Showing Revision 5 created 05/24/2016 by Udacity Robot.

1. Title:
   Welchs Two-Sample t-Test - Intro to Data Science
2. Description:

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   Let's talk more about the two sample t-test, since we'll want
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   to compare two different samples in our class project. There are
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   a few different versions of the t-test that one might employ
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   ,and they depend on really on what assumptions we make about the
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   data. So we might want to ask questions such as ,do our
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   samples have the same size ?,and do they have the same
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   variance? . Let's discuss a variant of the t-test called Welch's
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    t-test in more depth. Since it's the most general. It doesn't assume
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    equal sample size ,or equal variance. In Welch's
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    t-test ,we compute a t-statistic using following equation.
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    T equals mu1 minus mu2, divided by the
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    square root of sigma1 squared over n1. Plus
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    sigma 2 squared over n2. Where mu I ,is the sample mean for the Ith sample.
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    Sigma squared I is the sample variance for
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    the Ith sample. And NI is the sample size
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    for the Ith sample. We'll also want to estimate the number of degrees
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    of freedom, nu, using the following equation.
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    Nu is approximately equal to. Quantity sigma1
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    squared ,over n1 ,plus sigma2 squared over n2 ,squared over sigma1 of the
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    4th over n1 squared nu1 ,plus sigma2 to the 4th ,over n2 squared nu2.
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    Where mu I is equal to mi minus one ,and
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    this is the degrees of freedom associated with the Ith variance
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    estimate. If you're unfamiliar with degrees of freedom again it might
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    be a good idea to brush up on your stats concepts
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    with the audacity's intro to stats course. A link is
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    provided in the instructor comments. All right so once we have
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    these two values, we can estimate the P value. Conceptually, the
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    P-value is the probability of obtaining the test statistic at least
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    as extreme as the one that was actually observed
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    ,assuming that the null hypothesis was true. The P
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    value is not the probability of the null hypothesis
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    is true given the data. So again, just as a
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    thought experiment. Say we were testing whether left handed
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    or right handed baseball players. Were better batters by looking
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    at their average batting average. If the P value
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    is .05, this would mean that ,even if there is
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    no difference between left handed and right handed batters, since
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    that's our null hypothesis. So, even if this was true,
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    we would see a value of t ,equal or greater
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    to the one that we saw 5% of the time.
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    When performing a statistical test like this, we usually set
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    some critical value of P. Let's call it P critical.
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    If P falls below P critical, then we would reject
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    the null hypothesis. In the two sample case, this is equivalent
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    to stating that the mean for our two samples
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    is not equal. Calculating this P value for a
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    given set of data can be kind of of tedious.
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    Thankfully, we seldom have to perform this calculation explicitly.