Question

What effect would increasing the sample size have on a confidence interval

• A. The confidence interval would increase in size.
• B. The confidence interval would decrease in size.
• C. The confidence interval is unaffected by sample size.
• D. The confidence interval would become zero.

Hint:

Larger sample size (\( n \)) \(\implies\) Smaller standard error \(\implies\) Narrower (smaller) confidence interval \(\implies\) Higher precision of the estimate.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A confidence interval (CI) is an estimated range of values that is likely to contain an unknown population parameter, calculated from a given set of sample data.
The width (or size) of the confidence interval represents the level of precision of our estimate.
Key Formula or Approach:
For a population mean, the confidence interval is calculated as:
\[ \text{CI} = \bar{x} \pm z^* \left( \frac{s}{\sqrt{n}} \right) \]
where \(\bar{x}\) is the sample mean, \(z^*\) is the critical value, \(s\) is the sample standard deviation, and \(n\) is the sample size.
The term \( \frac{s}{\sqrt{n}} \) represents the standard error of the mean (SEM).

Step 2: Detailed Explanation:
The margin of error (half-width of the confidence interval) is given by:
\[ \text{Margin of Error} = z^* \left( \frac{s}{\sqrt{n}} \right) \]
From this formula, we can see that the margin of error is inversely proportional to the square root of the sample size (\( \sqrt{n} \)).
As the sample size (\( n \)) increases, the denominator in the standard error term increases, which causes the overall standard error of the mean to decrease.
A smaller standard error reduces the margin of error, making the confidence interval narrower (decreasing its size).
This reflects the fact that larger samples provide more precise estimates of the population parameter.
Therefore, increasing the sample size causes the confidence interval to decrease in size.

Step 3: Final Answer:
Increasing the sample size will cause the confidence interval to decrease in size.