Question

What is the effect of sample size on the width of a confidence interval?

Short answer

A larger sample size leads to a narrower confidence interval.

Step-by-step solution

Understanding Confidence Intervals

A confidence interval gives a range of values that is likely to contain the population parameter (e.g., mean or proportion). It is usually given as: \[\text{Confidence Interval} = \text{estimate} \pm \text{margin of error}\]The margin of error depends on the standard error and a critical value from a statistical distribution.

Defining Margin of Error

Margin of error can be expressed as the product of a critical value (based on confidence level) and the standard error, which is influenced by the sample size. Usually:\[\text{Margin of Error} = Z \frac{\sigma}{\sqrt{n}}\]where \(Z\) is the z-score for the given confidence level, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Analyzing the Sample Size Effect

Notice the term \(\sqrt{n}\) in the denominator of the standard error: as the sample size \(n\) increases, \(\sqrt{n}\) increases as well. This means the standard error decreases with a larger sample size, thereby decreasing the margin of error.

Concluding the Impact

Since the confidence interval's width equals twice the margin of error, a smaller margin of error directly leads to a narrower confidence interval. Thus, as sample size increases, the confidence interval becomes narrower, assuming a constant confidence level and population variance.

Key concepts

Understanding Sample Size

Sample size refers to the number of individual data points collected in a study or survey. It's an essential component of statistical analyses, particularly when aiming to estimate a population parameter, such as a mean or proportion, from a sample.

Derived from the population, a larger sample size can offer a more accurate reflection of that population.

• Larger sample sizes generally lead to more precise estimates.
• They tend to reduce the influence of outliers.
• They result in a smaller standard error, which positively impacts the confidence interval width.

In statistical terms, increasing the sample size helps enhance the reliability and validity of the results, providing a clearer view of the population being studied.

Exploring the Margin of Error

The margin of error represents the range in which the true population parameter is expected to lie with a given level of confidence.

It accounts for the variability in the data and helps us understand the uncertainty around our estimate. Often expressed as a plus-minus value around the sample estimate, the margin of error is calculated using:\[\text{Margin of Error} = Z \frac{\sigma}{\sqrt{n}}\]where:

• \(Z\) is the z-score correlating to the desired confidence level.
• \(\sigma\) is the population standard deviation.
• \(n\) is the sample size.

A smaller margin of error indicates a more precise estimate, leading to a narrower confidence interval.

Key factors that influence the margin of error include the desired confidence level (which determines the z-score) and the standard deviation of the population data.

Understanding Standard Error

Standard error is a measure of how much a sample mean is likely to vary if you were to take multiple samples from the same population. It is crucial for constructing confidence intervals and conducting hypothesis tests.

Mathematically, standard error is defined as:\[\text{Standard Error} = \frac{\sigma}{\sqrt{n}}\]In this formula:

• \(\sigma\) represents the population standard deviation.
• \(\sqrt{n}\) is the square root of the sample size.

Key aspects of standard error include:

• As the sample size increases, the standard error decreases.
• A smaller standard error indicates more accuracy in estimating the population mean.

It provides insights into the variability and precision of the sample estimates, thus influencing decisions based on statistical data.

Decoding Z-Score

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values.

It quantifies how far a data point is from the mean, factoring in the standard deviation of the data set. In the context of confidence intervals, the z-score determines the critical value that influences the margin of error.

• For a given confidence level, a specific z-score is used. For example, a 95% confidence level corresponds approximately to a z-score of 1.96.
• The z-score scales the standard error to produce the margin of error.

Higher confidence levels require higher z-scores, affecting the width of the confidence interval.

In practice, understanding the z-score is vital for interpreting results accurately, ensuring the estimates reflect the desired level of confidence.