When a thread-cutting machine is operating properly, only 2...

When a thread-cutting machine is operating properly, only $2 \%$ of the units produced are defective. Since the machine was bumped by a forklift truck, however, the quality-control manager is concerned that it may require expensive downtime and a readjustment. The manager would like to set up a right-tail test at the 0.01 level of significance, then identify the proportion of defectives in a sample of 400 units produced. His null hypothesis is $H_0: \pi \leq 0.02$, with $H_1: \pi>0.02$. a. What critical value of $z$ will be associated with this test? b. For the critical $z$ determined in part (a), identify the sample proportion that this $z$ value represents, then use this value of $p$ in stating a decision rule for the test. c. What is the probability that the quality-control manager will fail to reject a false $H_0$ if the actual population proportion of defectives is $\pi=0.02$ ? If $\pi=0.03$ ? If $\pi=0.04$ ? If $\pi=0.05$ ? If $\pi=0.06$ ? d. Making use of the probabilities calculated in part (c), plot the power and operating characteristic curves for the test.

When a thread-cutting machine is operating properly, only $2 \%$ of the units produced are defective. Since the machine was bumped by a forklift truck, however, the quality-control manager is concerned that it may require expensive downtime and a readjustment. The manager would like to set up a right-tail test at the 0.01 level of significance, then identify the proportion of defectives in a sample of 400 units produced. His null hypothesis is $H_0: \pi \leq 0.02$, with $H_1: \pi>0.02$.
a. What critical value of $z$ will be associated with this test?
b. For the critical $z$ determined in part (a), identify the sample proportion that this $z$ value represents, then use this value of $p$ in stating a decision rule for the test.
c. What is the probability that the quality-control manager will fail to reject a false $H_0$ if the actual population proportion of defectives is $\pi=0.02$ ? If $\pi=0.03$ ? If $\pi=0.04$ ? If $\pi=0.05$ ? If $\pi=0.06$ ?
d. Making use of the probabilities calculated in part (c), plot the power and operating characteristic curves for the test.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves setting up a null hypothesis (a statement of no effect or status quo) and an alternative hypothesis (representing the theory or effect to be tested), and using sample data to determine whether to reject the null hypothesis in favor of the alternative.

Decision Rule

The decision rule in hypothesis testing is a clear statement about when to reject the null hypothesis. It typically involves comparing the test statistic, such as a z-score, to the critical value: if the test statistic exceeds the critical value, the null hypothesis is rejected, and if it does not, the null hypothesis is not rejected.

Sampling Distribution and Standard Error

The sampling distribution of the sample proportion approximates a normal distribution for large samples, with a mean equal to the true population proportion and a standard error calculated using the formula ?[?(1-?)/n]. This concept allows one to quantify the variability expected in the sample proportion from one sample to the next.

Critical Value

The critical value is a point on the test distribution beyond which the null hypothesis is rejected at the chosen significance level. In a right-tail test, this is the z-score that corresponds to (1 - ?) of the cumulative probability. It defines the boundary of the rejection region for the test.

Operating Characteristic Curve

The operating characteristic curve (OC curve) shows the probability of not rejecting the null hypothesis (i.e., the probability of a Type II error) for various true values of the parameter. It provides a graphical illustration of the test’s performance, showing how the test is likely to behave under different scenarios of the actual population parameter.

Significance Level

The significance level, denoted by ?, is the probability threshold used to determine when to reject the null hypothesis. It represents the risk of making a Type I error, which is rejecting the null hypothesis when it is in fact true. In this context, a 0.01 level of significance means there is a 1% risk of incorrectly rejecting a true null hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement that there is no effect or that a parameter is less than or equal to a specified value, while the alternative hypothesis (H?) posits that the parameter is greater than the specified value. These hypotheses form the basis for deciding if observed data provide sufficient evidence to challenge the accepted baseline assumption.

Right-tail Test

A right-tail test is used when the alternative hypothesis suggests an increase or a value greater than a specified threshold. In this case, the rejection region for the test is located on the right side of the probability distribution, meaning that extreme values on the higher end of the scale lead to the rejection of the null hypothesis.

Type II Error and Power of a Test

A Type II error occurs when the test fails to reject a false null hypothesis. The power of a test, which is 1 minus the probability of a Type II error, indicates the probability that the test will correctly reject a false null hypothesis. Calculating power at various alternative values of the parameter helps in understanding the test's sensitivity to detect deviations from the null hypothesis.

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