When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.
b. Increase in the sample size

Verified step by step guidance

1. Understand the relationship between sample size and the width of a confidence interval: The width of a 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.
2. Recall the formula for the margin of error in a confidence interval: \( \text{Margin of Error} = z^* \cdot \frac{\sigma}{\sqrt{n}} \), where \( z^* \) is the critical value, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
3. Analyze the effect of increasing the sample size \( n \): Since \( n \) appears in the denominator under a square root, increasing \( n \) will decrease the value of \( \frac{\sigma}{\sqrt{n}} \), which in turn reduces the margin of error.
4. Conclude how this impacts the confidence interval: A smaller margin of error results in a narrower confidence interval, meaning the range of values within which the population parameter is estimated becomes more precise.
5. Summarize the effect: Increasing the sample size reduces the width of the confidence interval, improving the precision of the estimate while keeping all other factors constant.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed with a certain level of confidence, typically 95% or 99%, indicating the degree of certainty about the estimate. The width of the interval reflects the precision of the estimate; narrower intervals suggest more precise estimates.

Sample Size

Sample size refers to the number of observations or data points collected in a study. A larger sample size generally leads to more reliable estimates of population parameters, as it reduces the impact of random variability. In the context of confidence intervals, increasing the sample size typically results in a narrower interval, indicating greater precision in estimating the population parameter.

Margin of Error

The margin of error is the amount of error that can be tolerated in the estimate of a population parameter. It is influenced by the sample size and the variability of the data. A larger sample size decreases the margin of error, which in turn reduces the width of the confidence interval, allowing for a more accurate representation of the population parameter.