Question

What is the difference in the computation of a confidence interval between cases in which you know the population standard deviation and cases in which you have to estimate it?

Short answer

The confidence interval uses Z-distribution if population standard deviation is known; otherwise, it uses t-distribution.

Step-by-step solution

Understanding the Problem

To answer this question, we need to understand how a confidence interval is computed in both situations: when the population standard deviation is known, and when it is estimated using the sample standard deviation.

Scenario 1: Known Population Standard Deviation

When the population standard deviation (\(\sigma\)) is known, we use the Z-distribution to compute the confidence interval. The formula is \(\bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(Z\) is the Z-score for the desired confidence level, and \(n\) is the sample size.

Scenario 2: Estimated Population Standard Deviation

When the population standard deviation is not known and must be estimated using the sample standard deviation (\(s\)), we use the t-distribution to compute the confidence interval. The formula is \(\bar{x} \pm t \cdot \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(t\) is the t-score based on the desired confidence level and degrees of freedom (\(n-1\)), and \(n\) is the sample size.

Key Difference

The key difference between the two scenarios is the distribution used: Z-distribution is used when the population standard deviation is known, and t-distribution is used when it is estimated from the sample.

Key concepts

Population Standard Deviation

When we talk about population standard deviation, we refer to the measure of variability or spread in a set of a population's data. This parameter is denoted by the symbol \(\sigma\). Knowing the true population standard deviation is exceptionally useful in many statistical scenarios because it allows you to compute precise confidence intervals using the Z-distribution.
If you know \(\sigma\), you assume that the sample accurately captures the characteristics of the entire population. Determining the population standard deviation precisely can be challenging and is often impossible unless you have access to data for every individual in the population. That is why we sometimes need to rely on samples and estimates.
When \(\sigma\) is known, it simplifies calculations, allowing statistical inferences to be made with greater confidence.

Z-Distribution

The Z-distribution is a crucial tool in statistics when the population standard deviation is known. It's a type of normal distribution that has a mean of 0 and a standard deviation of 1, often referred to as the standard normal distribution. The Z-score tells you how many standard deviations away a data point is from the average.
The Z-distribution plays a vital role in hypothesis testing and constructing confidence intervals. For example, to calculate a confidence interval of a population mean when \(\sigma\) is known, the formula \(\bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}}\) is used. Here, \(\bar{x}\) is the sample mean, \(Z\) is the Z-score corresponding to your confidence level, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
This method assumes data is normally distributed and that the central limit theorem applies, which is reasonable with a sufficiently large sample size.

T-Distribution

The t-distribution becomes crucial when the population standard deviation is unknown. Instead, the sample standard deviation is calculated and used to estimate \(\sigma\). The t-distribution is similar in shape to the normal distribution but has fatter tails, which accounts for more variability due to estimating \(\sigma\) from a sample.
The formula to calculate a confidence interval using the t-distribution is \(\bar{x} \pm t \cdot \frac{s}{\sqrt{n}}\). Here, \(\bar{x}\) is the sample mean, \(t\) is the t-score found using your confidence level and degrees of freedom \((n-1)\), \(s\) is the sample standard deviation, and \(n\) is the sample size.
The degrees of freedom reflect the sample size minus one and are crucial in determining the exact shape of the t-distribution used. For smaller samples, the t-distribution shows more variability, reflecting the uncertainty from a smaller data set.

Sample Standard Deviation

Sample standard deviation is a statistic that estimates the standard deviation of a population based on a sample. Denoted by \(s\), it helps in measuring the spread of sample data around the sample mean.
Calculation of sample standard deviation follows the formula: \(s = \sqrt{\frac{\Sigma (x_i - \bar{x})^2}{n-1}}\), where \(x_i\) represents each value in the sample, \(\bar{x}\) is the sample mean, and \(n\) is the number of observations in the sample. It is essential to remember the \(n-1\) denominator known as Bessel's correction, which corrects bias in estimating a population's variance and standard deviation from a sample.
The sample standard deviation is crucial when the population standard deviation is unknown, as it allows the use of the t-distribution to compute confidence intervals, providing a way to make reliable inferences about the population from the sample.