Question 1206624
## Understanding the Problem
We're tasked with analyzing a hypothesis testing scenario where:

* **Null Hypothesis (H₀):** The computer is good.
* **Alternative Hypothesis (H₁):** The computer is defective.

The decision rule is to reject H₀ (declare the computer defective) if it fails 3 or more tests.

## Part a: Calculating the p-value
The p-value is the probability of observing a test statistic as extreme or more extreme than the observed one, assuming the null hypothesis is true. 

In this case, the observed test statistic is 2 failed tests. So, we need to calculate the probability of 2 or more failed tests, given that the computer is good.

From the table:
* P(0 failures | Good) = 0.80
* P(1 failure | Good) = 0.12
* P(2 failures | Good) = 0.02

So, the p-value = P(2 or more failures | Good) = 0.02 + 0 = **0.02**.

## Part b: Type I Error
**Type I Error:** Rejecting the null hypothesis when it's actually true.

**Consequence for the company:** A good computer is incorrectly classified as defective and not shipped, leading to unnecessary losses.

**Probability of Type I Error:** This is the significance level, α. In this case, α = P(Reject H₀ | H₀ is true) = P(3 or more failures | Good) = 0.03 + 0.02 + 0 = **0.05**.

## Part c: Type II Error
**Type II Error:** Failing to reject the null hypothesis when it's false.

**Consequence for the company:** A defective computer is incorrectly classified as good and shipped, potentially leading to customer dissatisfaction and product liability issues.

**Probability of Type II Error (β):** This is more complex to calculate directly. It would involve summing the probabilities of failing 2 or fewer tests, given that the computer is defective. However, we can estimate it by analyzing the table and understanding that β is related to the power of the test (1-β). 

A higher power means a lower probability of Type II error. In this case, the power of the test is relatively high, as the probability of a defective computer failing 2 or fewer tests is quite low. 

## Part d: Evaluating the Decision Rule
The decision rule of rejecting H₀ for 3 or more failures seems reasonable. 

* **Low Type I Error:** The probability of incorrectly rejecting a good computer is relatively low (5%).
* **High Power:** The test has a high probability of correctly identifying defective computers, reducing the risk of Type II errors.

However, it's important to balance the risks of both types of errors. If the cost of a Type II error is significantly higher than a Type I error, a more stringent decision rule (e.g., rejecting for 2 or more failures) might be considered.

Ultimately, the optimal decision rule depends on the specific costs associated with each type of error and the desired level of risk.