To answer the questions about the sampling distribution of means, we need to apply concepts from statistics. These include the Central Limit Theorem (CLT) and formulas related to the sampling distribution of means.

However, some information is missing: the population standard deviation $\sigma$, the population mean $\mu$, and the sample size $n$. We will discuss the answers assuming these details, but I’ll ask for clarification where needed.

(a) What is the shape of the sampling distribution of means?

The Central Limit Theorem (CLT) states that the sampling distribution of sample means will approach a normal distribution if the sample size is large enough, regardless of the population distribution, provided the population has a finite standard deviation.

• If the population distribution is normal or the sample size $n$ is large ($n \geq 30$), the sampling distribution will also be normal.

So, the correct answer is:

Answer: normal

(b) What is the center of the sampling distribution of means?

The mean of the sampling distribution of sample means is equal to the population mean $\mu$.

$\mu_{\text{sample means}} = \mu$

• To answer this question, I would need the value of $\mu$. If the population mean is provided, we can simply state that as the center of the sampling distribution. If not, please provide it, and I'll round the answer to four decimal places.

(c) What is the standard error of the sampling distribution of means?

The standard error (SE) of the sampling distribution of means is given by the formula:

$SE = \frac{\sigma}{\sqrt{n}}$

Where:

• $\sigma$ is the population standard deviation

• $n$ is the sample size

• To complete this part, I would need both the population standard deviation $\sigma$ and the sample size $n$. Once provided, I can calculate and round the answer to four decimal places.

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If you provide the missing details (population mean, population standard deviation, and sample size), I can finalize the solutions for parts (b) and (c).

Would you like further details or clarifications?

Related Questions:

1. How does increasing the sample size affect the shape of the sampling distribution of means?
2. Why does the Central Limit Theorem apply even when the population distribution is not normal?
3. What is the difference between the population standard deviation and the standard error of the mean?
4. If the population is skewed, at what sample size does the sampling distribution become approximately normal?
5. How would you interpret a right-skewed sampling distribution?

Tip: The larger your sample size, the smaller your standard error, leading to more precise estimates of the population mean.