An automobile manufacturer is considering using robots for part of its assembly process. Converting to robots is an expensive process, so it will be undertaken only if there is strong evidence that the proportion of defective installations is less for the robots than for human assemblers. Let p denote the actual proportion of defective installations for the robots. It is known that the proportion of defective installations for human assemblers is 0.02.
(a) Which of the following pairs of hypotheses should the manufacturer test?
$H_0: p = 0.02$ versus $H_a: p < 0.02$
or
$H_0: p = 0.02$ versus $H_a: p > 0.02$
Explain your answer.
$H_0: p = 0.02$ versus $H_a: p > 0.02$, as the conversion will only happen if the proportion of defective installations is higher for robots.
$H_0: p = 0.02$ versus $H_a: p < 0.02$, as the conversion will only happen if the proportion of defective installations is higher for robots.
$H_0: p = 0.02$ versus $H_a: p > 0.02$, as the conversion will only happen if the proportion of defective installations is lower for human assemblers.
$H_0: p = 0.02$ versus $H_a: p < 0.02$, as the conversion will only happen if the proportion of defective installations is lower for robots.
(b) In the context of this exercise, describe Type I and Type II errors. (Select all that apply.)
A Type II error would be obtaining convincing evidence that the proportion of defective installations for the robots is less than 0.02 when in fact it is (at least) 0.02.
A Type II error would be not obtaining convincing evidence that the proportion of defective installations for the robots is less than 0.02 when in fact it is less than 0.02.
A Type I error would be obtaining convincing evidence that the proportion of defective installations for the robots is less than 0.02 when in fact it is (at least) 0.02.
A Type I error would be not obtaining convincing evidence that the proportion of defective installations for the robots is less than 0.02 when in fact it is less than 0.02.
(c) Would you prefer a test with $\alpha = 0.01$ or $\alpha = 0.1$? Explain your reasoning.
0.01, as a Type II error results in people losing their jobs to a robot system that doesn't improve accuracy.
0.01, as a Type I error results in people losing their jobs to a robot system that doesn't improve accuracy.
0.1, as a Type II error results in people losing their jobs to a robot system that doesn't improve accuracy.
0.1, as a Type I error results in people losing their jobs to a robot system that doesn't improve accuracy.
