Constructing A 95% Confidence Interval For Difference Of Means

In the realm of statistical analysis, comparing the means of two populations is a common and crucial task. We often encounter scenarios where we need to determine if there's a significant difference between the average values of two groups. For example, we might want to compare the average test scores of students taught using two different methods or the average income of people in two different cities. To achieve this, we use confidence intervals for the difference of means, which provide a range of plausible values for the true difference between the population means. In this comprehensive guide, we will walk through the process of constructing a 95% confidence interval for the difference of two population means (${\mu_1 - \mu_2}$), given sample statistics from two independent samples drawn from normally distributed populations.

When comparing two population means (${\mu_1}$ and ${\mu_2}$), the goal is to estimate the difference between these means using sample data. A confidence interval provides a range of values within which the true difference is likely to lie. The 95% confidence level indicates that if we were to repeat the sampling process multiple times and construct confidence intervals each time, approximately 95% of these intervals would contain the true difference between the population means. This statistical tool is invaluable in various fields, including healthcare, engineering, and social sciences, allowing researchers and analysts to make informed decisions based on data.

Understanding the Fundamentals: Sample Statistics and Confidence Levels

Before diving into the calculation, let's define some key terms. Sample statistics are values calculated from the sample data, used to estimate population parameters. In this case, we have two samples with sizes ${n_1}$ and ${n_2}$, sample means ${\bar{x}_1}$ and ${\bar{x}_2}$, and sample standard deviations ${s_1}$ and ${s_2}$. The sample means are the averages of the data points in each sample, while the sample standard deviations measure the spread or variability within each sample. The confidence level, typically expressed as a percentage (e.g., 95%), represents the probability that the confidence interval contains the true population parameter. A higher confidence level results in a wider interval, reflecting a greater degree of certainty.

In the given scenario, we have two independent samples drawn from normal populations. Sample 1 has a size of ${n_1 = 11}$ and a sample mean of ${\bar{x}_1 = 4.8}$. Sample 2 has a size of ${n_2 = 18}$ and a sample mean of ${\bar{x}_2 = 7.9}$. Additionally, we have the sample standard deviations: ${s_1 = 1.7}$ and ${s_2 = 2.9}$. Our objective is to construct a 95% confidence interval for the difference between the population means, ${\mu_1 - \mu_2}$. This interval will provide a range of plausible values for the difference, helping us determine if there is a statistically significant difference between the two population means. The confidence interval is built around the point estimate, which is the difference between the sample means, and its width depends on the standard error of the difference and the desired confidence level.

Steps to Construct the Confidence Interval

To construct the 95% confidence interval for the difference in means, we will follow a step-by-step approach. This process involves identifying the appropriate formula, calculating the necessary statistics, and interpreting the results. We need to determine whether to use a z-interval or a t-interval, based on whether the population standard deviations are known or unknown. In this case, since we are given sample standard deviations and the population standard deviations are unknown, we will use a t-interval. The t-interval is more appropriate when dealing with small sample sizes or unknown population standard deviations, as it accounts for the additional uncertainty introduced by estimating the standard deviations from the samples.

Step 1: Determine the appropriate formula

Since the population standard deviations are unknown, we use the t-distribution. The formula for the confidence interval for the difference of two means with unknown population standard deviations is:

${(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2, df} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Where:

• ${\bar{x}_1}$ and ${\bar{x}_2}$ are the sample means.
• ${s_1}$ and ${s_2}$ are the sample standard deviations.
• ${n_1}$ and ${n_2}$ are the sample sizes.
• ${t_{\alpha/2, df}}$ is the t-critical value with ${df}$ degrees of freedom and a significance level of ${\alpha/2}$.
• The degrees of freedom (df) are calculated using the Welch-Satterthwaite equation, which accounts for the unequal variances and sample sizes:

${df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}}$

This formula provides a more accurate estimate of the degrees of freedom when the sample sizes and variances are unequal, ensuring that the resulting confidence interval is reliable. The degrees of freedom play a crucial role in determining the shape of the t-distribution, which in turn affects the critical value used in the confidence interval calculation. A lower degrees of freedom results in a wider interval, reflecting the increased uncertainty associated with smaller sample sizes.

Step 2: Calculate the degrees of freedom

First, we need to calculate the degrees of freedom (${df}$) using the Welch-Satterthwaite equation:

${df = \frac{\left(\frac{1.7^2}{11} + \frac{2.9^2}{18}\right)^2}{\frac{\left(\frac{1.7^2}{11}\right)^2}{11 - 1} + \frac{\left(\frac{2.9^2}{18}\right)^2}{18 - 1}}}$

${df = \frac{\left(\frac{2.89}{11} + \frac{8.41}{18}\right)^2}{\frac{\left(\frac{2.89}{11}\right)^2}{10} + \frac{\left(\frac{8.41}{18}\right)^2}{17}}}$

${df = \frac{\left(0.2627 + 0.4672\right)^2}{\frac{\left(0.2627\right)^2}{10} + \frac{\left(0.4672\right)^2}{17}}}$

${df = \frac{\left(0.7299\right)^2}{\frac{0.0689}{10} + \frac{0.2183}{17}}}$

${df = \frac{0.5328}{0.00689 + 0.01284}}$

${df = \frac{0.5328}{0.01973}}$

${df \approx 27.0056}$

We round the degrees of freedom down to the nearest whole number, so ${df = 27}$.

Calculating the degrees of freedom accurately is crucial because it determines the shape of the t-distribution, which in turn affects the critical value used in the confidence interval. The Welch-Satterthwaite equation provides a more precise estimate of the degrees of freedom when the sample sizes and variances are unequal, ensuring that the resulting confidence interval is reliable. By rounding down to the nearest whole number, we err on the side of caution, which leads to a slightly wider and more conservative confidence interval. This approach is preferred because it reduces the risk of underestimating the uncertainty in our estimate of the difference between the population means.

Step 3: Find the t-critical value

For a 95% confidence interval, the significance level ${\alpha = 1 - 0.95 = 0.05}$, so ${\alpha/2 = 0.025}$. We look up the t-critical value in the t-distribution table for ${df = 27}$ and ${t_{0.025, 27}}$. The t-critical value is approximately 2.052.

The t-critical value is a key component in constructing the confidence interval, as it determines the margin of error. This value is obtained from the t-distribution table, which provides critical values for various degrees of freedom and significance levels. The t-distribution is similar to the standard normal distribution but has heavier tails, which accounts for the additional uncertainty when estimating population standard deviations from sample data. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution. The t-critical value of 2.052 indicates how many standard errors we need to extend from the point estimate (the difference in sample means) to capture the true difference between the population means with 95% confidence.

Step 4: Calculate the confidence interval

Now we can plug the values into the formula:

${(4.8 - 7.9) \pm 2.052 \sqrt{\frac{1.7^2}{11} + \frac{2.9^2}{18}}}$

${-3.1 \pm 2.052 \sqrt{\frac{2.89}{11} + \frac{8.41}{18}}}$

${-3.1 \pm 2.052 \sqrt{0.2627 + 0.4672}}$

${-3.1 \pm 2.052 \sqrt{0.7299}}$

${-3.1 \pm 2.052 \times 0.8543}$

${-3.1 \pm 1.753}$

Therefore, the 95% confidence interval is:

${-3.1 - 1.753 = -4.853}$

${-3.1 + 1.753 = -1.347}$

So, the 95% confidence interval for ${\mu_1 - \mu_2}$ is (-4.853, -1.347).

By plugging the values into the formula, we obtain the lower and upper bounds of the confidence interval. The margin of error, calculated as the product of the t-critical value and the standard error, determines the width of the interval. In this case, the margin of error is 1.753, which is added to and subtracted from the point estimate (-3.1) to obtain the confidence interval. The resulting interval (-4.853, -1.347) provides a range of plausible values for the true difference between the population means. This interval is crucial for making inferences and decisions about the populations being compared.

Interpreting the Confidence Interval

The 95% confidence interval for the difference between the means ${\mu_1 - \mu_2}$ is (-4.853, -1.347). This means we are 95% confident that the true difference between the population means lies within this range. Since the interval does not contain 0, we can conclude that there is a statistically significant difference between the two population means at the 0.05 significance level. Specifically, since both bounds of the interval are negative, we can infer that ${\mu_1}$ is likely smaller than ${\mu_2}$.

Interpreting the confidence interval correctly is essential for drawing meaningful conclusions from the data. The fact that the interval does not include zero provides strong evidence that the two population means are different. If the interval had included zero, we would not have sufficient evidence to conclude that the means are different. In practical terms, the confidence interval helps us quantify the magnitude of the difference between the means. In this case, we can say with 95% confidence that the mean of population 1 is between 1.347 and 4.853 units smaller than the mean of population 2. This information can be used to inform decisions and guide further research.

Conclusion

Constructing a confidence interval for the difference of means is a powerful statistical tool for comparing two populations. By following the steps outlined above, we can accurately calculate and interpret the interval, providing valuable insights into the differences between population means. In this specific example, we constructed a 95% confidence interval for ${\mu_1 - \mu_2}$ and found it to be (-4.853, -1.347), indicating a statistically significant difference between the two population means.

Understanding confidence intervals and their construction is crucial for anyone working with data and making data-driven decisions. The ability to compare means and quantify the uncertainty in those comparisons allows for more informed judgments and conclusions. The process of constructing a confidence interval involves several steps, each requiring careful attention to detail. From determining the appropriate formula to calculating the degrees of freedom and finding the critical value, each step contributes to the accuracy and reliability of the final result. By mastering these steps, analysts and researchers can effectively use confidence intervals to make meaningful inferences about population parameters.

This comprehensive guide has provided a detailed explanation of how to construct a 95% confidence interval for the difference of means when population standard deviations are unknown. By following these steps, you can confidently analyze data and draw meaningful conclusions about the populations you are studying.