# Interactive and Relevant Applications of Statistics

Hawkes statistics courses include games and simulations that help students apply key concepts to the world outside of the classroom. Check these out below! If you’re an instructor who would like to explore these games and simulations yourself, sign up for free student access today.

## GAMES

### Relevant Application:
This lesson helps students apply the concept of the expected value of a random variable to winning or losing games. Students develop a rational approach to analyzing decisions that involve risk. After all, many business decisions—such as purchasing new equipment, hiring additional employees, and expanding into new markets—involve some kind of risk, and students need to assess these situations as best as they can.

Learn Key Concepts:

• Basic probability distribution
• Binomial distribution function
• Hypergeometric distribution function

### Relevant Application:
Even in today’s digital world, direct mail marketing remains one of the most viable and proven strategies to connect with customers.

Active Learning Approach:
By assuming the role of a direct mail marketing manager, students start off with \$20,000. They are then tasked with developing a strategy by finding mailing lists that will produce sufficient sales, using confidence intervals to determine which lists to use to reach their \$40,000 goal.

They win when they correctly formulate which questions they need to solve, collect the data, and analyze the data to evaluate potential risk and profitability for each mailing list.

Learn Key Concepts:
The game provides an environment in which students apply statistical concepts while making business decisions. They also learn the following:

• Confidence intervals
• Experimentation
• Statistical analysis
• Inference

### Relevant Application:
Students might not realize at first how many decisions involve measurements of a population attribute. For example, television stations base advertising charges on
ratings that reflect the percentage of viewers who watch a particular show. Political analysts are concerned with the fraction of voters who prefer a certain candidate. No
matter the field, estimating population proportions gives us greater insight into the data given to us.

Active Learning Approach:
In the game, students see a box filled with red and blue balls, and are asked to estimate the proportion of red balls in the population. They can draw sample sizes of 20, 50, or
100 to help them estimate the population proportion.

Learn Key Concepts:

• Determine the minimum sample size for a particular confidence level.
• Construct a confidence interval for a population proportion.

### 4. Central Limit Theorem with Proportions Relevant Application:
In many decisions, the variable of interest is a proportion. A university may want to know the fraction of first-year students with low grades in order to provide more support and resources for them. Manufacturers may be concerned with the fraction of parts that are defective.

Active Learning Approach:
Students see a box of red and blue balls, then draw three samples to calculate the sample proportions for each sample taken. Students draw samples again after being informed that samples of first 20 balls and then 40 balls were drawn 200 times to determine the proportion of the number of red balls to the total number of balls chosen. Students then view the data, including tables and histograms, to understand that the sampling distribution of the sampling proportion is approximately normal.

Learn Key Concepts:
Determine p-hat using the Central Limit Theorem for population proportions.

## SIMULATIONS

### Relevant Application:
This concept builder strengthens analytical skills in distribution recognition and data analysis. By detecting symmetric or skewed data, students will begin to understand how to apply this knowledge in the real world.

Active Learning Approach:
Students are asked to identify the type of distribution from a given histogram, frequency/relative frequency distribution, statistics table, or set of sample data. They
can increase the number of intervals on the histogram or frequency distribution, view different sample displays, or choose to view a hint before submitting their answer.

Learn Key Concepts:

• Analyze the histogram, frequency, statistics, and sample data of a distribution.
• Identify different distribution types: uniform, normal, exponential, chi-square, Poisson, and mystery.

### 2. Central Limit Theorem Relevant Application:
This simulation shows students how to use samples to make useful predictions about a population. Since many population sizes are too large to have their data collected and analyzed, we turn to the Central Limit Theorem for help.

The visual nature of this simulation lets students truly comprehend how the sample means from any population are normally distributed, regardless of the original
population’s distribution.

Active Learning Approach:
Students select a parent distribution and set the sample sizes and the burst rate. They choose the desired distribution type: exponential, chi-square, normal, Poisson, or bi-modal. Students can decide to run the simulation a set number of times or automatically, which will keep the simulation running.

Learn Key Concepts:

• Sample population
• Mean
• Variance
• Standard deviation
• Distribution type

### 3. Type II Error Relevant Application:
Understanding hypothesis testing and type II error is essential to fields like evidence-based medicine, quality engineering, and reliability engineering, among others.

Active Learning Approach:
The variance, hypotheses, and critical values are given. Students can increase or decrease the level of significance (α), true mean (μ), and sample size to see how these
changes affect the other factors involved.

Learn Key Concepts:

• Examine the interrelationship between α, sample size, and β (the probability of making a type II error).
• Develop an understanding of the concept of type II errors and the calculation of beta.
• Explore the relationship between α and β.

Are you an instructor who would like to explore these lessons further?