Question 1171395: ""Untitled," by Stephen Chen
I've often wondered how software is released and sold to the public. Ironically, I work for a company that sells products with known problems. Unfortunately, most of the problems are difficult to create, which makes them difficult to fix. I usually use the test program X, which tests the product, to try to create a specific problem. When the test program is run to make an error occur, the likelihood of generating an error is 1%.
So, armed with this knowledge, I wrote a new test program Y that will generate the same error that test program X creates, but more often. To find out if my test program is better than the original, so that I can convince the management that I'm right, I ran my test program to find out how often I can generate the same error. When I ran my test program 50 times, I generated the error twice. While this may not seem much better, I think that I can convince the management to use my test program instead of the original test program. Am I right?
Conduct a hypothesis test at the 5% level.
Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)"
I need help stating the distribution to use for the test. (I must round my answers to four decimal places.)
format: P'~ N (____, ____)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this hypothesis test and determine the appropriate distribution.
**1. Define the Problem:**
* **Null Hypothesis (H0):** The new test program Y is not better than the original program X. The error rate of Y is the same as or less than 1% (0.01).
* **Alternative Hypothesis (H1):** The new test program Y is better than the original program X. The error rate of Y is greater than 1% (0.01).
* **Significance Level (α):** 5% or 0.05.
**2. Data:**
* **Original Program X:** Error rate (p0) = 0.01
* **New Program Y:**
* Number of trials (n) = 50
* Number of errors (x) = 2
* Sample proportion (p̂) = x/n = 2/50 = 0.04
**3. Choose the Appropriate Distribution:**
* Since we're dealing with proportions and a relatively large sample size (n=50), we can use the **normal distribution** to approximate the binomial distribution.
**4. Calculate the Standard Error:**
* The standard error (SE) for a proportion is calculated as:
* SE = √(p0(1 - p0) / n)
* SE = √(0.01(1 - 0.01) / 50)
* SE = √(0.01(0.99) / 50)
* SE = √(0.0099 / 50)
* SE = √0.000198
* SE ≈ 0.01407
**5. State the Distribution:**
* P' ~ N (p0, SE^2)
* P' ~ N (0.01, 0.01407^2)
* P' ~ N (0.01, 0.000198)
Therefore, the distribution to use for the test is:
P' ~ N (0.0100, 0.000198)
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