Which will make the confidence interval for a mean for 1 population narrower? Select all that apply.

The formula is  or  depending on if  is known.

Group of answer choices

Decrease confidence level

Decrease sample size

To determine which factors will make the confidence interval narrower, let's analyze the formula for a confidence interval for a population mean.

The two formulas are:

1. If the population standard deviation \( \sigma \) is known:
\[
\text{CI} = \bar{x} \pm Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}
\]
Where:
- \( \bar{x} \) is the sample mean,
- \( Z_{\alpha/2} \) is the critical value from the standard normal distribution for the desired confidence level,
- \( \sigma \) is the population standard deviation, and
- \( n \) is the sample size.

2. If \( \sigma \) is not known, and we use the sample standard deviation \( s \), the formula becomes:
\[
\text{CI} = \bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}
\]
Where \( t_{\alpha/2} \) is the critical value from the t-distribution.

### Now, let’s evaluate the two options:

1. **Decrease confidence level:**
   - A lower confidence level (e.g., from 95% to 90%) reduces the critical value (\( Z_{\alpha/2} \) or \( t_{\alpha/2} \)), which in turn decreases the margin of error, making the confidence interval **narrower**.

2. **Decrease sample size:**
   - Decreasing the sample size \( n \) will increase the margin of error (since \( \frac{1}{\sqrt{n}} \) becomes larger), making the confidence interval **wider**, not narrower.

### Correct Answer:
- **Decrease confidence level** will make the confidence interval narrower.
- **Decrease sample size** will not make the interval narrower, it will make it wider.

Would you like further details or have any questions?

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Here are five related questions:
1. How does the critical value change with different confidence levels?
2. How does the width of a confidence interval depend on the sample standard deviation?
3. What is the impact of increasing the population size on the confidence interval?
4. How do you choose between the z-distribution and t-distribution for confidence intervals?
5. How does the shape of the t-distribution compare to the normal distribution?

**Tip:** Increasing the sample size is one of the most effective ways to narrow a confidence interval.

Which will make the confidence interval for a mean for 1 population narrower? Select all that apply. The formula is or depending on if is known. Group of answer choices: Decrease confidence level, Decrease sample size

Understand which factors make the confidence interval for a population mean narrower. This problem focuses on the effect of decreasing confidence level and sample size, analyzing key statistical formulas for confidence intervals. Suitable for college-level statistics students.

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