Question

Assume you have two populations \(\mathrm{N}\left(\mu_{1}, \sigma^{2}\right)\) and \(\mathrm{N}\left(\mu_{2}, \sigma^{2}\right)\). The distributions have the same, but unknown, variance \(\sigma^{2}\). Derive a method for determining a confidence interval for $$ \mu_{1}-\mu_{2} $$

Short answer

To derive a confidence interval for \(\mu_{1} - \mu_{2}\), calculate the sample means, pooled variance, degrees of freedom, standard error, and t-value for the desired confidence level. Then, compute the margin of error and use it to find the confidence interval: \((\bar{x}_{1} - \bar{x}_{2}) \pm \mathrm{ME}\).

Step-by-step solution

Identify sample means and sample sizes

First, we need to identify the sample means and sample sizes from the two populations. Let's say we have two random samples from the two populations. Let their sample means be \(\bar{x}_{1}\) and \(\bar{x}_{2}\), and their sample sizes be \(n_{1}\) and \(n_{2}\), respectively.

Calculate the pooled variance

We will calculate the pooled variance, which is the weighted average of the two sample variances. The sample variances are denoted as \(s_{1}^{2}\) and \(s_{2}^{2}\). The pooled variance formula is given by: \[ s_{p}^{2} = \frac{(n_{1} - 1)s_{1}^{2} + (n_{2} - 1)s_{2}^{2}}{n_{1} + n_{2} - 2} \] Compute the pooled variance using the given sample variances.

Calculate the degrees of freedom

The degrees of freedom (df) are calculated using the formula: \[ \mathrm{df} = n_{1} + n_{2} - 2 \] Compute the degrees of freedom using the given sample sizes.

Calculate the standard error

Next, we need to calculate the standard error of the difference in means. The formula for the standard error is given by: \[ \mathrm{SE}(\bar{x}_{1} - \bar{x}_{2}) = \sqrt{\frac{s_{p}^{2}}{n_{1}} + \frac{s_{p}^{2}}{n_{2}}} \] Compute the standard error using the pooled variance and the sample sizes.

Find the t value for the desired confidence level

In order to calculate the confidence interval, we need to find the t value that corresponds to the desired confidence level and the calculated degrees of freedom. Use a t-table or a statistical software to find the appropriate t value.

Calculate the margin of error

With the standard error and the t value, we can calculate the margin of error, which will give us the range of our confidence interval. The formula for the margin of error is: \[ \mathrm{ME} = t \cdot \mathrm{SE}(\bar{x}_{1} - \bar{x}_{2}) \] Compute the margin of error.

Calculate the confidence interval

Finally, to calculate the confidence interval of the difference in population means, use the sample means and the margin of error. The formula for the confidence interval is: \[ (\bar{x}_{1} - \bar{x}_{2}) \pm \mathrm{ME} \] Compute the confidence interval using the above formula, which will give you the range in which you can be confident that the true difference in population means lies.

Key concepts

Pooled Variance

When comparing two populations that are assumed to have the same variance, a key step in determining the confidence interval for the difference in their means is calculating the pooled variance. This value is a weighted average of the individual sample variances. It represents an estimate of the common variance of the two populations.

To calculate the pooled variance, we use the formula:
\[ s_{p}^{2} = \frac{(n_{1} - 1)s_{1}^{2} + (n_{2} - 1)s_{2}^{2}}{n_{1} + n_{2} - 2} \]

This formula considers the variances from both samples, denoted as \(s_{1}^{2}\) and \(s_{2}^{2}\), and the sample sizes minus one, which accounts for the degrees of freedom in each sample. By pooling the variances, researchers attain a more accurate estimate of the underlying variability when the true population variances are unknown but assumed equal.

Standard Error

Once we've established the pooled variance, the next step is to determine the standard error of the difference in sample means. This standard error serves as a measure of how much variance or 'noise' there is within our estimate of the difference between the two population means.

The formula for the standard error of the difference in means is:
\[ \mathrm{SE}(\bar{x}_{1} - \bar{x}_{2}) = \sqrt{\frac{s_{p}^{2}}{n_{1}} + \frac{s_{p}^{2}}{n_{2}}} \]

This equation shows that the standard error is influenced by the size of the samples – larger samples will typically lead to a smaller standard error, indicating a more precise estimate of the population mean difference.

Degrees of Freedom

The concept of degrees of freedom (df) is fundamental in the calculation of the confidence interval as it affects the calculation of the t-value, which we'll discuss shortly. Degrees of freedom are essentially the number of independent values or quantities which can be assigned to a statistical distribution.

In the context of estimating a confidence interval for the difference in means with pooled variances, the formula for degrees of freedom is:
\[ \mathrm{df} = n_{1} + n_{2} - 2 \]

The degrees of freedom here are based on the aggregate sample sizes of both groups minus two. This subtraction accounts for the estimated parameters (one mean from each group). The df is crucial for determining the critical t-value from the t-distribution that corresponds to our desired level of confidence.

T-Value

To create a confidence interval, we rely not only on the standard error but also on a multiplier known as the t-value. This is derived from the t-distribution, which is used especially when dealing with small sample sizes or unknown population variances.

To find the t-value, refer to a t-table or statistical software, looking up the critical value that corresponds to the desired confidence level (e.g., 95%) and the calculated degrees of freedom. The t-value, in conjunction with the standard error, gives us a measure of how far from the sample mean difference we expect the true mean difference to lie, assuming our estimated pooled variance is accurate.

Margin of Error

The margin of error is an expression of the extent of uncertainty in our estimate of the difference in population means. It represents the radius of the confidence interval, thus providing a range around our observed sample mean difference within which we expect the true population mean difference to fall.

It is calculated using the following formula:
\[ \mathrm{ME} = t \cdot \mathrm{SE}(\bar{x}_{1} - \bar{x}_{2}) \]

Here, the t-value dictates how wide our confidence range should be for a given level of confidence. Larger margins of error generally indicate less precise estimates of the population mean difference. The final confidence interval is then constructed by adding and subtracting the margin of error from the sample mean difference.