I have been working on the Type I error and a Type II error problem but now I can not figure out the Power of the test?

A manufacturer of computer monitors receives shipments of LCD panels from a suppliers overseas. It is not cost effective to inspect each LCD panel for defeats, so a sample is taken from each shipment. A significance test is conducted to determine whether the proportion of defective LCD panels is greater then the acceptable limit of 1%. If it is, the shipment will be taken back to the supplier. The hypothesis for this test is Ho:p=0.01 and Ha:p>0.01, where p is the true proportion of defective panels in the shipment.
If a Type I error were to be committed, we would conclude that there are more the 1% defective panels when there really not. The shipment would be returned when it was not authorized. If a Type II error were not committed, we would conclude that there are no more than 1% of defected panels when there really were. Then defective panels would be accepted from the supplier.
The supplier would think that the Type I error is more serious because they would be receiving LCD panels back that work fine. The computer monitor manufacturer would think that the Type II error is more serious because they would be receiving panels of poor quality.

What would the Power of the test be?

The probability of making a Type II error is equal to beta. A Type II error is failure to reject the null (Ho) when it is false. The power of the test is 1-beta and is the correct decision of rejecting the null when it is false. The alpha level directly affects the power of a test. The higher the level, the more powerful the test. Sample size also affects power. As the alpha level gets smaller, the probability of a Type II error increases and power decreases. You want to reject the null when it is truly false to have high power in statistical tests.

I hope this will help.

To determine the power of a statistical test, we need to first understand the concept of power. Power is the probability of correctly rejecting a false null hypothesis, or in other words, the probability of detecting an effect or difference when it truly exists.

In this scenario, the null hypothesis (Ho) is that the proportion of defective LCD panels is equal to or less than 1% (p ≤ 0.01), and the alternative hypothesis (Ha) is that the proportion of defective panels is greater than 1% (p > 0.01).

The power of the test is the probability of correctly rejecting the null hypothesis (Ho) when the alternative hypothesis (Ha) is true. In this case, it means correctly identifying a shipment as having a proportion of defective panels greater than 1%.

To calculate the power of the test, we need to know several factors:

1. The significance level (α): This is the probability of committing a Type I error, which is the error of rejecting the null hypothesis when it is true. It is often set to a standard value like 0.05 or 0.01.

2. The effect size: This represents the magnitude of the difference or effect we are trying to detect. It is usually measured using standardized effect sizes like Cohen's d or the proportion of variance explained (R^2).

3. Sample size: The size of the sample used in the test.

4. The distribution of the test statistic: Depending on the specific hypothesis test used (e.g., Z-test for proportions), we need to consider the distribution of the test statistic under both the null and alternative hypotheses.

Without specific values for the effect size and sample size, it is not possible to provide an exact calculation of the power of the test. However, I can explain the general procedure to calculate it.

To calculate the power of your test, you would typically perform a power analysis using statistical software or tables. This analysis takes into account the specific details of your hypothesis test, including the sample size, significance level, and effect size.

Here is an overview of the steps involved in a power analysis:

1. Specify the significance level (α): Typically, this is set to a standard value like 0.05 or 0.01, depending on the desired level of significance.

2. Choose an effect size: You need to decide on a meaningful effect size that represents the difference or effect you expect to detect. This can be based on prior research, expert knowledge, or practical considerations.

3. Determine the required sample size: Based on the significance level and effect size, you can estimate the necessary sample size to achieve a desired power level. This involves using statistical formulas or software.

4. Perform the power analysis: Using the specified significance level, effect size, and sample size, you can perform the power analysis to calculate the power of the test. This can be done using statistical software or tables specific to your hypothesis test.

It is important to note that power analysis is a proactive procedure conducted before data collection to determine the sample size needed for the study to achieve a desired power level. It helps in planning studies and ensures that they have an adequate sample size to detect meaningful effects.

Without specific information about the effect size and sample size, it is not possible to provide an exact calculation of the power of the test in this scenario. However, by following the steps outlined above and providing the necessary values, you can estimate the power of your test using appropriate statistical techniques.