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When we talk about "difference in means," we are discussing the comparison of the average values between two different groups. Imagine you have two sets of data – perhaps scores from two different classes on a math test. Each set of data will have an average or mean score. The key interest often lies in understanding if there's a significant difference in those average scores.

In the context of the exercise, we calculated the difference between the means of two sample groups. The first group had an average (mean) of 256, and the second had an average of 242. The difference in means is simply 256 minus 242, which gives us 14.

This value helps in determining if the groups really are differing from each other in terms of their averages or if any observed difference could be due to random chance. By examining this difference, analysts and researchers can determine if interventions or changes are needed or if the groups are inherently different.

The "standard error" is an important statistical measure that reflects how much the sample mean (average) of the data is expected to vary from the true population mean. It gives us an insight into the reliability of the sample mean as an estimate of the population mean.

In simpler terms, the standard error helps to understand how precise the mean from our sample is likely to be. A smaller standard error indicates that the sample mean is a more accurate reflection of the actual population mean.

In our exercise, we used a given standard error of 6.70. This standard error was derived using a bootstrap distribution, which suggests that we can expect the means not to deviate too much—our sample mean calculation is quite reliable. It's this standard error that was utilized to construct our confidence interval for the difference in means.

The "bootstrap distribution" is a statistical method used to estimate the sampling distribution of a statistic by sampling with replacement from the original data. This method provides a way to perform statistical inference without making strong assumptions about the initial sample data.

Think of it like creating multiple fake datasets from your original data by randomly taking samples (with replacement). Each time you take a sample, you're trying to mimic taking a new sample from the population. This helps in approximating the distribution of the sample mean or other statistics to understand better how they might behave across different samples.

In our confidence interval exercise for the difference in means, the standard error is derived from such a bootstrap distribution. This is particularly useful when the actual distribution is unknown or when dealing with small sample sizes. The approximately normal distribution of our bootstrap approach permits us to apply the z-score in calculating the confidence interval accurately, offering more assurance in the results derived from the data points.