| subject | actual | desired | difference |
|---|---|---|---|
| 1 | 265 | 225 | 40 |
| 2 | 150 | 150 | 0 |
| 3 | 137 | 150 | -13 |
| 4 | 159 | 125 | 34 |
| 5 | 145 | 125 | 20 |
Two sample inference
Hypothesis tests and intervals for comparing two population means
Two-sample inference
If \(x_1, \dots, x_{58}\) are the 1976 observations and \(y_1, \dots, y_{65}\) are the 1978 observations:
- \(\bar{x}\) is a point estimate for \(\mu_{1976}\) with standard error \(SE(\bar{x}) = \frac{s_x}{\sqrt{n}}\)
- \(\bar{y}\) is a point estimate for \(\mu_{1978}\) with standard error \(SE(\bar{y}) = \frac{s_y}{\sqrt{n}}\)
Inference uses a new \(T\) statistic:
\[ T = \frac{\bar{x} - \bar{y} - \delta_0}{SE(\bar{x} - \bar{y})} \]
- \(\delta_0\) is the hypothesized difference in means
- \(SE(\bar{x} - \bar{y}) = \sqrt{SE(\bar{x})^2 + SE(\bar{y})^2}\)
- \(t_\nu\) model approximates the sampling distribution when each sample meets assumptions for one-sample inference
Checking assumptions
The two-sample test is appropriate whenever two one-sample tests would be.
In other words, the test assumes that both samples are either:
- sufficiently large; or
- have little skew and few outliers
To check, simply inspect each histogram.
- both distributions unimodal
- both a bit left skewed
- no extreme outliers
- large sample sizes (58, 65)
Checking assumptions (alternative)
The two-sample test is appropriate whenever two one-sample tests would be.
In other words, the test assumes that both samples are either:
- sufficiently large; or
- have little skew and few outliers
Could also check side-by-side boxplots for:
- approximate symmetry of boxes
- outliers far from whiskers
This is also a nice visualization of differences between samples.
Body temperatures (again)
Does mean body temperature differ between men and women?
Test \(H_0: \mu_F = \mu_M\) against \(H_A: \mu_F \neq \mu_M\)
Welch Two Sample t-test
data: body.temp by sex
t = 1.7118, df = 34.329, p-value = 0.09595
alternative hypothesis: true difference in means between group female and group male is not equal to 0
95 percent confidence interval:
-0.09204497 1.07783444
sample estimates:
mean in group female mean in group male
98.65789 98.16500 Suggestive but insufficient evidence that mean body temperature differs by sex.
Notice: estimated difference (F - M) is 0.493 °F (SE 0.2879)
A statistical trap
If you collect enough data, you can detect an arbitrarily small difference in means almost always.
So keep in mind:
- statistical significance \(\neq\) practical significance
- always check your point estimates
The equal-variance \(t\)-test
If it is reasonable to assume the (population) standard deviations are the same in each group, one can gain a bit of power by using a different standard error:
\[SE_\text{pooled}(\bar{x} - \bar{y}) = \sqrt{\frac{\color{red}{s_p^2}}{n_x} + \frac{\color{red}{s_p^2}}{n_y}} \quad\text{where}\quad \color{red}{s_p} = \underbrace{\sqrt{\frac{(n_x - 1)s_x^2 + (n_y - 1)s_y^2}{n_x + n_y - 2}}}_{\text{weighted average of } s_x^2 \;\&\; s_y^2}\]
Implement by adding var.equal = T as an argument to t.test().
- larger df is used, hence more frequent rejections
- avoid unless you have a small sample
