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Confidence intervals and hypothesis tests

sophie waddington 6 questions Updated 25 Dec 2018

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  1. 1Central [blank_start]limit[blank_end] theorem = say that whatever the original distribution of the data, the distribution of the sample means tends to a [blank_start]normal[blank_end] distribution as the sample size, n, [blank_start]increases[blank_end]
    1. Alimit
    2. Bnormal
    3. Cincreases
  2. 2inferring [blank_start]population[blank_end] means from sample means ( a small sample)

    it is reasonable to assume that IQs comes from a [blank_start]normal[blank_end] distribution
    we need a way of showing how accurate our estimates our; how similar is our [blank_start]sample[blank_end] mean to our population mean
    - would be better to construct a "[blank_start]confidence[blank_end] interval" for our mean; i.e. a rang that will contain the true value most of the time

    so how did our sample mean arise?
    > by the [blank_start]central[blank_end] limit theorem, it came from a normal distribution with mean and standard deviation
    > standard deviation = mean / [blank_start]square[blank_end] root of sample size

    an important property of the normal distribution = [blank_start]95[blank_end]% of the data, lies within [blank_start]1.96[blank_end] (2) standard deviations of the mean either size

    we can say that we would be 95% confident that the population mean lies between: 1.96 x (standard deviation; population mean/(square root sample size) of the sample mean

    suppose you were to know what the standard [blank_start]deviation[blank_end] of population (sigma) was, then you could use this to construct a 95% confidence interval for the population.

    that would be the sample mean + or [blank_start]-[blank_end] (1.96 x standard deviation/square root n)

    this tells us: from the sample we took, we can be 95% sure that the population [blank_start]mean[blank_end] is between the ranges calculated

    PROBLEM IN REAL LIFE WE DO NOT KNOW WHAT SIGMA (STANDARD [blank_start]DEVIATION[blank_end] OF POPULATION IS)

    1. Apopulation
    2. Bnormal
    3. Csample
    4. Dconfidence
    5. Ecentral
    6. Fsquare
    7. G95
    8. H1.96
    9. Ideviation
    10. J-
    11. Kmean
    12. LDEVIATION
  3. 3PROBLEM = in real life, we don't know the standard [blank_start]deviation[blank_end] of the population

    solution: we estimate it by using the [blank_start]sample[blank_end] standard deviation which we call s

    unfortunately this introduces a bit more uncertainty and it means we have to replace our [blank_start]normal[blank_end] distribution in favour of a [blank_start]t[blank_end]-distribution (don't need to worry as to why, it has wider tails away from the mean due to uncertainty)
    1. Adeviation
    2. Bsample
    3. Cnormal
    4. Dt
  4. 4the difference between the [blank_start]t[blank_end] distribution and the normal distribution is that for a 95% confidence interval we need to replace the key number, 1.96, with a [blank_start]bigger[blank_end] number, which depends on the same size

    if n is the sample size, we always use the t-distribution with n-1 degrees of [blank_start]freedom[blank_end]
    - so if 16 in sample, degrees of freedom = 16-1 = [blank_start]15[blank_end]

    as we have estimated our population standard deviation, the confidence interval gets [blank_start]wider[blank_end]/bigger > for we have [blank_start]more[blank_end] uncertainty

    centred around the sample [blank_start]mean[blank_end], = the confidence interval

    luckily minitab can do it for us > stat > basic statistics > 1 sample t
    1. At
    2. Bbigger
    3. Cfreedom
    4. D15
    5. Ewider
    6. Fmore
    7. Gmean
  5. 5a 95% confidence interval will contain the true population mean [blank_start]95[blank_end]% of the time
    - we can work out a 99% confidence interval, this would be [blank_start]wider[blank_end]/bigger
    1. A95
    2. Bwider
  6. 6the p-value is a measure of probability, it tells us how likely it would be to see a sample as extreme or more extreme, as the one we have actually observed, if, in fact, the [blank_start]null[blank_end] hypothesis is true

    [blank_start]small[blank_end] p values tell us that the sample we have observed would be unlikely to have occurred if the [blank_start]null[blank_end] hypothesis is really true
    - [blank_start]small[blank_end] p value constitutes evidence against the null hypothesis
    1. Anull
    2. Bsmall
    3. Cnull
    4. Dsmall

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Description
Stats Quiz on Confidence intervals and hypothesis tests, created by sophie waddington on 25/12/2018.
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Created on 25 Dec 2018 · updated on 25 Dec 2018
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