Question

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 lower for the robots than for human assemblers. Let \(p\) denote the proportion of defective installations for the robots. It is known that human assemblers have a defect proportion of .02 . a. Which of the following pairs of hypotheses should the manufacturer test: \(H_{0}: p=.02\) versus \(H_{a}: p<.02\) or \(H_{0}: p=.02\) versus \(H_{a}: p>.02\) Explain your answer. b. In the context of this exercise, describe Type I and Type II errors. c. Would you prefer a test with \(\alpha=.01\) or \(\alpha=.1 ?\) Explain your reasoning.

Short answer

a. The correct pair of hypotheses to test is \(H_{0}: p=0.02\) versus \(H_{a}: p<0.02\) \n b. A Type I error would imply a false positive: concluding robots reduce defects when they do not. A Type II error is a false negative: wrongly concluding that robots do not reduce defects when they do. \n c. A test with \(\alpha=0.01\) is preferred to reduce the chance of a costly Type I error.

Step-by-step solution

Select Correct Pair of Hypotheses

The manufacturer wants to prove that defective installations are lower when using robots than humans. This implies they want to prove that the proportion \(p\) for defective installations for robots is potentially less than 0.02. Hence, the correct pair of hypotheses to test are: \(H_{0}: p=0.02\) versus \(H_{a}: p<0.02\)

Describe Type I and II errors

In the context of this exercise: \n- A Type I error would occur if the manufacturer rejects the null hypothesis \(H_{0}: p=0.02\) when it is true, meaning that they would incorrectly conclude that the robots have a lower defective installation rate when they do not. This could lead to unnecessary investment in robots. \n - A Type II error would occur if the manufacturer fails to reject the null hypothesis when the alternative hypothesis \(H_{a}: p<0.02\) is true. It indicates incorrectly inferring that robots do not have a lower defective installation rate when they do, potentially forgoing an opportunity to reduce defects.

Selecting Alpha

In this context, it is crucial to avoid a Type I error because, if incorrectly concluded, it can lead to large unnecessary expenditures, i.e., investing in robots when they don’t actually reduce defects. Therefore, it is preferred to have a test with a lower Type I error rate (\(\alpha\)), making the test with \(\alpha=0.01\) more desirable than \(\alpha=0.1\). This crafts a more stringent test, reducing the chance of a Type I error.

Key concepts

Null Hypothesis

When conducting hypothesis testing in statistics, the null hypothesis () serves as the default position or baseline that asserts no effect or no difference in the case study. It's a statement that we aim to test against the alternative hypothesis. For instance, in the automobile manufacturer exercise, the null hypothesis would be that the proportion of defective installations for robots, denoted as \(p\), is equal to the standard defect proportion of human assemblers, which is 0.02. So, the null hypothesis is established as \(H_{0}: p=0.02\).

By setting this as a starting point, we can use statistical methods to determine if the evidence is strong enough to reject this hypothesis, which would suggest that using robots indeed makes a significant difference. Having a clearly defined null hypothesis is crucial as it provides a concrete claim that can be challenged by the test statistics.

Alternative Hypothesis

Complementary to the null hypothesis, the alternative hypothesis () represents what we suspect or wish to prove is true. It's a statement that contradicts the null hypothesis and is denoted as \(H_{a}\) or \(H_{1}\). In our automobile manufacturing example, the alternative hypothesis posits that robots have a lower defective installation rate than human assemblers. Hence, the alternative hypothesis would be \(H_{a}: p<0.02\).

The choice of the alternative hypothesis is guided by the specific aim of the study or experiment. If the actual proportion of defects is indeed less than 0.02 for the robots, then we have enough evidence to reject the null hypothesis in favor of the alternative. In this sense, the alternative hypothesis articulates the manufacturer's goal in the testing process.

Type I and Type II Errors

In hypothesis testing, errors of inference can occur, and they are categorized as Type I and Type II errors. Type I error, often symbolized as , is the incorrect rejection of a true null hypothesis, also known as a 'false positive'. In the automobile manufacturer's context, this would mean deciding that the robots produce less defective installations than the humans when they do not. This would have financial implications such as unwarranted investment in robotics.

Conversely, Type II error, represented as , occurs when the null hypothesis is not rejected even though the alternative hypothesis is true; this is known as a 'false negative'. In our example, it would mean missing out on the chance to reduce defects because we didn't recognize the robots' actual effectiveness. Balancing these potential errors is a central challenge in designing an experiment, as they can have substantial practical consequences.

Significance Level ()

The significance level, also known as , plays a pivotal role in hypothesis testing. It's the threshold for deciding when to reject the null hypothesis. The lower the value of , the stronger the evidence must be to reject the null hypothesis. A common significance level is 0.05, but it can be set more stringently, like 0.01, or less stringarily, such as 0.1, depending on the situation's sensitivity to Type I errors. The exercise questions whether an of 0.01 or 0.1 is preferable. Since Type I errors can lead to significant unnecessary costs in this scenario, a more stringent level of 0.01 would be wiser to use. It reduces the risk of a costly false positive, thus, making the hypothesis test more robust against committing a Type I error.