Sampling distributions of differences

Three quick questions from earlier lessons.

Two independent sample means have SDs 3 and 4. What is the SD of their difference?

Differences between independent samples.

For independent samples:

\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2

,

\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\dfrac{p_1(1 - p_1)}{n_1} + \dfrac{p_2(1 - p_2)}{n_2}}

; and

\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}

.

$p_1 = 0.5$ , $p_2 = 0.3$ , $n_1 = n_2 = 100$ .

\mu = 0.2

Center.

\sigma = \sqrt{0.0046} \approx 0.068

0.0025 + 0.0021.

Your turn: $\sigma_1^2/n_1 = 9$ and $\sigma_2^2/n_2 = 16$ . Find $\sigma_{\bar{x}_1 - \bar{x}_2}$ .

5

√25.

Independent samples of size 4 come from populations with SDs 36 and 48. Find the SD of x̄₁ − x̄₂.
Independent samples of size 4 come from populations with SDs 18 and 24. Find the SD of x̄₁ − x̄₂.

Independent random samples of 100 are taken from two populations with proportions 8/10 and 1/10. Find the SD of p̂₁ − p̂₂.

When are the variances of x̄₁ and x̄₂ added to get the variance of x̄₁ − x̄₂ (samples of 66)?
1. When the two samples are independent
2. Only when the sample sizes are equal
3. Always, even for paired measurements

Three questions to check you have got it.

Independent samples of size 4 come from populations with SDs 36 and 48. Find the SD of x̄₁ − x̄₂.
Independent samples of size 4 come from populations with SDs 18 and 24. Find the SD of x̄₁ − x̄₂.
Independent samples of size 4 come from populations with SDs 54 and 72. Find the SD of x̄₁ − x̄₂.

Differences

Independent samples: means subtract, variances add.

Watch out: paired data are not independent.

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