We have data \(X_1,..,X_n\sim N(\mu_x,\sigma)\) and \(Y_1,..,Y_m\sim N(\mu_y,\sigma)\), \(\sigma\) unknown. (This is the so called two sample problem). Say we want to test
\[H_0:\mu_x=\mu_y\text{ vs. }H_0:\mu_x\ne\mu_y\]
The classical test is based on the test statistic
\[T=\frac{\bar{x}-\bar{y}}{s_p\sqrt{1/n+1/m}}\] where \(s_p^2=\frac{(n-1)s_x^2+(m-1)s_y^2}{n+m-2}\) is called the pooled standard deviation. Under the null hypothesis \(T\sim t(n+m-2)\) and the the test rejects the null hypothesis if \(|T|>qt_{\alpha/2,n+m-2}\).
Derive the two sample problem as a special case of ANOVA.
In an experiment an industrial engineer studied the effect of the type of coating and its thickness on the durability of a certain type of paint. There were three types of coatings labeled 1, 2 and 3, and three thickness levels labeled thin, medium and thick. The durability was measured in days. The data is
| thin | medium | thick | |
|---|---|---|---|
| 1 | 166, 154, 155, 156, 149 | 167, 171, 166, 165, 185 | 181, 185, 178, 178, 174 |
| 2 | 219, 241, 216, 220, 220 | 263, 241, 246, 245, 224 | 242, 258, 257, 242, 250 |
| 3 | 277, 276, 277, 278, 280 | 309, 281, 309, 302, 314 | 350, 348, 359, 340, 342 |
Analyze this data to see what effect(s) if any the type of coating and the thickness have on the durability. Specifically, which factor-level combination(s) is/are statistically significantly the best?