What is power? You manufacture and sell a liquid product...

What is power? You manufacture and sell a liquid product whose electrical conductivity is supposed to be 5. You plan to make 6 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 $=0.23^{\prime \prime}$ 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.

What is power? You manufacture and sell a liquid product whose electrical conductivity is supposed to be 5. You plan to make 6 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 $=0.23^{\prime \prime}$ 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

Effect Size

Effect size refers to the magnitude of the difference between the true parameter value and the value specified in the null hypothesis. A larger effect size generally makes it easier to detect a significant difference, resulting in a higher power for the test.

Sample Size

Sample size is a crucial factor that influences the power of a test. Larger samples tend to reduce the standard error, making it easier to detect differences from the hypothesized value, thereby increasing the statistical power. Conversely, smaller sample sizes often lead to lower power.

Type II Error and Its Relationship to Power

A Type II error occurs when a test fails to reject a false null hypothesis. The probability of making a Type II error is denoted by ?, and power is calculated as 1 - ?. Thus, minimizing ? increases power, improving the test's effectiveness in identifying true effects.

Statistical Hypothesis Testing

This is the framework in which we make decisions about population parameters based on sample data. It involves formulating a null hypothesis (the status quo) and an alternative hypothesis (what we suspect may be true instead), and then using a test statistic and a predefined significance level to determine if there is enough evidence to reject the null hypothesis.

Power of a Test

Power is the probability that the test correctly rejects a false null hypothesis. In other words, it is the test's ability to detect a true effect when one exists. A high power means there's a high likelihood of identifying a difference if the true parameter deviates from the value stated in the null hypothesis.

Significance Level (?) and Type I Error

The significance level, denoted by ?, is the threshold for rejecting the null hypothesis. It represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. Adjusting ? changes the tradeoff between being too conservative or too lenient in detecting effects.

Transcript

00:02 We know that power is the probability of making a correct decision to the correct decision of rejecting the null hypothesis when the null hypothesis is false.

00:12 So if we're given that the null hypothesis has an average of five, the alternative is that is not five.

00:20 We know that the alpha is 0 .05 and we're going against this idea that if we have a 5 .1, the liquid is not suitable for, its intended use.

00:33 And so we're told that the power is 0 .23.

00:37 What this really means is that if the true mean electrical conductivity for a liquid produced was 5 .1 instead of our null hypothesis of 5.

00:49 There is a 0 .23 probability that the sample will provide convincing evidence that the true mean is different from 5, which is our alternative hypothesis.

01:23 So again, because power is 2 .3, it means that there's a 2 .3 probability that the sample will provide convincing evidence that the true mean is different from 5.

01:40 So if we get a higher power against the same alternative with the same alpha by changing the number of measurements, what should we do? should we take fewer or more to increase our power? well, what we want to do is we want to make more measurements because ultimately increasing the sample size means that we have more information, which means that we have more information to make the correct decision.

02:16 So we want to make more measurements.

02:18 More measurements means more information, which means more information means more information to make more information to make.

02:30 The correct decision.

02:34 So given the option, you always want to take as many measurements as you can.

02:40 Our third part of our question, it asks if you could decide to change the significance level to an alpha of point one in place of a 0 .05 with no other changes to the test, will the power increase or decrease? well, if you change the alpha to a 0 .1, instead of a 0 .05, your power is going to increase...

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