# Sampling Distribution Question

• hsd
In summary, the conversation discusses the properties of a normal distribution, specifically the sampling distribution of a random variable Z and a test statistic for a null hypothesis. It also addresses how the results need to be modified if the population variance is unknown. It is suggested to refer to a textbook or other sources for a better understanding of the topic before asking for specific explanations.
hsd
I've added a screenshot (https://imgur.com/a/isQXZ) and the text below for your convenience.

Please show steps if possible to help my understanding. Thank you.

Consider a random variable that is Normally distributed, with population mean mu = E [X] and population variance sigma^2 = var [X]. Assume that we have a random sample of size N from this distribution. Let x-bar be the usual sample average.

(a) Using the properties of the Normal distribution, derive explicitly the sampling distribution of the random variable: Z = (sqrt(N)) ((x-bar - mu)/(sigma))

(b) Assume that we know sigma^2, but not mu. For the Null hypothesis H0 : mu = mu0, derive a test statistic, characterize its distribution under H0, and describe a test with the property that the probability of committing a type I error is alpha.

(c) How do you need to modify the results in (b) if sigma^2 is unknown.

hsd said:
I've added a screenshot (https://imgur.com/a/isQXZ) and the text below for your convenience.

Please show steps if possible to help my understanding. Thank you.

Consider a random variable that is Normally distributed, with population mean mu = E [X] and population variance sigma^2 = var [X]. Assume that we have a random sample of size N from this distribution. Let x-bar be the usual sample average.

(a) Using the properties of the Normal distribution, derive explicitly the sampling distribution of the random variable: Z = (sqrt(N)) ((x-bar - mu)/(sigma))

(b) Assume that we know sigma^2, but not mu. For the Null hypothesis H0 : mu = mu0, derive a test statistic, characterize its distribution under H0, and describe a test with the property that the probability of committing a type I error is alpha.

(c) How do you need to modify the results in (b) if sigma^2 is unknown.

Are you using a textbook? If so, all you need should be in there.

You are asking us to take the time to write things out and explain them to you, but many other people have already done that and put it into books and on web pages. So: I suggest you start looking elsewhere first, and if you have made a genuine effort---and are still confused about very specific issues---then come back here. General questions like yours will never be well received by most helpers..

## 1. What is a sampling distribution?

A sampling distribution is a probability distribution that shows the possible values of a sample statistic (such as mean or proportion) for all possible samples of a given size from a population.

## 2. Why is the concept of sampling distribution important?

The concept of sampling distribution is important because it allows us to make inferences about the population based on the characteristics of a sample. It also helps us understand the variability and reliability of our sample statistics.

## 3. How is a sampling distribution different from a population distribution?

A population distribution shows the frequency of values in the entire population, while a sampling distribution shows the frequency of values in a sample from that population. A sampling distribution is based on a random sample, while a population distribution represents the entire population.

## 4. What is the central limit theorem and how does it relate to sampling distribution?

The central limit theorem states that as the sample size increases, the sampling distribution of the mean will become increasingly normal, regardless of the shape of the population distribution. This means that we can use the normal distribution to make inferences about the population mean, even if the population distribution is not normal.

## 5. How does the sample size affect the shape of the sampling distribution?

The sample size affects the shape of the sampling distribution in two ways. First, as the sample size increases, the sampling distribution becomes more normal. Second, the standard error of the sampling distribution decreases as the sample size increases, resulting in a narrower and taller distribution.

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