What happens to the power of the z test, when the standard deviation increases?
The deviation from the mean must be larger to obtain the same significance level. I'm not familiar with the term "power" (we don't use that term in physics), but I would guess that this means that the power becomes less.
If standard deviation increases, power will decrease.
When the standard deviation increases, the power of a z-test decreases. The power of a statistical test measures the ability of the test to correctly reject the null hypothesis when it is false. In other words, it measures the probability of detecting an effect or difference if one truly exists.
To understand why the power decreases when the standard deviation increases, consider that the z-test compares the difference between the observed sample mean and the hypothesized population mean (under the null hypothesis) in terms of the standard deviation. A larger standard deviation implies more variability in the data, which makes it more difficult to detect a significant difference between the observed sample mean and the hypothesized population mean. Consequently, the power of the test decreases.
To verify this relationship between the standard deviation and the power of the z-test, one can perform a power analysis. A power analysis involves determining the necessary sample size or effect size to achieve a desired level of statistical power. By manipulating the standard deviation in the power analysis, you can observe how changes in the standard deviation affect the resulting power of the test.