Question

Data from two samples gave the following results: $$\begin{array}{|lcc|}\hline & \text { Sample 1 } & \text { Sample 2 } \\\\\hline n & 22 & 21 \\\\\bar{y} & 1.7 & 2.4 \\\\\text { SE } & 0.5 & 0.7 \\\\\hline \end{array}$$ Compute the standard error of \(\left(\bar{Y}_{1}-\bar{Y}_{2}\right)\).

Short answer

The standard error of \(\left(\bar{Y}_{1} - \bar{Y}_{2}\right)\) is approximately 0.86.

Step-by-step solution

Understand the concept of standard error of the difference between two means

The standard error of the difference between two independent means can be found by using the formula: \( SE_{\bar{Y}_{1} - \bar{Y}_{2}} = \sqrt{SE_{\bar{Y}_{1}}^2 + SE_{\bar{Y}_{2}}^2} \). This formula assumes that the two samples are independent of each other.

Plug values into the formula

Take the standard errors for each sample provided in the data and substitute them into the formula: \( SE_{\bar{Y}_{1} - \bar{Y}_{2}} = \sqrt{0.5^2 + 0.7^2} \).

Calculate the standard error of the difference

Calculate the square of each standard error, add them together, and take the square root of the sum to obtain the standard error of the difference between the two sample means: \( SE_{\bar{Y}_{1} - \bar{Y}_{2}} = \sqrt{0.5^2 + 0.7^2} = \sqrt{0.25 + 0.49} = \sqrt{0.74} \approx 0.86 \).

Key concepts

Standard Error

The standard error (SE) is a critical statistic that provides insight into the precision of a sample mean as an estimate of the population mean. It's fundamentally the measure of the variability within a sampling distribution of the sample mean. Understanding SE aids students in grasping how sampling variability influences their statistical estimates.

In the context of two independent samples, each with its own sample mean and standard error, the calculation of the SE for the difference between the two means is a practical extension. The formula you worked on in the exercise, \( SE_{\bar{Y}_{1} - \bar{Y}_{2}} = \sqrt{SE_{\bar{Y}_{1}}^2 + SE_{\bar{Y}_{2}}^2} \), combines the individual standard errors in a manner reflecting their cumulative impact on the difference between sample means. For those less math-inclined, think of the standard error as a gauge for the 'wiggle-room' in your estimates.

Independent Samples

Independent samples refer to distinct groups that have been measured once and have no overlap between them. Each individual in Sample 1 is unrelated to any individual in Sample 2. Why does this matter? If samples are not independent (i.e., they are related), the analysis would require other, more complex statistical methods.

In your exercise, the notion of independence ensures that the calculation of the standard error for the difference between means doesn't need to account for any potential correlation between the groups. This simplicity allows for the straightforward formula used to find the SE of the difference between means. When conducting research or experimenting, always verify that your samples are indeed independent to avoid analytical errors.

Sample Means

Sample means (\(\bar{Y}\)) are, quite simply, the averages obtained from individual samples. These averages are powerful statistics that provide a snapshot of a given dataset, representing the central tendency. When comparing two sets of data, the differences between their sample means can reveal variations that might exist between the populations they're drawn from.

The exercise provided gives two sample means: 1.7 for Sample 1 and 2.4 for Sample 2. Calculating the standard error of the difference between these means offers a perspective on the variability we might expect if we were to repeatedly calculate the difference between means from similar samples. The sample mean holds significance as a stepping-stone towards broader conclusions about the populations of interest.

Statistical Inference

Statistical inference is the process through which we make predictions or decisions about population parameters based on sample statistics. By utilizing the sample means and their standard errors, you're engaging in statistical inference when you estimate whether the observed difference between sample means reflects a true difference in the population.

In your exercise, after finding the standard error of the difference between two means, this SE can then be used to create confidence intervals or conduct hypothesis tests, furthering our understanding of the populations at hand. The ability to draw conclusions about a population, based on sample data, is the cornerstone of statistical science, and is what makes this exercise more than just a mathematical endeavor — it's a window into a larger reality.