Question

Data from two samples gave the following results: $$ \begin{array}{|lcc|}\hline & \text { Sample 1 } & \text { Sample 2 } \\\\\hline \bar{y} & 96.2 & 87.3 \\\\\mathrm{SE} & 3.7 & 4.6 \\\\\hline\end{array}$$ $$ \text { Compute the standard error of }\left(\bar{Y}_{1}-\bar{Y}_{2}\right) $$.

Short answer

The standard error of the difference between the two sample means is approximately 5.9.

Step-by-step solution

Understand the Concept of Standard Error for Difference in Means

The standard error (SE) of the difference between two sample means is used to measure the spread of the sampling distribution of that difference. The formula to calculate the standard error of the difference between two independent sample means is \( SE(\bar{Y}_1 - \bar{Y}_2) = \sqrt{SE_1^2 + SE_2^2} \), where \( SE_1 \) and \( SE_2 \) are the standard errors of the two samples, respectively.

Plug in the Given Values

Use the standard errors given for Sample 1 and Sample 2 to calculate the standard error of the difference between the two sample means. Given \( SE_1 = 3.7 \) and \( SE_2 = 4.6 \), plug these values into the formula: \( SE(\bar{Y}_1 - \bar{Y}_2) = \sqrt{3.7^2 + 4.6^2} \).

Calculate the Standard Error of the Difference

Perform the calculation: \( SE(\bar{Y}_1 - \bar{Y}_2) = \sqrt{3.7^2 + 4.6^2} = \sqrt{13.69 + 21.16} = \sqrt{34.85} \approx 5.9 \). Round to an appropriate number of significant figures based on the context.

Key concepts

Sampling Distribution

When conducting experiments or research, understanding the concept of a sampling distribution is crucial. Imagine you draw a multitude of samples, each one the same size, from a particular population. Each of these samples has its own mean, and if we were to plot all these sample means, we’d get what is called the sampling distribution of the mean. In essence, it is a probability distribution of all possible means that could be obtained from a given sample size.

Importantly, the central limit theorem tells us that, regardless of the population's shape, the sampling distribution of the mean will tend to be normally distributed if the sample size is large enough. This holds true especially if the samples are independent of each other and are drawn from a population with a finite level of variance. By knowing this, we can move on to understanding why the standard error is so important in statistical analysis.

Sample Means

The sample mean, often denoted as \( \bar{y} \), is simply the average of all the measurements in a sample. It serves as an estimator of the population mean. For example, if you take a random sample of people and measure their heights, the sample mean height would give you a good idea of the average height in the entire population if your sample is truly representative.

However, we have to consider that different samples will yield different means. This variability among sample means can tell us a lot about the population's characteristics and the precision of our sample mean as an estimate of the population mean. The more samples we take, the more we can be confident about our estimates; that's where the concept of the standard error comes into play, which measures the variation of the sample means.

Standard Error Calculation

The standard error is a statistical term that measures the accuracy with which a sample distribution represents a population by using standard deviation. In simpler terms, it tells us how far off we can expect our sample mean to be from the true population mean. The calculation for the standard error of a single sample mean is \( \frac{\sigma}{\sqrt{n}} \) where \( \sigma \) is the population standard deviation and \( n \) is the sample size.

For the case involving two sample means, the standard error of the difference between these means helps us understand how much variation exists between the two. It's important when comparing two groups. You calculate it using the formula provided in the solution you mentioned: \( SE(\bar{Y}_1 - \bar{Y}_2) = \sqrt{SE_1^2 + SE_2^2} \). This calculation is essential when we're looking to gauge whether a difference between two sample means is statistically significant or if it might just be due to random chance.