Lesson 4-4: Use Multiple Samples to Describe Accuracy

Property

The mean is the sum of the values in a data set divided by the number of values in the set.

Property

To understand sampling variability, multiple random samples can be drawn from the same population. Each sample, while random, will likely consist of different members of the population. Consequently, statistics calculated from each sample, such as the sample mean or proportion, will vary from sample to sample.

Examples

A teacher wants to estimate the average study time of their 20 students. They decide to draw multiple random samples of size $n=4$.

• Sample 1: Students {A, G, M, T} with a mean study time of 4.5 hours/week.
• Sample 2: Students {D, E, P, R} with a mean study time of 5.2 hours/week.
• Sample 3: Students {B, F, K, S} with a mean study time of 4.8 hours/week.

From a bag containing 50 red marbles and 50 blue marbles, three random samples of 10 marbles are drawn.

• Sample 1: 6 red, 4 blue (Proportion of red = 0.6)
• Sample 2: 3 red, 7 blue (Proportion of red = 0.3)
• Sample 3: 5 red, 5 blue (Proportion of red = 0.5)

Explanation

Generating multiple random samples is a method used to observe and understand sampling variability. By taking several different samples from the same population, we can see how sample statistics, like the mean or proportion, naturally vary. This process demonstrates that while any single random sample provides an estimate, the collection of multiple sample estimates tends to cluster around the true population parameter. This reinforces the idea that random sampling is a reliable process, even though individual sample results will differ.

Property

To estimate the overall population mean ($\mu$), you can calculate the mean of multiple sample means. If you have $k$ different sample means ($\bar{x}_1, \bar{x}_2, \dots, \bar{x}_k$), their mean is calculated as:

Property

Variability is a measure of how much samples or data differ from each other.
Understanding variability in samplings allows students the opportunity to estimate or even measure the differences.
Gauging how far off an estimate or prediction might be is a way to address issues of variation in samples.

Examples

• One random sample of 10 students reads an average of 3 books a month. A second sample of 10 students reads an average of 4 books. The difference is due to sampling variability.
• A pollster asks two separate random groups of 100 people if they like a new movie. In the first group, 65% say yes. In the second group, 70% say yes. This 5% difference shows variability.
• If you roll a die 12 times, you expect to get two 4s. But one time you might get one 4, and the next you might get three. This fluctuation is variability.

Explanation

Variability means that different samples from the same population won't be exactly alike. It's the natural, expected difference between samples. Understanding it helps you know how much you can trust your predictions and how much they might change.