Trying to finish an assignment: What happens to the power of the z test when the standard deviation increases? and Is the shape of a sampling distribution always the same as that of the null hypothesis population? Thanks!
As the SD increases, the exactness of the estimate of a population mean is less.
The null hypothesis usually assumes a normal distribution. As I mentioned in your later post, assu7ming that the distribution isdue to chance factors, the sampling distribution will approximate a normal distribution more closely as the sample size increases.
I hope this helps a little more. Thanks for asking.
To fully understand the relationship between the power of a z-test and the standard deviation, let's first clarify some concepts.
The power of a statistical test is the probability of correctly rejecting the null hypothesis when it is false. In other words, it measures the ability of the test to detect a true effect. A higher power means a higher likelihood of correctly rejecting the null hypothesis.
In a z-test, the test statistic is calculated by taking the difference between the sample mean and the hypothesized population mean, and dividing it by the standard error (which is the standard deviation of the sample divided by the square root of the sample size). The test statistic follows a standard normal distribution under the null hypothesis assumption.
Now, let's consider the effect of increasing the standard deviation on the power of the z-test. As the standard deviation increases, the variability within the population increases. This means that the difference between the sample mean and the hypothesized population mean is less likely to be significant because it is more likely to occur due to random chance.
To put it in simpler terms, increasing the standard deviation makes the population more spread out, which in turn increases the overlap between the null hypothesis distribution (assumed to be normal) and the alternative hypothesis distribution. This overlap reduces the power of the test because it becomes harder to distinguish between the two distributions.
In summary, as the standard deviation increases, the power of the z-test decreases. This is because the variability within the population makes it more difficult to detect a significant difference between the sample mean and the hypothesized population mean.
Regarding the shape of the sampling distribution, the shape is not always the same as that of the null hypothesis population. The shape of the sampling distribution depends on the sample size and the underlying population distribution.
If the sample size is large (usually considered to be at least 30), the central limit theorem states that the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution. This means that even if the population is not normally distributed, as long as the sample size is large enough, the shape of the sampling distribution will be close to normal.
However, if the sample size is small, the shape of the sampling distribution may still resemble the shape of the population distribution. For example, if the population distribution is skewed, the sampling distribution may also be skewed, especially for small sample sizes.
In conclusion, the shape of the sampling distribution may or may not be the same as that of the null hypothesis population, depending on the sample size and the underlying population distribution.