MHB How Is the Sampling Distribution of a Sample Mean Determined?

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The sampling distribution of a sample mean is determined using the formula $$\bar{x}\sim N\left(\mu,\frac{\sigma}{\sqrt{n}}\right)$$ when the population standard deviation is known. In cases where the population mean and standard deviation are unknown, the sample mean can be approximated with $$\bar{x}$$, and the student-$t$ distribution should be used instead of the normal distribution. Understanding these concepts is crucial for solving related problems effectively. The discussion emphasizes the importance of knowing whether the population parameters are available for accurate calculations. Mastery of these principles is essential for tackling sampling distribution questions.
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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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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?
 
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