Lesson 2: Making Inferences from Data

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

An inference is a conclusion about a population based on sample data. Valid inferences can only be made from unbiased samples that are representative, random, and sufficiently large.

Examples

Property

Inferences about a population based on a random sample are estimates, not exact values, due to sampling variability. Therefore, conclusions should be stated using approximate language rather than as precise values.

Examples

• If the mean weight of 20 randomly sampled apples is $152$ grams, a valid inference is: "The mean weight of all apples from this orchard is about $152$ grams."
• If 3 out of 50 randomly selected light bulbs are defective, a valid inference is: "It is likely that around $6\%$ of all light bulbs produced are defective."

Explanation

When we use a sample to learn about a larger population, the results will vary from one sample to another. This is called sampling variability. Because of this, we cannot state our conclusions with absolute certainty. Using words like "about," "around," or "most likely" acknowledges this uncertainty and makes our inferences more accurate and reasonable.