Question 7.3: Refer to Example 7.2. Can you conclude that the mean diamete...
Refer to Example 7.2. Can you conclude that the mean diameter for carbon dioxide welds (μ_Y ) exceeds that for argon welds (μ_X) by more than 0.015 μm?
The null and alternate hypotheses are
H_0: \mu_X-\mu_Y \geq-0.015\quad \quad \text{versus}\quad \quad H_1: \mu_X-\mu_Y<-0.015
We observe \bar{X}=0.37, \bar{Y}=0.40, s_X=0.25, s_Y=0.26, n_X=544, and n_Y=581. Under H_0, we take \mu_X-\mu_Y=-0.015. The null distribution of \bar{X}-\bar{Y} is given by expression (7.3) to be
\bar{X}-\bar{Y}\sim N\left(\Delta_0, \frac{\sigma_X^2}{n_X}+\frac{\sigma_Y^2}{n_Y}\right) (7.3)
\bar{X}-\bar{Y}\sim N\left(-0.015, \ 0.01521^2\right)
We observe \bar{X}-\bar{Y}=0.37-0.40=-0.03. The z-score is
z=\frac{-0.03-(-0.015)}{0.01521}=-0.99
This is a one-tailed test. The P-value is 0.1611. We cannot conclude that the mean diameter of inclusions from carbon dioxide welds exceeds that of argon welds by more than 0.015 μm.