's
  • 00:11:25
    attached to the formula sheets and all
  • 00:11:26
    that stuff so you don't have to use this
  • 00:11:29
    if you totally understand how to use the
  • 00:11:31
    calculator some kids actually like to
  • 00:11:33
    teach art better so let me show you one
  • 00:11:34
    here's an example of the exact teacher
  • 00:11:36
    you will be given on the AP stats test
  • 00:11:39
    it says T distribution critical values
  • 00:11:41
    so what you do on the left hand side is
  • 00:11:43
    you look up your degrees of freedom and
  • 00:11:45
    you could look up your tail probability
  • 00:11:47
    that's the area at the bottom across the
  • 00:11:49
    top or if you scroll the bottom you'll
  • 00:11:51
    notice it actually has the level of
  • 00:11:53
    confidence for example 95 percent
  • 00:11:55
    confident which has a tail probability
  • 00:11:58
    of point O 2 5 as we mentioned did all
  • 00:12:01
    you got to do is match that up with your
  • 00:12:02
    degrees of freedom so let's just say you
  • 00:12:04
    have a sample size of 16 that gives you
  • 00:12:06
    15 degrees of freedom you follow the 15
  • 00:12:09
    over to the 95% confidence column and
  • 00:12:12
    you have 2 point 1 3 1 as your t star so
  • 00:12:17
    a lot of kids like this just because
  • 00:12:18
    they don't have to waste their time
  • 00:12:19
    typing things in the calculator they can
  • 00:12:21
    just simply look up their T star now
  • 00:12:24
    this actually also has these stars on it
  • 00:12:26
    that they don't tell you that the bottom
  • 00:12:28
    row is identified with an infinity now
  • 00:12:31
    remember the T model is just like the
  • 00:12:33
    normal model for really really big
  • 00:12:34
    samples so for a sample that's infinite
  • 00:12:36
    in size that's essentially going to be
  • 00:12:39
    the Z model or Z stars so if you look
  • 00:12:41
    across the bottom row
  • 00:12:43
    those are your Z stars for example 95%
  • 00:12:45
    confidence you may remember we found
  • 00:12:47
    1.96 for 90% confidence we were 1.645
  • 00:12:52
    actually we said 1 point 6 4 for 9 but
  • 00:12:55
    the only round to 3 decimals here 99%
  • 00:12:58
    confident would be the two point five
  • 00:13:00
    seven six so that bottom row there is
  • 00:13:02
    your Z stars so instead of using your
  • 00:13:04
    calculator you are more than welcome to
  • 00:13:06
    look up your critical values on that T
  • 00:13:09
    chart all right the last thing I want to
  • 00:13:12
    mention before we move on is what does
  • 00:13:14
    the level of confidence mean since we're
  • 00:13:16
    on the topic so when we say that we're
  • 00:13:18
    95% confident what does that really mean
  • 00:13:21
    first and foremost it's not a
  • 00:13:23
    probability most kids think that means
  • 00:13:25
    oh there's a 95% chance that the truth
  • 00:13:29
    is in our interval not at all that is
  • 00:13:31
    not at all what it represents it
  • 00:13:34
    represents the fact that you have to
  • 00:13:37
    understand that we built our interval
  • 00:13:39
    based on our one sample we went a little
  • 00:13:42
    bit above it we went a little bit below
  • 00:13:44
    it but if somebody found a mother sample
  • 00:13:47
    they might have a slightly different
  • 00:13:49
    interval or another sample a slightly
  • 00:13:52
    different interval or another sample a
  • 00:13:54
    slightly different interval so what 95%
  • 00:13:57
    kauffman is actually talking about it's
  • 00:13:59
    talking about the fact that there are
  • 00:14:00
    many many many samples out there tons of
  • 00:14:03
    samples out there just like yours of the
  • 00:14:06
    same size from the same population if
  • 00:14:09
    you think about all those samples the
  • 00:14:13
    idea is that 95% of those samples will
  • 00:14:17
    create intervals that contain the truth
  • 00:14:19
    so it's not about 95% probability or 95%
  • 00:14:23
    chance or even 95% of the time it's
  • 00:14:26
    about 95% of intervals 95% of intervals
  • 00:14:30
    created just like yours will contain the
  • 00:14:32
    truth it goes back to that picture I
  • 00:14:34
    drew earlier we know that according to a
  • 00:14:37
    sampling distribution the true mean is
  • 00:14:39
    smack dab in the middle so as long as we
  • 00:14:42
    get a sample near it when we build that
  • 00:14:44
    interval we capture it so as long as we
  • 00:14:47
    are one of those many many many many
  • 00:14:49
    many many many samples that are near the
  • 00:14:52
    truth our interval should capture and
  • 00:14:54
    the idea when we say we're 95% confident
  • 00:14:57
    is that 95% of those intervals do
  • 00:14:59
    contain the truth obviously there is
  • 00:15:01
    that chance that we get an interval down
  • 00:15:03
    here I'm sorry a sample down here sorry
  • 00:15:06
    down here or up here now those would be
  • 00:15:09
    really really unlikely samples and their
  • 00:15:11
    intervals would not contain the truth
  • 00:15:13
    but that's the remaining 5% so 95% of
  • 00:15:17
    intervals will contain the truth not 5%
  • 00:15:20
    will not that's what the level of
  • 00:15:22
    confidence means all right let's talk
  • 00:15:26
    more specifically now about estimating a
  • 00:15:28
    population proportion so let's say we
  • 00:15:30
    want to
  • 00:15:31
    I'm the true proportion of all boys in
  • 00:15:34
    high school that wear glasses all right
  • 00:15:36
    that's what I'm trying to find how am I
  • 00:15:38
    gonna do it
  • 00:15:38
    well the first thing I'm going to do is
  • 00:15:40
    I'm going to start with a sample
  • 00:15:41
    proportion maybe I found in a sample
  • 00:15:44
    that 32 percent of boys wear glasses
  • 00:15:46
    okay that's my sample proportion from
  • 00:15:49
    there I'm gonna go up and down by my
  • 00:15:51
    margin of error which is a combination
  • 00:15:53
    of your critical value the Z star that
  • 00:15:57
    we just talked about finding based on
  • 00:15:59
    how confident you are times the standard
  • 00:16:02
    deviation of the statistic and this is
  • 00:16:03
    where we run into a slight problem the
  • 00:16:05
    standard deviation of a sample
  • 00:16:08
    proportion is the square root of P times
  • 00:16:12
    1 minus P divided by the sample size n
  • 00:16:15
    the problem is I don't know what the
  • 00:16:18
    true P is remember that's what we're
  • 00:16:20
    trying to estimate so I can't possibly
  • 00:16:23
    calculate the standard deviation of the
  • 00:16:25
    sample proportion if I don't know the
  • 00:16:28
    truth so we do something better we use
  • 00:16:31
    some well I take that back not
  • 00:16:32
    necessarily something better but we use
  • 00:16:34
    an estimate we use something called
  • 00:16:36
    standard air standard air is very
  • 00:16:40
    similar to standard deviation in every
  • 00:16:42
    aspect but it's simply an estimate so
  • 00:16:45
    since we cannot use standard deviation
  • 00:16:47
    because we don't know the true p
  • 00:16:48
    standard air good news is the exact same
  • 00:16:51
    formula but instead of using P it uses P
  • 00:16:55
    hat because we do know P hat that's
  • 00:16:58
    obviously our sample proportion so to
  • 00:17:01
    calculate an or to estimate a population
  • 00:17:03
    proportion with an interval we start off
  • 00:17:06
    with our sample proportion P hat we go
  • 00:17:08
    up and down by Z star that's our
  • 00:17:10
    critical value times the standard error
  • 00:17:12
    and to calculate the standard error we
  • 00:17:14
    use this formula right here which is the
  • 00:17:16
    same formula standard deviation but
  • 00:17:18
    because we don't know the true P we have
  • 00:17:20
    to use P easy all right
  • 00:17:24
    what about estimating a population mean
  • 00:17:26
    what if I want to know the true average
  • 00:17:29
    time it takes a high school or to get to
  • 00:17:32
    school in the morning what is the true
  • 00:17:33
    mean well what I can do is I could take
  • 00:17:37
    a sample mean maybe my sample mean shows
  • 00:17:39
    that the average of a group of you know
  • 00:17:42
    40 kids is ten point three minutes great
  • 00:17:45
    so now I'm gonna take that mean from my
  • 00:17:47
    sample and I'm gonna go up and down by a
  • 00:17:50
    critical value which we just got done
  • 00:17:52
    talking about is eighty star based on
  • 00:17:55
    your level of confidence times once
  • 00:17:58
    again the standard deviation of the
  • 00:18:00
    sample mean so what's the formula for
  • 00:18:02
    the standard deviation of the sample
  • 00:18:04
    mean well it is the square it is Sigma
  • 00:18:06
    the standard deviation of the entire
  • 00:18:07
    population divided by the square root of
  • 00:18:10
    your sample size n but wait a minute I
  • 00:18:13
    don't know the standard deviation of the
  • 00:18:15
    entire population heck I don't even know
  • 00:18:17
    the mean of the entire population that's
  • 00:18:19
    what I'm trying to estimate so how could
  • 00:18:21
    I possibly know the standard deviation
  • 00:18:24
    of the entire population well we don't
  • 00:18:26
    you're absolutely correct so what I'm
  • 00:18:28
    gonna have to use is something called
  • 00:18:30
    standard air once again standard air is
  • 00:18:34
    just like standard deviation in every
  • 00:18:35
    way but instead of using the standard
  • 00:18:37
    deviation of the population on top it
  • 00:18:40
    uses s that is the standard deviation
  • 00:18:43
    from our sample divided by the square
  • 00:18:47
    root of n so again instead of using
  • 00:18:49
    Sigma we use s the standard deviation of
  • 00:18:51
    our sample because I know my sample for
  • 00:18:54
    crying out loud and I'm gonna use that
  • 00:18:56
    as an estimate for the standard
  • 00:18:58
    deviation and that is where we gets in
  • 00:19:00
    an air from so that is how simple it is
  • 00:19:02
    to estimate a population mean all right
  • 00:19:07
    what about trying to estimate the
  • 00:19:08
    difference between two population
  • 00:19:11
    proportions for example what if I want
  • 00:19:13
    to know hey I wonder is there a
  • 00:19:15
    difference or what is the difference
  • 00:19:16
    between the proportion of boys that wear
  • 00:19:20
    glasses and the proportion of girls that
  • 00:19:22
    wear glasses
  • 00:19:23
    what could the difference be well I have
  • 00:19:26
    to first start off with the difference
  • 00:19:28
    between two samples so I'm going to take
  • 00:19:31
    a sample of the boys minus the
  • 00:19:34
    proportion from a sample of girls so I
  • 00:19:37
    have to first use my sample proportions
  • 00:19:40
    right now you know obviously give you a
  • 00:19:42
    specific problem with boys and girls but
  • 00:19:44
    I want to keep things very generic I
  • 00:19:45
    would just be one and two right the
  • 00:19:47
    sample one verse sample two looking at
  • 00:19:50
    the difference between them so I'm going
  • 00:19:51
    to calculate my difference between my
  • 00:19:53
    sample proportions the
  • 00:19:56
    once again I'm gonna go up and down by a
  • 00:19:58
    margin of error to build my interval I
  • 00:20:00
    once again need my critical value Z star
  • 00:20:04
    based on how confident I would like to
  • 00:20:05
    be times well once again I would like to
  • 00:20:09
    use standard deviation but I can't I
  • 00:20:11
    have to use standard error
  • 00:20:12
    now the only issue now is that I have
  • 00:20:15
    two proportions not one so what is the
  • 00:20:17
    formula for standard error well it's a
  • 00:20:20
    little bit of a different formula but if
  • 00:20:21
    you pay attention in class I explain
  • 00:20:23
    where it came from
  • 00:20:23
    it's a giant square root you're gonna do
  • 00:20:26
    P Hat 1 times the opposite of P Hat 1
  • 00:20:30
    divided by the sample size for group 1
  • 00:20:33
    plus P hat 2 times 1 minus P hat 2 all
  • 00:20:40
    divided by the sample size for group 2
  • 00:20:43
    and combining all of that together that
  • 00:20:46
    gets your standard error for the
  • 00:20:48
    difference so it's kind of a weird
  • 00:20:50
    formula but we did talk about where it
  • 00:20:51
    came from and I want to get into too
  • 00:20:53
    much in class we're basically combining
  • 00:20:54
    the two together and it is P hat 1 times
  • 00:20:57
    the opposite one minus P hat 1 divided
  • 00:20:59
    my sample size plus P hat 2 times 1
  • 00:21:02
    minus P 2 divided by sample size and
  • 00:21:04
    then a giant squared around all that do
  • 00:21:06
    all that together that again that
  • 00:21:07
    calculates our standard error you can
  • 00:21:09
    find your interval very very easy all
  • 00:21:12
    right one more to go here estimating the
  • 00:21:14
    difference between two population means
  • 00:21:16
    so is there a difference between how
  • 00:21:19
    long it takes a boy to get to school and
  • 00:21:21
    the average amount of time it takes a
  • 00:21:23
    girl to get to school what's the
  • 00:21:25
    difference well the first thing I do is
  • 00:21:28
    got to look at some samples I'm gonna
  • 00:21:29
    look at a sample of boys and I'm gonna
  • 00:21:31
    look at the difference between the
  • 00:21:33
    sample of boys and the sample of girls
  • 00:21:35
    now I could use a being a G here but I'm
  • 00:21:36
    gonna use the one in two to keep a
  • 00:21:38
    generic basically I have two samples and
  • 00:21:40
    I'm gonna look at the difference between
  • 00:21:42
    them now that's the difference between
  • 00:21:44
    my samples that's not necessarily the
  • 00:21:46
    difference between the true values so
  • 00:21:48
    what I'm going to do is I'm going to go
  • 00:21:49
    up and down by a margin of error in
  • 00:21:51
    hopes of locating what that true
  • 00:21:53
    difference could be I need a t-star
  • 00:21:55
    because when you work with means you
  • 00:21:57
    need T star we talked about how to find
  • 00:21:58
    that critical value already times once
  • 00:22:01
    again a standard error okay I have two
  • 00:22:04
    samples those so what's the standard
  • 00:22:06
    error for two samples well it's kind of
  • 00:22:09
    again
  • 00:22:09
    a formula but here it is it's a giant
  • 00:22:11
    square root it's the standard deviation
  • 00:22:13
    from your first group squared divided by
  • 00:22:16
    the standard by the sample size of your
  • 00:22:18
    first group plus the standard deviation
  • 00:22:21
    squared of your second group divided by
  • 00:22:25
    the sample size of your second group so
  • 00:22:28
    again it's a giant square root inside of
  • 00:22:30
    that square root is the standard
  • 00:22:31
    deviation of Group one squared divided
  • 00:22:33
    by sample size plus the stand deviation
  • 00:22:35
    of group two squared divided by sample
  • 00:22:37
    size that will help calculate the
  • 00:22:39
    standard error you need to build that
  • 00:22:40
    interval and it's really that easy all
  • 00:22:43
    right let's take a look at a couple
  • 00:22:44
    examples that way we can make sure we
  • 00:22:46
    know how to actually use these intervals
  • 00:22:48
    so a ninety-eight percent confidence
  • 00:22:52
    interval for the population mean amount
  • 00:22:54
    of time for a high school student to get
  • 00:22:56
    to school in the morning is eight point
  • 00:22:57
    three to 18 point seven so what is this
  • 00:23:00
    interval meet let's interpret this
  • 00:23:02
    interval the first thing I have to start
  • 00:23:03
    off with is saying I'm 98% confident I'm
  • 00:23:06
    98% confident what okay use the problem
  • 00:23:09
    right I'm 90% confident that the true
  • 00:23:12
    population mean about the time for high
  • 00:23:14
    school to get to school is somewhere in
  • 00:23:16
    the interval between eight point three
  • 00:23:18
    minutes and eighteen point seven minutes
  • 00:23:20
    so I don't know exactly what the true
  • 00:23:22
    average amount of time it takes on high
  • 00:23:24
    school to get to school is but based on
  • 00:23:25
    my sample it's somewhere in that
  • 00:23:27
    interval I'm 98% confident it's in that
  • 00:23:30
    interval all right what is my point s
  • 00:23:32
    that's oftentimes a multiple choice
  • 00:23:34
    question maybe even in frq if you have
  • 00:23:37
    the interval what was the sample mean
  • 00:23:39
    well remember how an interval is created
  • 00:23:41
    you take your sample mean and you go up
  • 00:23:44
    and down by the margin of error that
  • 00:23:46
    should put the sample mean smack dab in
  • 00:23:49
    the middle so all you have to do to find
  • 00:23:52
    that sample mean is add together 8.3 and
  • 00:23:56
    18.7 and then divide by two the dead
  • 00:24:00
    center will be your sample mean that
  • 00:24:02
    means my sample mean must have been
  • 00:24:05
    thirteen point five minutes that was the
  • 00:24:08
    mean of my sample and then I calculated
  • 00:24:10
    my margin of error by the way we could
  • 00:24:13
    also find our margin of error because
  • 00:24:15
    all I got to do is take the 18 point
  • 00:24:17
    seven and subtract the center
  • 00:24:25
    so I thought that mistake there and that
  • 00:24:28
    gives me a margin of error 5.2 or I
  • 00:24:30
    could take the bottom 8.3 and subtract
  • 00:24:33
    the way the center and that would give
  • 00:24:35
    me the same margin of error but negative
  • 00:24:37
    because again you go up and down so my
  • 00:24:39
    margin of error would be plus or minus
  • 00:24:41
    five point two minutes all right
  • 00:24:44
    let's make sure we once again because
  • 00:24:46
    I'm this is really important so I'm
  • 00:24:47
    going over twice is what does 98 percent
  • 00:24:49
    confident means well this was my
  • 00:24:52
    interval if people were to conduct more
  • 00:24:56
    samples of the same size from the same
  • 00:24:58
    population every sample mean would
  • 00:25:01
    technically create its own interval now
  • 00:25:04
    98% of intervals will contain the true
  • 00:25:08
    population mean I just hope that mine is
  • 00:25:12
    one of them so it's not about a
  • 00:25:14
    probability again the truth is either in
  • 00:25:16
    my interval or it's not but I'm 98%
  • 00:25:18
    confident that's in my interval because
  • 00:25:20
    I know that 98% of intervals just like
  • 00:25:23
    mine will contain victories all right
  • 00:25:27
    let's look at this example now all right
  • 00:25:30
    this often comes up on the AP stats test
  • 00:25:32
    and I want to make sure I'm very clear
  • 00:25:33
    on this I've seen this statement worded
  • 00:25:36
    many times I want to make sure if I
  • 00:25:37
    understand so let's go back to our
  • 00:25:38
    interval eight point three to eighteen
  • 00:25:40
    point seven this is a statement that's
  • 00:25:42
    often I've seen on multiple choice so
  • 00:25:44
    say something like this if the true mean
  • 00:25:45
    was outside our interval then the
  • 00:25:48
    probability of getting our sample mean
  • 00:25:50
    would be very low okay that's a great
  • 00:25:53
    statement but a lot of kids don't
  • 00:25:54
    understand why well here's the deal
  • 00:25:56
    one ninety-eight percent confident that
  • 00:25:58
    the truth is somewhere in this interval
  • 00:26:00
    so all I'm trying to say is that if the
  • 00:26:03
    truth was not in that interval let's
  • 00:26:05
    just say the truth was 20 minutes
  • 00:26:07
    obviously my sample did not contain it
  • 00:26:10
    all that means is that my sample mean of
  • 00:26:13
    13.5 must have been very very
  • 00:26:17
    significantly low that is why the
  • 00:26:20
    interval around it did not contain the
  • 00:26:22
    truth it all comes back to that sampling
  • 00:26:25
    distribution if 20 minutes really is the
  • 00:26:28
    truth then my interval should have
  • 00:26:30
    contained 20 minutes but the fact that
  • 00:26:33
    my interval
  • 00:26:34
    it means I must have had an interval I
  • 00:26:36
    must have had a sample way down here
  • 00:26:38
    that when I built my interval I missed
  • 00:26:42
    the truth all that tells me is that my
  • 00:26:45
    sample mean was a very unlikely sample
  • 00:26:48
    now unlikely samples do happen
  • 00:26:50
    unfortunately mine's one of them and
  • 00:26:52
    again this is why we say we're only 98
  • 00:26:55
    percent confident that the truth is in
  • 00:26:56
    our interval but it does bring up a good
  • 00:26:59
    point that if the true mean was outside
  • 00:27:01
    of our interval then the probability of
  • 00:27:03
    getting our sample would have been very
  • 00:27:05
    low but again we don't necessarily
  • 00:27:08
    believe in low probability events
  • 00:27:10
    occurring that is why again we're very
  • 00:27:12
    confident that the truth is not twenty
  • 00:27:14
    minutes and that it really is in our
  • 00:27:16
    interval all right let's do another
  • 00:27:18
    example now with proportions this time
  • 00:27:20
    in an attempt to estimate the proportion
  • 00:27:23
    of households in his city that own at
  • 00:27:25
    least one dog mark instructed the
  • 00:27:27
    following 95% confidence interval
  • 00:27:30
    his interval is 0.15 2.27 all right
  • 00:27:33
    let's interpret this interval I'm 95%
  • 00:27:36
    confident that the true proportion of
  • 00:27:38
    households in his city that at own at
  • 00:27:39
    least one dog is somewhere between 15%
  • 00:27:42
    and 27% all right now a very popular
  • 00:27:46
    question often a multiple choice
  • 00:27:47
    sometimes I'm for response is hey if you
  • 00:27:50
    know your interval what was your sample
  • 00:27:53
    size this is a great question all right
  • 00:27:55
    this isn't it take me a minute to
  • 00:27:56
    explain it but I need a lot of
  • 00:27:58
    information first the first thing I need
  • 00:28:00
    is my sample proportion which again
  • 00:28:03
    should be smack dab in the center so
  • 00:28:06
    once again if I go to my calculator if I
  • 00:28:08
    take the point one five plus point two
  • 00:28:11
    seven add them together divide by 2
  • 00:28:13
    that gets my sample to be point two one
  • 00:28:17
    okay the other thing I need is my margin
  • 00:28:20
    of error my margin of error is how much
  • 00:28:23
    I go up and down from my sample if my
  • 00:28:26
    sample was point two one then I must
  • 00:28:28
    have went up and down by 0.06 because if
  • 00:28:31
    you go up 106 you get 27 percent down
  • 00:28:34
    point O six you get 15% all right so
  • 00:28:37
    what was my sample size now there's one
  • 00:28:39
    more thing I need and that is my Z star
  • 00:28:41
    and we know if you're 95% confident you
  • 00:28:45
    could simply look it up 95% cough
  • 00:28:47
    your Z star would be 1.96 or you can use
  • 00:28:51
    your calculator okay
  • 00:28:52
    now I'm all ready to go remember that
  • 00:28:54
    the back part the back part is your Z
  • 00:28:57
    star times the standard error that back
  • 00:28:59
    part to the formula is your margin of
  • 00:29:02
    error so now I'm just going to fill in
  • 00:29:04
    all the blanks the margin of error is
  • 00:29:06
    0.06 the Z star is 1.96 the formula for
  • 00:29:12
    standard error is the square root of P
  • 00:29:14
    hat 0.2 1 times 1 minus P hat which
  • 00:29:18
    would be 0.79 all divided by the sample
  • 00:29:22
    size and that is what I don't know that
  • 00:29:25
    is what I have to solve for so I needed
  • 00:29:28
    the margin of error
  • 00:29:29
    I needed the Z star I needed my P hat so
  • 00:29:31
    I could find my 1 minus P hat as well
  • 00:29:33
    and now I could solve for n first step
  • 00:29:36
    is to divide both sides by the 1.96
  • 00:29:44
    second step is to get rid of the square
  • 00:29:46
    root by squaring both sides that cancels
  • 00:29:49
    out the square root so I get 0.06
  • 00:29:52
    divided by 1 point 9 6 squared equals
  • 00:29:55
    point 2 1 times point 7 9 divided by n
  • 00:29:58
    next step would be to instead of Multan
  • 00:30:01
    stead of dividing by n to multiply that
  • 00:30:03
    end up to here and then the final step
  • 00:30:07
    would be to divide by all of that stuff
  • 00:30:09
    in parentheses so I'm going to take the
  • 00:30:10
    point 2 1 times point 7 9 and divide by
  • 00:30:13
    the point O 6 divided by 1 point 9 6 all
  • 00:30:17
    squared ok so that's a lot hopefully
  • 00:30:21
    you're good at algebra this is not as
  • 00:30:22
    hard as the algebra gets 0.2 one times
  • 00:30:24
    point seven nine okay there's that and
  • 00:30:28
    I'm going to divide that by in
  • 00:30:29
    parentheses 0.06 divided by 1.96 and
  • 00:30:36
    then don't forget to make sure that
  • 00:30:37
    squares on the outside those parentheses
  • 00:30:39
    so I get a sample of about one hundred
  • 00:30:41
    seventy seven point zero three how they
  • 00:30:43
    recommend always rounding up because
  • 00:30:45
    that will make you a little bit more
  • 00:30:46
    accurate remember bigger sample is
  • 00:30:48
    always better so I would say
  • 00:30:49
    approximately 178 households must have
  • 00:30:53
    been his sample size that's a very
  • 00:30:56
    popular question hopefully you remember
  • 00:30:57
    how to do that all right
  • 00:31:00
    to the last topic which was making sure
  • 00:31:02
    you can understand how comfortable can
  • 00:31:03
    help us understand significance test a
  • 00:31:05
    95 percent confident a mode for the
  • 00:31:08
    difference between the proportion of
  • 00:31:09
    teenage girls that wear glasses and the
  • 00:31:11
    proportion teams boys and wear glasses
  • 00:31:12
    is negative point oh eight two point oh
  • 00:31:15
    four so let's make sure we know how to
  • 00:31:16
    interpret this I'm 95% confident that
  • 00:31:19
    the true difference between the
  • 00:31:20
    proportion of boys and girls that wear
  • 00:31:22
    glasses is anywhere from negative eight
  • 00:31:25
    percent to positive four percent that
  • 00:31:27
    means girls are eight percent higher to
  • 00:31:32
    four percent lower than boys or I can
  • 00:31:35
    look at that the other way around
  • 00:31:36
    boys are eight percent lower to four
  • 00:31:38
    percent higher the fact that this
  • 00:31:40
    interval is negative on one side and
  • 00:31:42
    positive on the other means that girls
  • 00:31:44
    could be eight percent more than boys in
  • 00:31:46
    terms of wearing glasses or girls can be
  • 00:31:48
    four percent less than boys because I'm
  • 00:31:51
    doing boy I'm excuse me I'm doing girls
  • 00:31:53
    - boys so you got to make sure you truly
  • 00:31:55
    understand how this interval works now
  • 00:31:57
    does this interval show that there is a
  • 00:32:00
    significant difference between boys and
  • 00:32:02
    girls well no it doesn't because this
  • 00:32:07
    interval actually leads me to three
  • 00:32:08
    conclusions girls could be more likely
  • 00:32:11
    to hourglasses
  • 00:32:12
    boys could be more likely to hourglasses
  • 00:32:14
    or you know what numbers in this
  • 00:32:16
    interval a big fat zero there can
  • 00:32:19
    actually be no difference between boys
  • 00:32:22
    and girls and wearing glasses so the
  • 00:32:25
    proportion of boys that were glasses and
  • 00:32:27
    the proportion of girls that wear
  • 00:32:28
    glasses can actually be exactly the same
  • 00:32:30
    and if that's true there's no difference
  • 00:32:33
    and that is a legitimate possibility
  • 00:32:34
    since zero falls in this interval so is
  • 00:32:37
    there really a difference
  • 00:32:38
    no the interval contains zero and the
  • 00:32:41
    interval contains a positive and
  • 00:32:42
    negative numbers so do girls wear that
  • 00:32:45
    wear more is there a larger proportion
  • 00:32:47
    of girls that wear glasses or a larger
  • 00:32:49
    proportion of boys that wear glasses I
  • 00:32:51
    don't know it could be larger for girls
  • 00:32:54
    it could be larger for boys or you know
  • 00:32:56
    what guys
  • 00:32:56
    there probably is no difference
  • 00:32:58
    whatsoever so when we think about a
  • 00:33:00
    significance s which there's a separate
  • 00:33:02
    video for our null is that the perp must
  • 00:33:06
    do something wrong there the proportion
  • 00:33:08
    of boys is equal to the proportion of
  • 00:33:10
    girls and the alternative would be
  • 00:33:12
    something like
  • 00:33:13
    the proportion of boys is more than the
  • 00:33:15
    proportion of girls who wear glasses
  • 00:33:17
    well my interval doesn't tell me that it
  • 00:33:19
    doesn't say that boys is more it says
  • 00:33:22
    boys could be more but boys could be
  • 00:33:23
    less or there could be no difference so
  • 00:33:26
    that is when we would actually fail to
  • 00:33:28
    reject the null because I don't have
  • 00:33:30
    enough evidence to say there really is a
  • 00:33:31
    difference because there could be no
  • 00:33:34
    difference so hopefully your teacher
  • 00:33:36
    went over that with you and hopefully
  • 00:33:38
    that actually makes sense to you because
  • 00:33:39
    a confidence interval oftentimes should
  • 00:33:41
    or should always support a significance
  • 00:33:44
    test all right guys that was a long
  • 00:33:46
    video but there is an awful lot to talk
  • 00:33:48
    about when it comes to confidence
  • 00:33:49
    intervals so hopefully this will help
  • 00:33:50
    you ace any questions on the tests that
  • 00:33:52
    cover them
Tags
  • interval de confiança
  • estadístiques AP
  • marge d'error
  • valors crítics
  • mostra aleatòria
  • mitjana de la mostra
  • proporció de la mostra
  • condicions
  • conclusions
  • significativitat