To answer this question, let's break it down step by step:

1. **Understanding Confidence Intervals**: A confidence interval provides a range of values that is likely to contain the population parameter. The width of the confidence interval depends on the sample size, the variability in the data, and the confidence level.

2. **Effect of Sample Size on Confidence Interval**: The width of the confidence interval is inversely related to the square root of the sample size. This means that as the sample size increases, the width of the confidence interval decreases.

3. **Mathematical Relationship**: The formula for the margin of error (which determines the width of the confidence interval) is:
$$
\text{Margin of Error} = z \times \frac{\sigma}{\sqrt{n}}
$$
where $ z $ is the z-score corresponding to the desired confidence level, $ \sigma $ is the population standard deviation, and $ n $ is the sample size.

4. **Comparing Sample Sizes**:
- Initial sample size ($ n_1 $) = 81
- New sample size ($ n_2 $) = 84

5. **Calculating the Effect**:
- For $ n_1 = 81 $:
$$
\text{Margin of Error}_1 = z \times \frac{\sigma}{\sqrt{81}} = z \times \frac{\sigma}{9}
$$
- For $ n_2 = 84 $:
$$
\text{Margin of Error}_2 = z \times \frac{\sigma}{\sqrt{84}} \approx z \times \frac{\sigma}{9.165}
$$

6. **Comparison**: Since $ \sqrt{84} $ is slightly larger than $ \sqrt{81} $, the denominator in the margin of error formula increases, leading to a smaller margin of error.

Therefore, when the sample size increases from 81 to 84, the confidence interval will narrow.

Question

To answer this question, let's break it down step by step:

1. **Understanding Confidence Intervals**: A confidence interval provides a range of values that is likely to contain the population parameter. The width of the confidence interval depends on the sample size, the variability in the data, and the confidence level.

2. **Effect of Sample Size on Confidence Interval**: The width of the confidence interval is inversely related to the square root of the sample size. This means that as the sample size increases, the width of the confidence interval decreases.

3. **Mathematical Relationship**: The formula for the margin of error (which determines the width of the confidence interval) is:
   $$
   \text{Margin of Error} = z \times \frac{\sigma}{\sqrt{n}}
   $$
   where $ z $ is the z-score corresponding to the desired confidence level, $ \sigma $ is the population standard deviation, and $ n $ is the sample size.

4. **Comparing Sample Sizes**: 
   - Initial sample size ($ n_1 $) = 81
   - New sample size ($ n_2 $) = 84

5. **Calculating the Effect**:
   - For $ n_1 = 81 $:
     $$
     \text{Margin of Error}_1 = z \times \frac{\sigma}{\sqrt{81}} = z \times \frac{\sigma}{9}
     $$
   - For $ n_2 = 84 $:
     $$
     \text{Margin of Error}_2 = z \times \frac{\sigma}{\sqrt{84}} \approx z \times \frac{\sigma}{9.165}
     $$

6. **Comparison**: Since $ \sqrt{84} $ is slightly larger than $ \sqrt{81} $, the denominator in the margin of error formula increases, leading to a smaller margin of error.

Therefore, when the sample size increases from 81 to 84, the confidence interval will narrow.

Knowee AI · Accepted Answer

To answer this question, let's break it down step by step:

1. **Understanding Confidence Intervals**: A confidence interval provides a range of values that is likely to contain the population parameter. The width of the confidence interval depends on the sample size, the variability in the data, and the confidence level.

2. **Effect of Sample Size on Confidence Interval**: The width of the confidence interval is inversely related to the square root of the sample size. This means that as the sample size increases, the width of the confidence interval decreases.

3. **Mathematical Relationship**: The formula for the margin of error (which determines the width of the confidence interval) is:
   $$
   \text{Margin of Error} = z \times \frac{\sigma}{\sqrt{n}}
   $$
   where $ z $ is the z-score corresponding to the desired confidence level, $ \sigma $ is the population standard deviation, and $ n $ is the sample size.

4. **Comparing Sample Sizes**: 
   - Initial sample size ($ n_1 $) = 81
   - New sample size ($ n_2 $) = 84

5. **Calculating the Effect**:
   - For $ n_1 = 81 $:
     $$
     \text{Margin of Error}_1 = z \times \frac{\sigma}{\sqrt{81}} = z \times \frac{\sigma}{9}
     $$
   - For $ n_2 = 84 $:
     $$
     \text{Margin of Error}_2 = z \times \frac{\sigma}{\sqrt{84}} \approx z \times \frac{\sigma}{9.165}
     $$

6. **Comparison**: Since $ \sqrt{84} $ is slightly larger than $ \sqrt{81} $, the denominator in the margin of error formula increases, leading to a smaller margin of error.

Therefore, when the sample size increases from 81 to 84, the confidence interval will narrow.

Suppose you compute a confidence interval with a sample size of 81. What will happen to the confidence interval if the sample size increases to 84?The confidence interval will widen.The confidence interval will narrow.The width of the confidence interval will stay the same.

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