A production line operates with a mean filling weight of 16...

A production line operates with a mean filling weight of 16 ounces per container. Overfilling or underfilling presents a serious problem and when detected requires the operator to shut down the production line to readjust the filling mechanism. From past data, a population standard deviation $\sigma=.8$ ounces is assumed. A quality control inspector selects a sample of 30 items every hour and at that time makes the decision of whether to shut down the line for readjustment. The level of significance is $a=.05$. a. State the hypothesis test for this quality control application. b. If a sample mean of $\bar{x}=16.32$ ounces were found, what is the $p$-value? What action would you recommend? c. If a sample mean of $\bar{x}=15.82$ ounces were found, what is the p-value? What action would you recommend? d. Use the critical value approach. What is the rejection rule for the preceding bypothesis vesting procedure? Repeat parts (b) and (c). Do you reach the same conclusion?

A production line operates with a mean filling weight of 16 ounces per container. Overfilling or underfilling presents a serious problem and when detected requires the operator to shut down the production line to readjust the filling mechanism. From past data, a population standard deviation $\sigma=.8$ ounces is assumed. A quality control inspector selects a sample of 30 items every hour and at that time makes the decision of whether to shut down the line for readjustment. The level of significance is $a=.05$.
a. State the hypothesis test for this quality control application.
b. If a sample mean of $\bar{x}=16.32$ ounces were found, what is the $p$-value? What action would you recommend?
c. If a sample mean of $\bar{x}=15.82$ ounces were found, what is the p-value? What action would you recommend?
d. Use the critical value approach. What is the rejection rule for the preceding bypothesis vesting procedure? Repeat parts (b) and (c). Do you reach the same conclusion?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In quality control contexts, the null hypothesis often represents the process being in control (e.g., the mean is equal to the target), and the alternative suggests that some deviation exists that might warrant corrective action.

p-value Interpretation

The p-value represents the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. It is used as a measure of the evidence against the null hypothesis; a small p-value indicates strong evidence against the null hypothesis, leading to its rejection in favor of the alternative hypothesis.

Critical Value Approach

The critical value approach involves determining cutoff values (or critical values) from the sampling distribution that correspond to the chosen significance level. If the test statistic exceeds these critical values, the null hypothesis is rejected. This approach provides a clear decision rule that relates the observed test statistic to the rejection region, ensuring consistency in decision making.

Decision Rules in Hypothesis Testing

Decision rules in hypothesis testing are based on comparing the test statistic with either the critical values or the p-value with the significance level. For example, if the p-value is less than the significance level (commonly 0.05), or if the test statistic falls in the rejection region defined by the critical values, the decision is to reject the null hypothesis. This systematic approach helps maintain the error rate set by the level of significance.

Z-Test for Population Mean

A Z-test for the population mean is used when the standard deviation of the population is known and the sample comes from a normally distributed population or the sample size is large enough. In this scenario, the test statistic is computed using the formula z = (sample mean - hypothesized mean) / (?/?n), which assesses how far the sample mean is from the target mean in terms of standard errors.

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