1. Sampling Distribution of the Difference of Means

Suppose we have two independent populations:

• Population 1: mean $\mu_1$, variance $\sigma_1^2$, sample size $n_1$
• Population 2: mean $\mu_2$, variance $\sigma_2^2$, sample size $n_2$

Sample means: $\bar{X}_1$ and $\bar{X}_2$

We are interested in the difference of sample means:

Properties

1. Mean of $D$:

   $$E(D) = E(\bar{X}_1 - \bar{X}_2) = \mu_1 - \mu_2$$

2. Variance of $D$ (assuming independence):

   $$Var(D) = Var(\bar{X}_1) + Var(\bar{X}_2) = \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}$$

3. Standard Error (SE) of Difference:

   $$SE(D) = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$$

4. Distribution Shape:

• If populations are normal → $D$ is exactly normal
• If sample sizes are large → $D$ is approximately normal (Central Limit Theorem)

Example:

• Population 1 mean = 50, $\sigma_1 = 10$, $n_1 = 25$
• Population 2 mean = 45, $\sigma_2 = 8$, $n_2 = 36$ $$SE(D) = \sqrt{\frac{10^2}{25} + \frac{8^2}{36}} = \sqrt{4 + 1.78} = \sqrt{5.78} \approx 2.41$$
• Difference in means = (50 - 45 = 5), SE = 2.41

2. Sampling Distribution of the Difference of Proportions

Suppose we have two independent populations with proportions of success:

• Population 1: $p_1$, sample size $n_1$
• Population 2: $p_2$, sample size $n_2$

Sample proportions: $\hat{p}_1$ and $\hat{p}_2$

We are interested in the difference of sample proportions:

Properties

1. Mean of $D_p$:

   $$E(D_p) = p_1 - p_2$$

2. Variance of $D_p$ (assuming independence):

   $$Var(D_p) = \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}$$

3. Standard Error (SE) of Difference:

   $$SE(D_p) = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$$

4. Distribution Shape:

• For large $n_1$ and $n_2$ → approximately normal

Example:

• Population 1 proportion $p_1 = 0.6$, $n_1 = 100$
• Population 2 proportion $p_2 = 0.4$, $n_2 = 120$ $$SE(D_p) = \sqrt{\frac{0.6*0.4}{100} + \frac{0.4*0.6}{120}} = \sqrt{0.0024 + 0.002} = \sqrt{0.0044} \approx 0.066$$
• Difference in proportions = (0.6 - 0.4 = 0.2), SE ≈ 0.066

3. Key Points to Remember

Important Notes:

1. Populations should be independent.
2. For small sample sizes, consider using t-distribution for means.
3. For proportions, normal approximation works if $n_1p_1, n_1(1-p_1), n_2p_2, n_2(1-p_2) \ge 5$.