You manufacture and sell a liquid product whose electrical...

You manufacture and sell a liquid product whose electrical conductivity is supposed to be $5 .$ You plan to make six measurements of the conductivity of each lot of product. If the product meets specifications, the mean of many measurements will be $5 .$ You will therefore test $$ \begin{array}{l} H_{0}: \mu=5 \\ H_{a}: \mu \neq 5 \end{array} $$ If the true conductivity is $5.1,$ the liquid is not suitable for its intended use. You learn that the power of your test at the $5 \%$ significance level against the alternative $\mu=5.1$ is 0.23. (a) Explain in simple language what "power $=0.23 "$ means in this setting. (b) You could get higher power against the same alternative with the same $\alpha$ by changing the number of measurements you make. Should you make more measurements or fewer to increase power? (c) If you decide to use $\alpha=0.10$ in place of $\alpha=0.05$, with no other changes in the test, will the power increase or decrease? Justify your answer. (d) If you shift your interest to the alternative $\mu=5.2$, with no other changes, will the power increase or decrease? Justify your answer.

You manufacture and sell a liquid product whose electrical conductivity is supposed to be $5 .$ You plan to make six measurements of the conductivity of each lot of product. If the product meets specifications, the mean of many measurements will be $5 .$ You will therefore test
$$
\begin{array}{l}
H_{0}: \mu=5 \\
H_{a}: \mu \neq 5
\end{array}
$$
If the true conductivity is $5.1,$ the liquid is not suitable for its intended use. You learn that the power of your test at the $5 \%$ significance level against the alternative $\mu=5.1$ is 0.23.
(a) Explain in simple language what "power $=0.23 "$ means in this setting.
(b) You could get higher power against the same alternative with the same $\alpha$ by changing the number of measurements you make. Should you make more measurements or fewer to increase power?
(c) If you decide to use $\alpha=0.10$ in place of $\alpha=0.05$, with no other changes in the test, will the power increase or decrease? Justify your answer.
(d) If you shift your interest to the alternative $\mu=5.2$, with no other changes, will the power increase or decrease? Justify your answer.

Key Concepts

Statistical Power

Statistical power is the probability that a test will correctly reject a false null hypothesis. It reflects the test's ability to detect an effect or difference when one truly exists, thus helping to avoid Type II errors. A power of 0.23 means that, when the alternative hypothesis is true, the test will only detect the effect 23% of the time.

Significance Level (?)

The significance level is a predetermined threshold set by the researcher that defines the maximum acceptable probability of making a Type I error, which is rejecting a true null hypothesis. It informs the critical region of the test statistic distribution, and changing ? (for example from 0.05 to 0.10) alters the balance between Type I and Type II errors.

Sample Size

The sample size is the number of observations or measurements taken. It plays a crucial role in determining the power of a test since larger sample sizes reduce the standard error of the estimate, making it easier to detect a true effect. Therefore, increasing the number of measurements typically increases the test’s power.

Alternative Hypothesis and Effect Size

The alternative hypothesis represents the postulated situation where there is a true difference from the null scenario. Effect size indicates how much the true parameter deviates from the null value. A larger deviation (or effect size) from the null hypothesis typically makes it easier for a test to detect the difference, thereby increasing the power.

Type I and Type II Errors

Type I error is the mistake of rejecting a true null hypothesis, controlled by the significance level (?), while Type II error is the failure to reject a false null hypothesis. The balance between these errors is central to the design of hypothesis tests, and improving power means reducing the probability of committing a Type II error when the alternative hypothesis is true.

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