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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.

Step by step solution

01

Understanding Power

Power in statistical testing is the probability that the test will correctly reject a false null hypothesis. In this context, a power of 0.23 means that there is a 23% chance of correctly identifying that the true mean conductivity is 5.1, rather than 5, using the current test setup.

02

Effect of More Measurements

Increasing the number of measurements can increase the power of a test. This is because more measurements provide a more accurate estimate of the true mean, reducing the standard error, which increases the likelihood of detecting a true difference from the null hypothesis value.

03

Effect of Changing Significance Level

Increasing the significance level \(\alpha\) from 0.05 to 0.10 results in an increased power. This occurs as a higher significance level makes it easier to reject the null hypothesis, thereby increasing the probability of correctly rejecting it when it is false.

04

Effect of Alternative Hypothesis Shift

If the alternative hypothesis is shifted to \(\mu = 5.2\), the power of the test generally increases. This is because a larger difference between the null hypothesis mean and the actual mean makes it easier to detect the difference, enhancing the test's ability to correctly reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether a claim or hypothesis about a population is supported by evidence from a sample.
It begins with a statement called the null hypothesis, denoted as \( H_0 \). In our context, \( H_0: \mu = 5 \) indicates that the mean conductivity of the liquid product is presumed to be 5.
The alternative hypothesis, \( H_a \), represents what we are trying to prove. Here \( H_a: \mu eq 5 \) suggests that the mean conductivity differs from 5.

**Key Steps in Hypothesis Testing:**

• Formulate the null and alternative hypotheses.
• Select a significance level (usually 0.05 or 0.10) that defines the threshold for rejecting the null hypothesis.
• Conduct the test using sample data and determine the p-value.
• Compare the p-value to the significance level to decide whether to reject \( H_0 \).

Understanding these basics will help you grasp the nuances of statistical power in hypothesis testing.

Significance Level

The significance level, often denoted by \( \alpha \), is a critical concept in hypothesis testing.
It represents the probability of rejecting the null hypothesis when it is actually true. In simpler terms, it’s the chance of making a Type I error.
Common values for \( \alpha \) are 0.05 or 0.10, meaning a 5% or 10% risk of erroneously rejecting \( H_0 \).

**Importance of Significance Level:**

• A lower \( \alpha \) means stricter criteria for rejecting \( H_0 \), reducing the risk of Type I error but potentially lowering the test power.
• A higher \( \alpha \) increases test power, making it easier to detect a true effect. However, it also raises the risk of a Type I error.

In our example, adjusting the \( \alpha \) from 0.05 to 0.10 resulted in increased power, highlighting the balance required between sensitivity and error risk.

Alternative Hypothesis

The alternative hypothesis \( H_a \) is a pivotal part of hypothesis testing as it embodies the outcome you want to test.
When considering changes to \( H_a \), such as in our example moving from \( \mu = 5.1 \) to \( \mu = 5.2 \), the power of the test can be affected.

**Why Change Matters:**

• Larger differences between \( \mu \) in \( H_0 \) and \( \mu \) in \( H_a \) make it easier to detect true effects, thus increasing test power.
• An accurate alternative hypothesis helps provide clearer evidence, supporting more confident decision-making.

In practice, specifying \( H_a \) accurately to reflect realistic differences is essential for conducting meaningful tests.

Sample Size

Sample size is integral to hypothesis testing as it directly influences the power of statistical tests.
Increasing the sample size can significantly enhance the ability to detect true differences between the null and alternative hypotheses.

**How Sample Size Affects Power:**

• More measurements reduce the margin of error, leading to a more precise estimate of the population mean.
• Larger samples increase test power by minimizing the chance of missing a real effect due to variability.

In the case of testing the conductivity of a liquid, more samples mean a better chance of detecting any deviation from the specified mean of 5.
Deciding on the right sample size involves balancing resource constraints with the need for reliable results.