Question 1206624: A computer company checks every computer it produces before shipping. Each computer undergoes a series of 100 tests. The number of tests it fails will be used to determine whether the computer is Good or Defective. If it fails more than a certain number, it will be classified as Defective and will not be shipped. From past history, the distribution of the number of tests failed is known for both Good and Defective computers. The probabilities associated with each outcome are listed in the table below:
NUMBER OF TEST FAILED 0 1 2 3 4 5 MORE THAN 5
GOOD % 80 12 2 3 2 1 0
DEFECTIVE% 0 10 70 5 4 1 10
The table indicates, for example, that 80% of Good computers will have exactly 0 failures in 100 tests, while 70% of Defective computers have exactly 2 failures in 100 tests.
This is a hypothesis-testing situation.
Null hypothesis: computer is Good
Alternative hypothesis: computer is Defective.
The computer will be declared Defective (reject null hypothesis) if it fails at least 3 tests (3 or more).
Part a)
Suppose we test a computer and it fails 2 tests. What is the associated p-value?
Part b)
In this example what would a type I error be?
What is a consequence for the company if they make a type I error?
What is the probability of a Type I error?
Part c)
In this example what would a type II error be?
What is a consequence for the company if they make a type II error?
What is the probability of a Type II error?
Part d)
Do you think that the standard of declaring a computer Defective if it fails at least 3 tests (3 or more) is reasonable? Use the risk of type I and type II error to justify your opinion.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! ## 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.
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