A production line operation is tested for filling weight...

A production line operation is tested for filling weight accuracy using the following hypotheses. $\begin{array}{ll}\text { Hypothesis } & \text { Conclusion and Action } \\ H_0: \mu=16 & \text { Filling okay; keep running } \\ H_{\mathrm{a}}: \mu \neq 16 & \text { Filling off standard; stop and adjust machine }\end{array}$ The sample size is 30 and the population standard deviation is $\sigma=.8$. Use $\alpha=.05$. a. What would a Type II error mean in this situation? b. What is the probability of making a Type II error when the machine is overfilling by 5 ounces? c. What is the power of the statistical test when the machine is overfilling by .5 ounces? d. Show the power curve for this hypochesis test. What information does it contain for the production manager?

A production line operation is tested for filling weight accuracy using the following hypotheses.
$\begin{array}{ll}\text { Hypothesis } & \text { Conclusion and Action } \\ H_0: \mu=16 & \text { Filling okay; keep running } \\ H_{\mathrm{a}}: \mu \neq 16 & \text { Filling off standard; stop and adjust machine }\end{array}$
The sample size is 30 and the population standard deviation is $\sigma=.8$. Use $\alpha=.05$.
a. What would a Type II error mean in this situation?
b. What is the probability of making a Type II error when the machine is overfilling by 5 ounces?
c. What is the power of the statistical test when the machine is overfilling by .5 ounces?
d. Show the power curve for this hypochesis test. What information does it contain for the production manager?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method that involves making a decision about a population parameter based on sample data. It starts by assuming a null hypothesis (H0) and comparing it with an alternative hypothesis (Ha) to determine whether there is enough evidence to reject H0 in favor of Ha. This process is fundamental in quality control and other decision-making contexts where uncertainty needs to be quantified.

Null and Alternative Hypotheses

The null hypothesis (H0) represents the status quo or a default claim (for example, that the production line is filling correctly), while the alternative hypothesis (Ha) represents a deviation from that state. Understanding these hypotheses is crucial since the entire testing procedure, including error types and power calculations, relies on the context provided by H0 and Ha.

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected when it is actually true. This error is often controlled by setting a significance level (alpha), which is the probability of making such an error. In quality control settings, a Type I error might mean stopping a production line that is actually operating as intended.

Type II Error

A Type II error happens when the null hypothesis is not rejected even though the alternative hypothesis is true. In other words, it's a failure to detect a real change or problem – for instance, not adjusting a machine that is overfilling. Recognizing and quantifying this error is critical for understanding the sensitivity of the test.

Significance Level

The significance level, often denoted by alpha, is a threshold set by the researcher to determine when to reject the null hypothesis. It represents the probability of committing a Type I error. A lower significance level means less risk of falsely rejecting H0, but it may increase the likelihood of a Type II error.

Power of a Test

The power of a statistical test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. Essentially, it is calculated as 1 minus the probability of a Type II error. In practical terms, high power means that the test is sensitive to actual changes, such as a deviation in the filling weight, and is less likely to miss a real issue.

Z-Test

A Z-test is a statistical test used when the population standard deviation is known and the sample size is sufficiently large. It involves comparing the test statistic, calculated from the sample mean, to the standard normal distribution. This test is widely used in quality control to assess whether a production process is operating at its intended specifications.

Power Curve

A power curve is a graphical representation that shows the power of a test across a range of true parameter values. It provides valuable information to decision makers, such as production managers, by illustrating how likely it is that the test will detect deviations from the nominal process value under different conditions. This tool aids in balancing the risk of Type II errors against the sensitivity required in the testing process.

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