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Topic What to use Z or T distribution?

What to use Z or T distribution?

Home Forums Old Forums General What to use Z or T distribution?

This topic contains 3 replies, has 4 voices, and was last updated by Profile photo of Chris Seider Chris Seider 13 years ago.

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  • #31962

    I have to appeal form you guys.  I was reading a lot of materials and it tells me that I should use Z distribution for hypothesis testing if:
         – Population Standard Deviation is KNOWN, sample size is either less than or more than 30
         – Population Standard Deviation is UNKNOWN, sample size is either more than or equal to 30
    I would use T distribution for hypothesis testing if:
         – Population Standard Deviation is UNKNOWN, sample size is less than 30
    Is this correct?  I am asking this because there seem to be a misunderstanding in my group.  Some are thinking that even if Population Standard Deviation is known, as long as sample size is less than 30, T distribution should be used. 
    What is correct?  Please help.  Thanks.


    I have seen the T distribution used in scenarios where there is an estimate of the population standard deviation and looking to perform inferential statistics with small confirmatory samples.  With a sample size that is less than 30, a conservative would use the T vs the Z since there is some added buffer in the wider tails of the T.  My recommendation would be to run both distributions and see how comparable the results are. Utilizing a Z vs T distribution results in a higher likelihood of committing a Type I error.  At the same level of significance, barring large sample sizes 30+, the critical value for the Z will be less than the corresponding T.  Another factor to consider is what is the alternate hypothesis (e.g. nondirectional vs. directional)  this will also have a huge impact on the results of your analysis.


    The rule is simple:
    If the population standard deviation is known, use the z-distribution.
    If the population standard deviation is estimated using the sample standard deviation, use the t-distribution.
    It so happens that the t-distribution tends to look quite normal as the degrees of freedom (n-1) becomes larger than 30 or so, so some users use this as a shortcut.


    Your question can be answered in two ways. 
    It depends on what statistic you are attempting to delve into.  It is difficult to determine what % of products would have a value of X or above–an individual distribution (shown as Z).  Note the t distributions assume a minimum df = 1, meaning an average of 2 items.  I would use a Z distribution if I wanted to know what % of items would be above some value assuming I had enough data (samples > 30) to approximate the population. 
    If you are comparing means and need to decide between the Z and t distributions, it does not matter if sigma (greek) is known or not if the sample is large enough in each group, greater than 30–you can use either Z or t because they become similar enough past n = 30.
    I quote “Introduction to Business Statistics” by Kvanli, Guynes, and Pavur (4th edition) which says on p.250 “Remember, however, that a more accurate confidence interval is always obtained using the t table when the sample standard deviation (s) is used in construction of this interval.
    I hope this helps.

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