Answer:
(a) The correct option is: <em>H₀</em>: <em>p</em> = 0.02 vs. <em>Hₐ</em>: <em>p</em> < 0.02.
(b) Explained below.
(c) The better value of <em>α</em> will be 0.10.
Step-by-step explanation:
An automobile manufacturer is considering using robots for part of its assembly process only if there is strong evidence that the proportion of defective installations is less for the robots than for human assemblers.
To test whether the proportion of defective installations is less for the robots than for human assemblers use a single-proportion <em>z</em>-test.
(a)
The hypothesis can be defined as:
<em>H₀</em>: The proportion of defective installations is same for both the robots and human assemblers, i.e. <em>p</em> = 0.02.
<em>Hₐ</em>: The proportion of defective installations is less for the robots than for human assemblers, i.e. <em>p</em> < 0.02.
The alternate hypothesis is the claim or the statement that is being tested.
In this case we need to test whether the proportion of defective installations is less for the robots than for human assemblers or not, so that the manufacturer can decide whether they want to apply the conversion.
Thus, the correct option is:
<em>H₀</em>: <em>p</em> = 0.02 vs. <em>Hₐ</em>: <em>p</em> < 0.02.
(b)
A type I error occurs when we discard a true null hypothesis and a type II error is made when we fail to discard a false null hypothesis.
In this case a type I error will be committed if conclude that the proportion of defective installations is less for the robots than for human assemblers when in fact it is not.
And a type II error will be committed if we fail to conclude that proportion of defective installations is less for the robots than for human assemblers.
(c)
The power of the test is the probability of rejecting a false null hypothesis.
The power of the test sis affected by the significance level of the test (<em>α</em>).
Lesser the significance level of the test the lesser is the power of the test.
If the value of <em>α</em> is reduced from 0.05 to 0.01 then the region of acceptance will increase. This implies that there is low probability of rejecting the null hypothesis even when it is false.
So higher the value of <em>α</em> the higher is the probability of making a correct decision.
Thus, the better value of <em>α</em> will be 0.10.
