Question

A factory produces smoke alarms. For an extended period, faults have been found in only one in every thousand smoke alarms produced. However, the factory's machinery has recently been overhauled, and the management has decided to test the product. A random sample of smoke alarms will be selected and tested, and if there is convincing evidence that the proportion of alarms that are defective is greater than 0.001, the machinery will be inspected in order to find out what is causing the greater proportion of defective alarms. Let p be the true proportion of defective alarms.

(a) Write the null and alternative hypotheses that will be used for the hypothesis test.

(b) Describe a Type I error in this context. What would be the consequences of a Type I error?

(c) Describe a Type II error in this context. What would be the consequences of a Type II error?

(d) A decision has to be made as to whether a 1% or 5% significance level will be used for the hypothesis test. Which of these significance levels will produce the smaller probability of a Type I error? Which will produce the smaller probability of a Type II error? Which significance level would you suggest be used? Explain your answer.

(e) What change would have to be made in the management's procedure in order to reduce the probability of a Type II error without affecting the probability of a Type I error? -0000,0

(f) Consider the following two scenarios: (A) The actual value of p is 0.002. (B) The actual value of p is 0.008. 2500.0- For which of these two scenarios is a Type II error less likely?

Answer 1

(d) The significance level that will produce the smaller probability of a Type I error=1%

The significance levels will produce the smaller probability of a Type II error=5%

Since here Type I error is more risky than Type II error so we use 1% significance level.

(e) Increase the probability of rejecting null hypothesis when it is false i.e. power of a test.

(f) Option B (since p for Option A is closed to  0.001 than Option B).