How Is the Sampling Distribution of a Sample Mean Determined?

In summary, the first problem is about the sampling distribution of the mean, where the normal distribution can be used if the population standard deviation is known. The formula for this is given by $\bar{x}\sim N\left(\mu,\frac{\sigma}{\sqrt{n}}\right)$, where $\mu$ is the given mean, $\sigma$ is the known population standard deviation, and $n$ is the sample size. However, in real-world situations, $\mu$ and $\sigma$ are usually unknown. In this case, the normal distribution cannot be used and the student-$t$ distribution must be used. Once the sampling distribution is known, the rest of the problem can be solved.
  • #1
CGuthrie91
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1. On this question I really have no idea how they got these answers so I just need someone to walk me through it step by step please

View attachment 42052. Part B on this question I don't know how to get the correct answer either

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  • #2
In your first problem, you're talking about the samping distribution of the mean. If the standard deviation of the population is known, you can use the normal distribution; that is,
$$\bar{x}\sim N\left(\mu,\frac{\sigma}{\sqrt{n}}\right),$$
where $\mu$ is as given, $\sigma$ is the population standard deviation (also given in this case), and $n$ is the sample size. As a side note: in most real-world applications, you don't know $\mu$ or $\sigma$. Not knowing the mean isn't such a big deal - just approximate with $\bar{x}$. But if you don't know $\sigma$, then you have to use the student-$t$ distributions instead of the normal distribution.

So, now that you know the sampling distribution, can you work out the rest of the problem?
 

FAQ: How Is the Sampling Distribution of a Sample Mean Determined?

What is a sampling distribution of sample mean?

A sampling distribution of sample mean is a theoretical probability distribution that represents the distribution of sample means that could be obtained from a population by drawing all possible samples of a fixed size. It provides information about the distribution of the sample mean and how it varies from sample to sample.

What is the central limit theorem and how does it relate to the sampling distribution of sample mean?

The central limit theorem states that as the sample size increases, the sampling distribution of sample mean approaches a normal distribution regardless of the shape of the population distribution. This means that regardless of the population distribution, as long as the sample size is large enough, the sample means will follow a normal distribution.

How do you calculate the mean and standard error of a sampling distribution of sample mean?

The mean of a sampling distribution of sample mean is equal to the mean of the population. The standard error is equal to the standard deviation of the population divided by the square root of the sample size.

Why is the sampling distribution of sample mean important?

The sampling distribution of sample mean is important because it allows us to make inferences about the population based on a sample. By understanding the distribution of sample means, we can estimate the population mean and make predictions about the population.

What factors can affect the shape of the sampling distribution of sample mean?

The shape of the sampling distribution of sample mean can be affected by the shape of the population distribution, the sample size, and the variability of the population. As the sample size increases, the distribution becomes more normal. As the population variability increases, the distribution becomes more spread out.

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