Suppose that a manufacturing process makes about 3 %...

Suppose that a manufacturing process makes about $3 \%$ defective items, which is considered satisfactory for this particular product. The managers would like to decrease this to about $1 \%$ and clearly want to guard against a substantial increase, say to $5 \%$. To monitor the process, periodically $n=100$ items are taken and the number $X$ of defectives counted. Assume that $X$ is $b(n=100, p=\theta)$. Based on a sequence $X_{1}, X_{2}, \ldots, X_{m}, \ldots$, determine a sequential probability ratio test that tests $H_{0}: \theta=0.01$ against $H_{1}: \theta=0.05 .$ (Note that $\theta=0.03$, the present level, is in between these two values.) Write this test in the form $$ h_{0}>\sum_{i=1}^{m}\left(x_{i}-n d\right)>h_{1} $$ and determine $d, h_{0}$, and $h_{1}$ if $\alpha_{a}=\beta_{a}=0.02$.

Suppose that a manufacturing process makes about $3 \%$ defective items, which is considered satisfactory for this particular product. The managers would like to decrease this to about $1 \%$ and clearly want to guard against a substantial increase, say to $5 \%$. To monitor the process, periodically $n=100$ items are taken and the number $X$ of defectives counted. Assume that $X$ is $b(n=100, p=\theta)$. Based on a sequence $X_{1}, X_{2}, \ldots, X_{m}, \ldots$, determine a sequential probability ratio test that tests $H_{0}: \theta=0.01$ against $H_{1}: \theta=0.05 .$ (Note that $\theta=0.03$, the present level, is in between these two values.) Write this test in the form
$$
h_{0}>\sum_{i=1}^{m}\left(x_{i}-n d\right)>h_{1}
$$
and determine $d, h_{0}$, and $h_{1}$ if $\alpha_{a}=\beta_{a}=0.02$.

Key Concepts

Decision Boundaries and Test Statistic Transformation

For ease of evaluation in sequential testing, the log-likelihood ratio is often rewritten in a form that transforms it into a sum of functions of the individual sample observations. This permits the SPRT decision rule to be expressed as comparing a cumulative statistic (for example, the sum of deviations from a specific benchmark) against two thresholds, h0 and h1. These boundaries are determined based on the acceptable levels of Type I and Type II errors and are used to make stopping decisions during the sequential test.

Binomial Distribution in Quality Control

In quality control, the number of defective items in a batch is modeled using the binomial distribution, which describes the number of successes in a fixed number of independent trials, each with a constant probability of success. Here, each item’s classification as defective or nondefective is analogous to a Bernoulli trial. This distribution provides the framework for calculating likelihoods under different defect probabilities.

Sequential Probability Ratio Test (SPRT)

The SPRT is a sequential analysis technique developed for testing simple hypotheses. Unlike fixed-sample tests, SPRT evaluates data as it becomes available and decides whether to accept one hypothesis or the other at each stage. The decision is based on comparing a cumulative test statistic—typically the log-likelihood ratio—to predetermined decision boundaries that control the Type I and Type II error probabilities.

Likelihood Ratio

The likelihood ratio is a fundamental concept in hypothesis testing that measures the relative support that the observed data provide for one hypothesis over another. In the context of the SPRT, the ratio of the likelihood functions under the two competing hypotheses is computed sequentially. The logarithm of this ratio is summed over observations to form the test statistic, which is then compared to the critical values to decide between the hypotheses.

Type I and Type II Errors

These error types are critical in setting up the operating characteristics of a hypothesis test. A Type I error (alpha) is the probability of incorrectly rejecting the null hypothesis, while a Type II error (beta) is the probability of failing to reject a false null hypothesis. In sequential testing, both error probabilities are controlled by appropriate choice of the decision boundaries, ensuring that the test meets the predefined performance criteria.

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