Difference of Two Means

Comparing two means

• To compare two population means, use the difference of two sample means 两个样本均值之差 $\bar{x}_1 - \bar{x}_2$.
• Its sampling distribution is centered at the true difference: $\mu_1 - \mu_2$.
• This statistic drives every two-sample $t$-procedure in Unit 7.
• It tells us whether two groups' averages genuinely differ.

Combining the two spreads

• The standard deviation combines both groups' variability:
• $$\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$$
• Again the variances add — each group contributes its own $\frac{\sigma^2}{n}$.
• The $+$ holds even though we're taking a difference of means.

The distribution of x̄₁ − x̄₂: centres subtract and variances add

Conditions for both samples

• Normality: each $\bar{x}$ must be normal — a normal population or a large enough $n$ (CLT) for each group.
• Independence: the two samples are independent of each other.
• The 10% condition for each if sampling without replacement.
• Verify these separately for group $1$ and group $2$.

Using the normal model

• With conditions met, $\bar{x}_1 - \bar{x}_2$ is approximately normal.
• Standardize: (observed difference $-$ true difference) $\div$ the combined SD.
• Read the probability from the standard normal.
• This foundation becomes the two-sample $t$-test and interval later.

Write from a difference of means equal to 4

• One population mean is 10 and the other is 6.
• The samples are independent. The lesson difference of means 5 is another pair.