(CO 4) In a situation where the standard deviation was decreased from 5.8 to 3.1, what would be the impact on the confidence interval?

Group of answer choices

It would become wider due to using the t distribution

It would become narrower due to using the z distribution

It would remain the same as the standard deviation does impact the interval

It would become narrower due to the change in standard deviation

The width of the confidence interval is directly affected by the standard deviation. Specifically, the formula for the confidence interval for a mean is given by:

\[
\text{CI} = \bar{x} \pm t_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right)
\]

Where:
- \(\bar{x}\) is the sample mean.
- \(t_{\alpha/2}\) is the critical value from the t-distribution.
- \(s\) is the sample standard deviation.
- \(n\) is the sample size.

In this formula, the term \(\left(\frac{s}{\sqrt{n}}\right)\) represents the standard error of the mean. If the standard deviation \(s\) decreases, the standard error of the mean will also decrease, assuming the sample size \(n\) remains the same.

When the standard error decreases, the margin of error (which is \(t_{\alpha/2} \times \text{standard error}\)) also decreases. A smaller margin of error means that the confidence interval will be narrower.

Therefore, if the standard deviation decreases from 5.8 to 3.1, the impact on the confidence interval would be:

**It would become narrower due to the change in standard deviation.**

Thus, the correct answer is:

\[
\boxed{\text{It would become narrower due to the change in standard deviation}}
\]

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How does the sample size affect the width of a confidence interval?
2. What is the relationship between the confidence level and the width of the confidence interval?
3. Can you explain the difference between standard deviation and standard error?
4. Why is the t-distribution used for small sample sizes?
5. What would happen to the confidence interval if the sample size were increased, keeping the standard deviation constant?
6. How is the margin of error calculated in the context of confidence intervals?
7. What assumptions are made when constructing a confidence interval for the mean?
8. How would you interpret a confidence interval in the context of a real-world scenario?

**Tip:** Understanding the factors that affect the width of a confidence interval—such as sample size, confidence level, and standard deviation—can help you make more accurate and reliable statistical inferences.

Understand how changes in standard deviation affect the width of confidence intervals. Explore a detailed explanation with formulas and practical examples.

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