# t-test: understanding alternative hypothesis, power, and noncentral distributions

1. While the alternative hypothesis $H_a: \mu_c \ne \mu_t$ covers an infinite number of possible values for $\mu_t$, the power of the test can only be defined based on a point value of $\mu_t$ - is this statement true?

2. If the alternative hypothesis $H_a: \mu_c \ne \mu_t$ were true, then does the t-statistic follow the non-central t-distribution, with noncentrality parameter, $\lambda$? (e.g., wiki, real-statistics)

3. Are all distributions under alternative hypotheses skewed/asymmetric? (e.g. wiki, real-statistics)

My understanding of a t-test

In general, I have heard that any test statistic can be summarised as (e.g., statisticshowto, duke.edu):

$$\frac{x_\mathsf{obs} - x_\mathsf{hyp}}{S.E} \mathsf{\ \ \ (1)}$$

i.e., I interpret this as the 'standardised' difference (i.e., in units of standard error) between observed measurements and hypothesised values.

For the comparison of two sample means, Eq. 1 can be expressed more specifically to give the value of the t-statistic (e.g., wiki, real-statistics):

$$t = \frac{\big(\bar{x_c} - \bar{x_t}\big) - \big(\mu_c - \mu_t\big)}{\sqrt{ \frac{s_c^2}{n_c} + \frac{s_t^2}{n_t} }} \mathsf{\ \ \ (2)}$$

where subscript $c$ and $t$ signify the control sample and test sample, respectively, and

• $\bar{x}$: mean of the observed sample
• $s$: the standard deviation of the observed sample
• $n$: the size of the observed sample
• $\mu$: the actual/assumed/hypothesised mean of the underlying population

In the case where the null hypothesis $H_0: \mu_c = \mu_t$, the second term in Eq. (2) becomes 0. If the null hypothesis were true, then the t-statistic follows the central t-distribution. The observed (sample-specific) value of $t$ (along with degrees of freedom, $\nu$) can then be used to find a $p$-value via:

$$p = f(t,\nu)$$

where $f$ is some function (e.g., T.DIST (Excel) or scipy.stats.ttest_ind (Python)) and $\nu$ is approximated by the Welch-Satterthwaite equation:

$$\frac{\Big( \frac{s_c^2}{n_c} + \frac{s_t^2}{n_t} \Big)^2}{ \frac{\Big(\frac{s_c^2}{n_c}\Big)^2}{n_c-1} + \frac{\Big(\frac{s_t^2}{n_t}\Big)^2}{n_t-1} }$$

• 1. Yes. 2. Only when the distributional assumptions are met. Non-normal data will have a non-central T limiting distribution. 3. Yes the general shape of a non-central T with non-zero NCP is skewed. – AdamO Apr 26 '18 at 14:19
• Interesting question! Welcome to CV. The "any test statistic can be summarized" sentence ignores omnibus test statistics (e.g., F tests), contingency-table test statistics (e.g., $\chi^{2}$ tests), test statistics formed using the two one-sided tests form, and likely others. – Alexis Apr 26 '18 at 15:15