Question

Compute the standard error of \(\left(\bar{Y}_{1}-\bar{Y}_{2}\right)\) for the following data: $$ \begin{array}{|ccc|} \hline & \text { Sample 1 } & \text { Sample 2 } \\ \hline n & 5 & 7 \\ \bar{y} & 44 & 47 \\ s & 6.5 & 8.4 \\ \hline \end{array} $$

Short answer

The standard error of \((\bar{Y}_{1}-\bar{Y}_{2})\) is approximately 4.303.

Step-by-step solution

Title - Understanding Standard Error of the Difference Between Two Means

The standard error of the difference between two sample means \(\bar{Y}_{1}-\bar{Y}_{2}\) is calculated using the formula \[\text{SE}(\bar{Y}_{1}-\bar{Y}_{2}) = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\] where \(s_1\) and \(s_2\) are the standard deviations of the two samples, and \(n_1\) and \(n_2\) are the sample sizes of the two samples.

Title - Plugging Given Values into the Formula

Use the given values from the table to plug into the formula for the standard error. For Sample 1: \(n_1 = 5\), \(s_1 = 6.5\); for Sample 2: \(n_2 = 7\), \(s_2 = 8.4\). Substituting into the formula gives us \[\text{SE}(\bar{Y}_{1}-\bar{Y}_{2}) = \sqrt{\frac{6.5^2}{5} + \frac{8.4^2}{7}}\].

Title - Calculating the Standard Error

Compute the standard error by solving the equation: \[\text{SE}(\bar{Y}_{1}-\bar{Y}_{2}) = \sqrt{\frac{6.5^2}{5} + \frac{8.4^2}{7}} = \sqrt{\frac{42.25}{5} + \frac{70.56}{7}} = \sqrt{8.45 + 10.08} = \sqrt{18.53} \approx 4.303\].

Key concepts

Understanding Standard Deviation

At its core, standard deviation is a measure of how spread out the numbers in a data set are. Think of it like the average distance each number lies from the mean - the middle point of all the numbers.

For instance, if we were looking at heights of different people, a small standard deviation would mean most people are about the same height, whereas a large standard deviation would indicate heights that vary greatly from really short to really tall.

Why is it important in statistics?

Standard deviation helps us understand the diversity and variability in our data. Imagine we are looking at the test scores of two classes. If one class has a low standard deviation, most students scored similarly, while a high standard deviation indicates that some students did really well, whereas others did not.

In the exercise given, the standard deviations for the two samples are 6.5 and 8.4, respectively. These values are key in measuring the reliability of the mean and are used in calculating the standard error of the difference between two means, which tells us how much confidence we can have in the difference observed between these two sample means.

The Role of Sample Size in Statistics

The sample size, denoted as 'n' in many statistical formulas, is basically the number of observations or measurements you have. Imagine you're trying to figure out the favorite ice cream flavor of kids in a school. If you only ask three kids, your sample size is tiny, and you can't be sure they represent all kids. But if you ask a hundred kids, you have more confidence in your conclusion.

When it comes to calculating statistics, like in our standard error example, a larger sample size usually gives more reliable results. This is because the effect of outliers or unusual data points is reduced.

How does sample size affect standard error?

The standard error decreases when the sample size increases because you're spreading the variability (standard deviation) across more data points. This means we can trust our estimates more. In the exercise we're looking at, the sample sizes are 5 and 7. These aren't large sample sizes, so our standard error will be larger than if we had, let's say, 50 or 70 data points.

Navigating Statistical Formulas

Statistical formulas are like recipes that help us cook up answers from our data ingredients. They guide us in how to process data to perform various analyses.

Some of these formulas can look daunting with symbols and variables everywhere, but once you get the hang of what each part represents, it's like following a map to treasure. Remember, every symbol has a meaning, and each formula serves a purpose, whether it's finding an average or understanding the variability of data.

Applying formulas to our exercise

In our exercise, we applied a specific statistical formula to calculate the standard error of the difference between two means. We had all our ingredients: the standard deviations (s1 and s2), and the sample sizes (n1 and n2). The formula helped us combine these ingredients to find out how much the difference between two sample means might vary simply due to chance.

Understanding the purpose of each variable in the formula and accurately placing our data into it brought us closer to measuring the reliability of our observed differences. Formulas are the toolbox of statistics, allowing us to extract meaningful information, make predictions, and ultimately, drive decision-making based on data.