Let
- $X_1,...,X_n\stackrel{i.i.d}\sim N(μ_0,σ_0^2)$
- $Y_1,...,Y_m\stackrel{i.i.d}\sim N(μ_1,σ_1^2)$
- the $X_i$'s are independent of the $Y_i$'s
where $μ_0,μ_1,σ_0^2,σ_1^2$ are unknown parameters.
Suppose the data collected are summarized as follows:
- Sample sizes: $n=m=5$;
- Sample means:
- $\bar{X}_{n}=1.2$
- $\bar{Y}_{m}=1.0$
- Sample variances: $\hat{\sigma}_1^2=\hat{\sigma}_2^2=0.5$.
Choosing from the following distributions that best approximates the distribution of the $t$-statistic $T_{n=5,m=5}$ under the null hypothesis $H_0 ∶μ_0=μ_1$:
- Normal distribution with mean $μ≠0$
- Standard normal distribution
- $t$-distribution
- $F$-distribution
The most "plausible" distribution would be the $t$-distribution. Seems about right?
My question is related to the degrees of freedom (assuming my choice of distribution is correct).
For the $t$-distribution, the $t$-statistic is given by
$$ T_n =\frac{\bar{X_{n}}-\bar{Y_{m}}}{\sqrt{\hat{\sigma}_1^2/m+\hat{\sigma}_0^2/n}}$$
The general formula is $df = n_1 + n_2 - 2$, where $n_1,n_2$ are $n,m$ respectively so $T_n \sim t_{n+m-2, \alpha/2}$ therefore $df=8$. On the other hand, according to Welch-Satterthwaite's (WS) formula (which calculates an approximation to the effective degrees of freedom), under $H_0$, $T_n$ is approximately distributed as $t_{40}$ where $df=40$.
Which $df$ value is the correct one?
The problem says "best approximates" so WS makes sense but I'm not completely sure.
Welch’s $t$-test is used when the population variances are assumed unequal.
You may conduct a test such as F-test, Levene’s test, Bartlett’s test, to assess the equality of variance assumption.
This may lead to inflated Type I error though, so it may be wise to specify a priori which test to use.
If the sample sizes are equal, student’s $t$-test is robust to unequal variances. (Welch’s $t$-test is robust regardless.)