Is a sampling distribution always the same as the curve for a null hypothesis?
How does the power of a z test change when sigma increases?
Assuming that they are caused solely by chance, sampling distributions will approximate a normal distribution more closely as the size of the sample increases. However, with small samples, there may be great differences.
I don't understand your second question. Sigma in a formula is used just to indicate that you add the terms that follow that symbol.
I hope this helps. If not, repost your new questions. Thanks for asking.
No, a sampling distribution is not always the same as the curve for a null hypothesis. A sampling distribution represents the distribution of a statistic (such as the mean) from all possible samples taken from a population. It gives us information about the sampling variability of that statistic. On the other hand, the curve for a null hypothesis represents the theoretical distribution of the statistic under the assumption that the null hypothesis is true. It is often a normal distribution centered around a specific value.
Regarding your second question, the power of a z test is the probability of correctly rejecting the null hypothesis when it is false. It is influenced by several factors, including the sample size, the effect size, and the significance level. The effect size, in the case of a z test, is typically represented by the difference between the population mean and the hypothesized mean, divided by the standard deviation.
When the sigma (standard deviation) increases, the power of the z test generally decreases. This is because a larger standard deviation leads to greater variability in the data, making it more difficult to detect a significant difference from the null hypothesis. In other words, a larger sigma reduces the signal-to-noise ratio, making it harder to distinguish the true effect from random variability.
In summary, the power of a z test decreases when the standard deviation increases. However, it's important to note that power is also influenced by other factors such as sample size and effect size.