Question about t statistic. Of these two, which formula is correct? t(x) = frac(bar(X) - mu )(sqrt(frac(1)(n)sum left ( X_(i) - bar(X)right )^(2)))sqrt(n)

Jayvion Caldwell 2022-07-22 Answered
Question about t statistic. Of these two, which formula is correct?
t ( x ) = X ¯ μ 1 n ( X i X ¯ ) 2 n 1
t ( x ) = X ¯ μ 1 n 1 ( X i X ¯ ) 2 n
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Answers (1)

Franklin Frey
Answered 2022-07-23 Author has 15 answers
They are mathematically equivalent, because
n 1 1 / n = n n 1 = n 1 / ( n 1 ) .
That said, the second formula is a better reflection of how the t statistic is calculated, because it is obtained by estimating the standard deviation of the population from the sample when it is unknown. Thus,
s 2 = 1 n 1 i = 1 n ( X i X ¯ ) 2
is the unbiased estimator for the variance σ 2 when X i are iid normal random variables with unknown mean μ and unknown variance σ 2 . Then the statistic
X μ s / n
is Student t distributed where s / n is the standard error of the mean.
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