two sample comparison

statistics

Ch 10 in Introductory Statistic book.

If the requirements aren’t met, look into Wilcoxon rank sum tests.

Definitions:

$D$ : diferences between two paired samples

$d_i$ : the ith observation in $D$

Understanding Means

Means w/ Large samples (z-score)

Requirements for large sample inferences about ${\mu }_1-{\mu }_2$:

1. samples are independent
2. samples sizes are both large (n >= 30)

Similar to the Confidence Intervals for a single mean (which is $\mu =\bar{x}\pm z_{\frac{\alpha }{2}}{\sigma }_{\bar{x}}$), we can compare by subtracting ${\mu }_1-{\mu }_2$.

$$\begin{array}{r}(\overline{x_1}-\overline{x_2})\pm z_{\frac{\alpha }{2}}{\sigma }_{(\overline{x_1}-\overline{x_2)}}\end{array}$$

The standard deviation of the sampling distribution (${\sigma }_{\overline{x_1}-\overline{x_2}}$ aka the standard error of the statistic) is:

$$\begin{array}{r}{\sigma }_{(\overline{x_1}-\overline{x_2)}=\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}}}\end{array}$$

Two populations means with known standard deviations Test statistic: $z_c=\frac{({\bar{x}}_1-{\bar{x}}_2)-D_0}{\sqrt{(\frac{{\sigma }^{2_1}}{n_1})+(\frac{{\sigma }^{2_2}}{n_2})}}$ where $D_0$ is the number in the hypothesis $H_0$ being tested.

Means w/ Small samples (t-score)

Requirements:

1. both are normally distributed
2. both have equal population variances
3. selected independently

b/c they have equal variances, we can use a “poled sample estimator ${\sigma }^2$ called $s_p^2$

$$\begin{array}{r}{s}_p^2=\frac{\Sigma (x_1-{\bar{x}}_1{)}^2+\Sigma (x_2-{\bar{x}}_2{)}^2}{n_1+n_2-2}\end{array}$$

Test statistic (aka t-score): $t_c=\frac{({\bar{x}}_1-{\bar{x}}_2)-D_0}{\sqrt{s_p^2(\frac{1}{n_1}+\frac{1}{n_2})}}$

paired difference experiment

This is a type of blocking.

Assumptions:

1. The population of differences in test scores is approximately normal.
2. sample differences are randomly selected from population differences.

This is when you pair up folks from the same population. We can’t use a t-test b/c they aren’t independent variables (the first one may be but the second is bucketed into the other side of the pair).

Instead, we take the difference of the pairs and use that as the sample.

Test statistic:

$$\begin{array}{r}t=\frac{{\bar{x}}_d-D_0}{\frac{s_d}{\sqrt{n_d}}}\end{array}$$

where ${\bar{x}}_d$ is the sample mean difference, n is the number of pairs, and s is the sample Standard Deviation of differences.

Examples:

1. pairing up gas prices based on location to account for regional differences
2. comparing men & women salaries, segmented by majors and GPAs.

Proportions

Confidence interval:

$$\begin{array}{r}({\hat{p}}_1-{\hat{p}}_2)\pm z_{\alpha /2}\sqrt{\frac{p_1{q}_1}{n_1}+\frac{p_2{q}_2}{n_2}}\end{array}$$

Test statistic:

$$\begin{array}{r}{z}_c=\frac{{\hat{p}}_1-{\hat{p}}_2}{\sqrt{\frac{p_1{q}_1}{n_1}+\frac{p_2{q}_2}{n_2}}}\approx \frac{{\hat{p}}_1-{\hat{p}}_2}{\sqrt{\hat{p}\hat{q}(\frac{1}{n_1}+\frac{1}{n_2})}}\end{array}$$

Requirements:

1. random selections
2. sample sizes are large (each proportion ($n\hat{p},n\hat{q}$) results in at least 15.