Question

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample drawn from a \(N\left(\mu, \sigma^{2}\right)\) distribution. As discussed in Example 4.2.1, the pivot random variable for a confidence interval is $$ t=\frac{\bar{X}-\mu}{S / \sqrt{n}} $$ where \(\bar{X}\) and \(S\) are the sample mean and standard deviation, respectively. Recall by Theorem \(3.6 .1\) that \(t\) has a Student \(t\) -distribution with \(n-1\) degrees of freedom; hence, its distribution is free of all parameters for this normal situation. In the notation of this section, \(t_{n-1}^{(\gamma)}\) denotes the \(\gamma 100 \%\) percentile of a \(t\) -distribution with \(n-1\) degrees of freedom. Using this notation, show that a \((1-\alpha) 100 \%\) confidence interval for \(\mu\) is $$ \left(\bar{x}-t^{(1-\alpha / 2)} \frac{s}{\sqrt{n}}, \bar{x}-t^{(\alpha / 2)} \frac{s}{\sqrt{n}}\right) $$

Short answer

A (1−α)100% confidence interval for μ is given by \( (\bar{X}-t^{(1−α/2)} \frac{S}{\sqrt{n}}, \bar{X}-t^{(α/2)} \frac{S}{\sqrt{n}}) \), where \( t^{(1−α/2)} \) and \( t^{(α/2)} \) are t-scores that cut off an area of α/2 in each tail of the Student's t-distribution with \( n-1 \) degrees of freedom. This interval captures the true population mean μ with (1-α)100% confidence.

Step-by-step solution

Understanding the Dispersion of Samples and Population

Start by understanding that the Student's t-distribution arises when estimating the mean of a normally distributed population, when the sample size is small and population standard deviation is unknown. Here, the pivot variable \(t\) is given by: \( t =\frac{\bar{X}-\mu}{S / \sqrt{n}} \) which follows the Student's t-distribution with \( n-1 \) degrees of freedom, where \( \bar{X} \) is sample mean and \( S \) is sample standard deviation.

Understanding Confidence Intervals

Next, consider the fact that a (1-α)100% confidence interval for μ is defined as a range of values that, with a certain level of certainty ((1-α)100%), contains the true value of a population parameter. The range is calculated such that there is a specific probability ((1-α)100%) that the range contains the true value of μ.

Manipulate the expression for Student's t-distribution

Rewrite the pivot variable \( t \) around the mean: \( \mu =\bar{X}-t \frac{S}{\sqrt{n}} \) . This is because the pivot, being a function of both sample and population parameters, has a known (here, t) distribution even when population parameters are unknown.

Form the confidence interval

If we want the middle area of the t-distribution, then we take the (1−α) proportion, splitting α symmetrically into the tails of the distribution. In other words, we find the t values at degrees of freedom \( (n-1) \) that cut off an area of α/2 in each tail of the t-distribution. This gives us \( t^{(1−α/2)} \) and \( t^{(α/2)} \). Substituting these values for t in our equation gives the limits of the the (1-α) confidence interval for μ. \( \mu =\bar{X}-t^{(1−α/2)} \frac{S}{\sqrt{n}} \) and \( \mu =\bar{X}-t^{(α/2)} \frac{S}{\sqrt{n}} \), these are lower and upper limit of the confidence interval, respectively.

Key concepts

Understanding Student's t-distribution

The Student's t-distribution is a probability distribution that is used to estimate population parameters when the sample size is small and the population standard deviation is unknown. Unlike the normal distribution which is symmetrical and bell-shaped, the t-distribution is slightly different, especially for small sample sizes where it takes on a more spread out and flatter shape.

This means that it accounts for more variability and has heavier tails, indicating a higher probability for extreme values. As the sample size increases, the t-distribution approaches the normal distribution. It's particularly useful because it allows us to make inferences about the population mean when the only data we have from the population is a small sample.

Deciphering Sample Mean

The sample mean, often denoted as \(\bar{X}\), is the average value of a sample, which is calculated by summing all the observations in a sample and dividing by the number of observations. It serves as an unbiased estimator of the population mean, which means that if we were to take many samples and calculate the mean of each, the average of those means would be equal to the true population mean.

The sample mean is key to many statistical procedures, including the calculation of confidence intervals. It is at the heart of the formula used to make estimates about the population mean and plays a pivotal role alongside the standard deviation in the Student's t-distribution.

Delving into Degrees of Freedom

Degrees of freedom are a concept in statistics that refer to the number of values in a calculation that are free to vary. When calculating a statistic such as the sample variance or the t-statistic, not all data points are free to vary once certain parameters (like the sample mean) are fixed.

For instance, in the context of the t-distribution, the degrees of freedom are equal to the sample size minus one \(n - 1\). This accounts for the fact that the sample mean is used in the calculation of the sample variance and standard deviation. The degrees of freedom affect the shape of the t-distribution; with smaller degrees of freedom, the distribution is more spread out, but as the degrees of freedom increase, it more closely resembles the standard normal distribution.