How Is the Sampling Distribution of a Sample Mean Determined?

[CGuthrie91]

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

View attachment 4206

 

[Ackbach]

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?