Confidence Intervals

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Intro to Confidence Intervals for One Mean (Sigma Known)
10:36 — by jbstatistics

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Key Questions

  • Answer:

    See explanation.

    Explanation:

    An estimate of a population parameter may be expressed in two ways:

    POINT ESTIMATION

    A point estimate of a population parameter is a single value of a statistic.

    Example,

    The sample mean #barx# is a point estimate of the population mean μ. Similarly, the sample proportion p is a point estimate of the population proportion P.

    INTERVAL ESTIMATION

    An interval estimate is defined by two numbers, between which a population parameter is said to lie.

    Example

    #a < x < b# is an interval estimate of the population mean #mu#. It indicates that the population mean is greater than a but less than #b#.

    In any estimation problem, we need to obtain both a point estimate and an interval estimate. The point estimate is our best guess of the true value of the parameter, while the interval estimate gives a measure of the accuracy of that point estimate by providing an interval that contains plausible values.

    http://www.biochemia-medica.com/

  • There are total 3 cases
    if sigma is known(doesn't matter n is large or small) then use this formula
    #barx- z_"(alpha/2)" *sigma/sqrt(n)# for lower confidence interval limit
    #barx+ z_"(alpha/2)" *sigma/sqrt(n)# for upper confidence interval limit

    if n is >30 and sigma is unknown then use this formula
    #barx- z_"(alpha/2)" *S/sqrt(n)# for lower confidence interval limit
    #barx+z_"(alpha/2)" *S/sqrt(n)#for upper confidence interval limit

    if n is <=30 sigma is unknown then use this formula
    #barx- t_"(alpha/2,df=n-1)" *s/sqrt(n)#for lower confidence interval limit
    #barx+ t_"(alpha/2,df=n-1)" *s/sqrt(n)# for upper confidence interval limit

    now consider the example for large sample size
    n=3,500
    xbar=174.4
    S=38.7
    The Z value for 95% confidence is Z=1.96
    #174.4-1.96*38.7/sqrt(3500)#
    #174.4+1.96*38.7/sqrt(3500)#
    hence
    #173.11<=mu<=175.68#

    now consider the example for small sample size
    n=10
    xbar=176
    s=33
    the degrees of freedom (df) = n-1 = 9. The t value for 95% confidence with df = 9 is t = 2.262.
    #176-2.262*33/sqrt(10)#
    #176+2.262*33/sqrt(10)#
    hence
    #152.39<=mu<=190.60#

  • Suppose the confidence interval is 95%.

    This interval then represents the range within which you would expect a sample mean to fall in 95% of random samples.

    It does not mean that the probability is 95%, or that a specific value will fall within this interval in 95% of cases.

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