Setting Up a Two-Mean Test

A four unit gap needs an uncertainty model

• Two fictional independent random samples have means 52 and 48 units. Each sample has 25 observations; their standard deviations are 6 and 8 units.
• The observed difference is four units. The inference target is the difference between the two population means, with the subtraction order group 1 minus group 2.

State a population equality claim

• Use $H_0:\mu_1-\mu_2=0$. The preplanned two-sided alternative is $H_a:\mu_1-\mu_2\ne0$.
• Choose a one-sided alternative only if that direction comes from the research question before examining the results.

Check each group and their relationship

• Check random design, independence between samples, the 10% condition when relevant, and the shape of each sample. Each group has fewer than thirty observations here.
• Matched or repeated measurements require a paired analysis. Separate group labels alone do not prove that their observations are independent.

Add the two variance contributions

• For the Welch two-sample method taught here, $SE=\sqrt{s_1^2/n_1+s_2^2/n_2}$. There is no equal-variance pooling assumption.
• Here $SE=\sqrt{36/25+64/25}=2$ units, so $t=(52-48)/2=2$. The variance contributions add even though the means are subtracted.

Use degrees of freedom from this procedure

• A two-sample t routine gives Welch degrees of freedom, which can be fractional. For these values, $df\approx44.51$.
• Do not automatically use $25+25-2=48$; that is the pooled equal-variance procedure. A t value of 2 does not alone determine its p-value.

Carry the statistic to the selected tail rule

• For this two-sided example, the p-value is approximately 0.0516 using $df\approx44.51$. At alpha 0.05, fail to reject population equality.
• The observed four unit difference remains an estimate; non-rejection does not prove equality. Report the uncertainty and use a prechosen direction for any one-sided test.

Write from equal population means

• Two independent groups are compared.
• The question is whether their population means differ. A sample gap of 4 units is observed.