Problem 45 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) are independent random samples from normal distributions with means \(\mu_{X}\) and \(\mu_{Y}\) and known standard deviations \(\sigma_{X}\) and \(\sigma_{Y}\), respectively. Derive a \(100(1-\alpha) \%\) confidence interval for \(\mu_{X}-\mu_{Y}\).

Step-by-Step Solution

Answer

The \(100(1-\alpha) \%\) confidence interval for \(D = \mu_{X} - \mu_{Y}\) is given by \((\overline{X} - \overline{Y}) \pm Z_{\alpha/2}\sqrt{\sigma_{X}^{2}/n + \sigma_{Y}^{2}/m}\).

Step 1: Identify known variables

We know that the random samples \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) are independent and they are taken from normal distributions with means \(\mu_{X}\) and \(\mu_{Y}\) and known standard deviations \(\sigma_{X}\) and \(\sigma_{Y}\), respectively.

Step 2: Calculate sample means

We need to calculate the sample means for both sets of samples. They are given by

Step 3: Define the difference in means

Define \(D = \mu_{X} - \mu_{Y}\), which is the difference in the population means we are interested in.

Step 4: Calculate standard error of the difference

The standard deviation of the difference in the sample means is given by \(\sigma_{D} = \sqrt{\sigma_{X}^{2}/n + \sigma_{Y}^{2}/m}\), where n and m are the sizes of the two samples.

Step 5: Derive confidence interval

The \(100(1-\alpha) \%\) confidence interval for \(D = \mu_{X} - \mu_{Y}\) is given by \((\overline{X} - \overline{Y}) \pm Z_{\alpha/2}\sigma_{D}\), where \(Z_{\alpha/2}\) is the z-value that captures the middle \(100(1-\alpha) \%\) area under the standard normal curve.

Key Concepts

Understanding Normal Distribution

A normal distribution is a common way to describe data that clusters around a central mean value. Here's why it's important:

• The normal distribution is symmetric, meaning it looks the same on both sides of the mean.
• Most of the data tends to be close to the mean, with fewer cases appearing as you move away.
• It follows a bell-shaped curve, known as the Gaussian curve.

In many natural phenomena, data is expected to cluster in this way. For example, heights, test scores, and other measurable traits often show a normal distribution.

This distribution makes statistical analysis simpler and more predictable, especially when making inferences about a population based on a sample.

The Importance of Sample Mean

The sample mean is a key value in statistics, representing the average of a set of data points collected from a larger population.

• It is calculated by summing all the data values and dividing by the number of values.
• The sample mean serves as an unbiased estimator of the population mean.

This value allows researchers to make educated guesses about the overall population. In the context of the problem, finding the sample mean for each dataset (\(\overline{X}\) for the sample of \(X\)s and \(\overline{Y}\) for the sample of \(Y\)s) helps compare their central tendencies.

The closer the sample mean is to the true population mean, the more accurate the estimate will be.

Exploring Standard Error

Standard error measures the precision of the sample mean when estimating the population mean.

• It is calculated as the standard deviation of the sampling distribution.
• For the difference in means in our problem, it is found using: \(\sigma_{D} = \sqrt{\sigma_{X}^{2}/n + \sigma_{Y}^{2}/m}\).

The standard error tells us how much we can expect the sample mean to fluctuate from the population mean. A smaller standard error indicates more reliable, precise estimates.

In practical terms, it helps establish the boundaries of our confidence interval, which we use to make predictions about the population difference \(D = \mu_{X} - \mu_{Y}\).

Understanding the Z-value

The Z-value is a critical part of calculating confidence intervals. It corresponds to a position under the standard normal distribution curve.

• For a given confidence level, the Z-value determines how far the sample mean is from the population mean.
• For a 95% confidence interval, \(Z_{0.025}\) is typically used, as it leaves 2.5% in each tail of the distribution.

This value helps quantify the certainty of the interval estimate, indicating how far we must look under the standard normal curve to enclose the desired coverage.

Thus, using the Z-value, we can derive bounds that reflect our confidence in how close the sample statistics are to the actual population parameters.