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The population mean is a crucial statistical measure that represents the average of every single value in a population. In our example, the population consists of the following video rental numbers for five housemates: 8, 14, 16, 10, and 11. To find the population mean, you add up all of these numbers and then divide by the total number of data points in the population, which in this case is 5.

Here's how to calculate it:

• First, sum up all the numbers: 8 + 14 + 16 + 10 + 11 = 59.
• Then, divide the sum by the total number of values (5): \[\text{Population Mean} = \frac{59}{5} = 11.8.\]

The population mean is useful for understanding the overall behavior of the data set as it provides a central value around which all the other data points are spread. This tells us that, on average, each housemate rented approximately 11.8 videos during the academic year.

The sample mean is an estimate of the population mean, based on a subset of the data. When you're dealing with large populations, it's often impractical to measure every data point, so you take a sample instead. For example, consider taking a random sample of size 2 from our population of video rentals.

You might randomly pick numbers like 14 and 16. To compute the sample mean for these selected values, add them up and divide by the sample size (which is 2 in this example).

• Add the selected numbers: 14 + 16 = 30.
• Compute the mean: \[\text{Sample Mean} = \frac{30}{2} = 15.\]

Sample means can vary depending on which elements of the population are selected. In the larger exercise, you would perform this calculation for 25 different samples, each time potentially getting a different sample mean. This variability provides insight into the distribution of sample means around the population mean, highlighting one of the fundamental concepts in statistics: while individual samples might vary, collectively, they provide a reliable estimate of the broader population.

A density histogram is a graphical tool that allows us to visualize the distribution of sample means, in this case, obtained from the 25 different samples, each of size 2. By plotting these sample means, you can see how they are distributed relative to each other and how they compare to the population mean of 11.8 computed earlier.

When creating a density histogram:

• First, mark the horizontal axis with the range of sample mean values.
• Then, on the vertical axis, indicate the frequency or density of each sample mean.
• Each bar in the histogram represents the number of sample means falling within a certain range or interval.

Analyzing the density histogram will help answer two key questions:

• Do most of the sample means cluster around the population mean of 11.8?
• Is there much variation among the sample means?

If the histogram shows most sample means near the population mean and with relatively low variability, it demonstrates that sample means tend to be consistent and reliable estimates of the population mean. This characteristic is a fundamental reason why sampling is such a powerful method in statistics.