Alex collects data for the Australian Bureau of Statistics. His boss asks him to do a cost benefit analysis of collecting a random sample for a sample mean that is three times the size of the original sample. Which of the following is a truthful statement Alex could say to his boss?
Group of answer choices

Increasing the sample size increases the volatility so it can't be worthwhile

Increasing the sample size has no impact on the sampling distribution. Thus, increasing the sample size is not worth the cost.

None of the answers are correct

If enough data is collected, it will become clear that the original data (not the sample mean) is Normally distributed

Let's analyze each of the statements to determine which is truthful:

1. **"Increasing the sample size increases the volatility so it can't be worthwhile."**

   This statement is false. Increasing the sample size generally decreases the volatility (variability) of the sample mean. Larger sample sizes lead to more stable estimates of the population parameter.

2. **"Increasing the sample size has no impact on the sampling distribution. Thus, increasing the sample size is not worth the cost."**

   This statement is also false. Increasing the sample size affects the sampling distribution of the sample mean by reducing its standard error. The standard error of the mean is inversely proportional to the square root of the sample size (\(SE = \frac{\sigma}{\sqrt{n}}\)). Therefore, increasing the sample size decreases the standard error, making the sampling distribution narrower and more precise.

3. **"None of the answers are correct."**

   This statement seems plausible as the other statements so far are incorrect.

4. **"If enough data is collected, it will become clear that the original data (not the sample mean) is Normally distributed."**

   This statement is misleading. The Central Limit Theorem (CLT) states that, regardless of the shape of the population distribution, the sampling distribution of the sample mean will tend to be normal if the sample size is large enough. However, collecting more data will not necessarily reveal that the original data (the population itself) is normally distributed unless the population is inherently normal.

### Conclusion:

- The correct choice is **"None of the answers are correct."** because the other statements about the effects of increasing sample size and the distribution of original data are incorrect or misleading.

Would you like more details on any of these points, or do you have any questions?

### Related Questions:
1. What is the Central Limit Theorem and how does it apply to sampling distributions?
2. How does increasing the sample size affect the margin of error in statistical estimates?
3. What are the benefits and costs of increasing the sample size in statistical analysis?
4. How is the standard error of the mean calculated, and why is it important?
5. Can the original population distribution ever become normal by increasing sample size?

**Tip:** When performing a cost-benefit analysis for data collection, consider both the reduction in variance and the increase in precision against the additional costs involved.

Alex collects data for the Australian Bureau of Statistics. His boss asks him to do a cost benefit analysis of collecting a random sample for a sample mean that is three times the size of the original sample. Which of the following is a truthful statement Alex could say to his boss?
Group of answer choices

- Increasing the sample size increases the volatility so it can't be worthwhile
- Increasing the sample size has no impact on the sampling distribution. Thus, increasing the sample size is not worth the cost.
- None of the answers are correct
- If enough data is collected, it will become clear that the original data (not the sample mean) is Normally distributed

Understand the cost-benefit analysis of increasing sample size in data collection. This problem explores key concepts like the sampling distribution, the Central Limit Theorem, and how increasing sample size affects the standard error. Perfect for college-level statistics courses.

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