Suppose that 100 items are drawn from a population of manufactured products and the weight, X, of each item is recorded. Prior experience has shown that the weight has a non- normal probability distribution with u - 3 ounces. Which of the following is true about the sampling of X? = 8 ounces and o a. Its mean is 8 ounces b. Its standard error is 0.3 ounces. c. Its shape is approximately normal d. All of these choices are true.

Given the scenario, let's break down each option:

1. **Option (a): Its mean is 8 ounces.**
   - This statement implies that the mean weight of the sampled items is 8 ounces. If the mean of the population is \(\mu = 8\) ounces, then the mean of the sample (\(\bar{X}\)) will also be approximately 8 ounces by the law of large numbers.

2. **Option (b): Its standard error is 0.3 ounces.**
   - The standard error (SE) of the sample mean \(\bar{X}\) is calculated using the formula:
     \[
     \text{SE} = \frac{\sigma}{\sqrt{n}}
     \]
     Where \(\sigma\) is the population standard deviation, and \(n\) is the sample size. Given \(\sigma = 3\) ounces and \(n = 100\), the SE would be:
     \[
     \text{SE} = \frac{3}{\sqrt{100}} = \frac{3}{10} = 0.3 \text{ ounces}
     \]
     So, this statement is true.

3. **Option (c): Its shape is approximately normal.**
   - The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean \(\bar{X}\) will be approximately normal, regardless of the shape of the population distribution, provided the sample size is large enough. In this case, with a sample size of 100, the distribution of \(\bar{X}\) should be approximately normal.

4. **Option (d): All of these choices are true.**
   - Since options (a), (b), and (c) are all true, this option is also correct.

**Conclusion**: **(d) All of these choices are true** is the correct answer.

Would you like more details or have any questions?

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Here are some related questions to consider:

1. What is the Central Limit Theorem, and why is it important?
2. How does the sample size affect the standard error?
3. Why is the normality of the sampling distribution important in statistical inference?
4. What would happen to the standard error if the sample size was reduced to 25?
5. How can the law of large numbers be explained in the context of this problem?
6. What would the implications be if the population standard deviation \(\sigma\) were unknown?
7. How would you calculate a confidence interval for the mean weight based on the sample?
8. What assumptions must hold true for the Central Limit Theorem to apply?

**Tip:** When working with sampling distributions, always check whether the sample size is large enough for the Central Limit Theorem to ensure the normality of the sampling distribution.

Explore how sampling from a non-normal distribution affects mean and standard error calculations. Learn about the Central Limit Theorem and its application in statistics.

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