Question

A manufacturer of plumbing fixtures has developed a new type of washerless faucet. Let p = P(a randomly selected faucet of this type will develop a leak within 2 years under normal use). The manufacturer has decided to proceed with production unless it can be determined that p is too large; the borderline acceptable value of p is specified as .10. The manufacturer decides to subject n of these faucets to accelerated testing (approximating 2 years of normal use). With X = the number among the n faucets that leak before the test concludes, production will commence unless the observed X is too large. It is decided that if p = .10 , the probability of not proceeding should be at most .10, whereas if p = .30 the probability of proceeding should be at most .10. Can n = 10 be used? n = 20?n = 25 ? What is the appropriate rejection region for the chosen n, and what are the actual error probabilities when this region is used?

Answer 1

To decide on using n=10, n=20, or n=25 for faucet testing, we must evaluate the binomial probabilities for Type I and II errors with a threshold of 0.10. We seek an n such that np and nq are both greater than 5, allowing us to approximate the binomial with a normal distribution and accurately determine the rejection region.

Step-by-step explanation:

To determine whether to use n = 10, n = 20, or n = 25 faucets in the test, we must consider the Binomial distribution. This considers two probabilities: a probability of false positive (Type I error) when p is actually 0.10 and a probability of false negative (Type II error) when p is actually 0.30. The acceptable threshold for each error is 0.10. By calculating the binomial probabilities for different values of n and determining the X that would be too large to commence production, we can find the appropriate sample size and the rejection region.

For the Binomial distribution, we use the formula P(X) with the mean μ = np and standard deviation σ = √npq, where q = 1 - p. For the case of p = 0.10 and n = 10, np > 5 and nq > 5, so the normal approximation could be used. However, with n = 10, it is highly likely that both types of errors would exceed the acceptable threshold of 0.10. Increasing n to 20 or 25 raises np and nq numbers above 5, which is preferable for a more accurate approximation to the normal distribution.

Calculations must be performed for each n to determine the X value at which production would stop (the rejection region). Software or a graphing calculator can be used to calculate exact binomial probabilities, or one could use the normal approximation to the binomial distribution once np and nq are both sufficiently large.]

Answer 2

The manufacturer can use n = 20 for the accelerated testing, but n = 10 and n = 25 are not suitable. The appropriate rejection region for n = 20 is to reject the null hypothesis if X (the number of faucets that leak) is greater than or equal to 4 or less than or equal to 1.

Step-by-step explanation:

To determine if n = 20 is appropriate, we can use the binomial distribution to find the critical values. The null hypothesis is that p = 0.10, and the manufacturer wants the probability of not proceeding to be at most 0.10 when p = 0.10. Using the binomial distribution, we find that the critical values for n = 20 are X ≥ 4 or X ≤ 1.

The error probabilities can be calculated by finding the probability of observing X values in the critical region under the null hypothesis. For the upper critical value (X ≥ 4), we calculate P(X ≥ 4) under the assumption that p = 0.10. Similarly, for the lower critical value (X ≤ 1), we calculate P(X ≤ 1) under the assumption that p = 0.10. The actual error probabilities can then be compared to the specified maximum probability of 0.10 to ensure that the test meets the manufacturer's criteria.