How to Apply Bayesian and Frequentist Methods in Comparative Sales Analysis Assignments

This blog offers a detailed theoretical guide to approaching such comparative simulation-based assignments, helping students build the conceptual foundation and logical reasoning needed to handle similar tasks confidently. Whether you're preparing for an exam or a term project, the methods discussed here will strengthen your approach to assignments where real-world comparisons meet rigorous statistical logic.

Understanding the Assignment Context

These types of assignments typically present a scenario such as:

“Determine which product (e.g., PlayStation 3 vs. Nintendo Wii) was more successful using global sales data over a fixed period.”

You're usually given:

• A choice between different statistical designs (e.g., comparing independent samples vs. paired samples),
• Hypothetical or real distributional assumptions (e.g., normality in log-transformed sales),
• Expectations on effect size (e.g., a log difference of 0.6),
• And a limited sample size (e.g., 54 games per console).

From here, you're expected to define hypotheses, choose priors, perform simulations, and offer data-driven insights—while remaining theoretically grounded.

Step 1: Choosing Between Independent and Paired Comparisons

Explain when to use an independent sample approach versus a paired design based on the structure of the dataset and research goals.

In most assignments like this, you are provided with two analytical approaches:

• Approach A: Compare mean global sales between the two products using independent samples.
• Approach B: Focus on only those games released on both platforms (i.e., a paired design).

This choice matters.

• Independent samples assume the games on PS3 and Wii are different in title and nature.
• Paired samples assume some games exist on both platforms and can be directly compared—game by game.

Key decision point: Use the paired design if overlap exists in the product lineup; otherwise, rely on independent sampling.

Step 2: Formulating Hypotheses

Detail how to frame null and alternative hypotheses for both independent and paired setups, using appropriate notation and assumptions.

In either approach, your first task is to write null and alternative hypotheses. Let’s define some notation:

• Let μ_PS3 be the mean log global sales of games on PlayStation 3.
• Let μ_Wii be the mean log global sales of games on Nintendo Wii.
• Let D = μ_PS3 − μ_Wii (the difference in means).

Independent Samples:

• H₀: μ_PS3 = μ_Wii (no difference in average sales)
• H₁: μ_PS3 ≠ μ_Wii (some difference exists)

Paired Samples:

• H₀: Mean of paired differences = 0
• H₁: Mean of paired differences ≠ 0

This is a purely theoretical setup, but it lays the groundwork for simulation and decision-making.

Step 3: Setting Prior Probabilities

Discuss how to choose prior probabilities (P(H₀), P(H₁)) in Bayesian analysis and how prior knowledge or assumptions influence these values.

Bayesian simulations require that you choose prior probabilities for each hypothesis (H₀ and H₁). This is where you incorporate subjective beliefs or past data.

For example:

• If historical data shows Sony games often outsell Nintendo’s, you might assign P(H₁) = 0.7 and P(H₀) = 0.3.
• If you remain skeptical or neutral, use uninformative priors: P(H₀) = 0.5 and P(H₁) = 0.5.

Theoretical reasoning behind priors is crucial. Don’t just pick numbers—explain them:

“Given that Sony had higher sales in previous generations, we assigned greater prior weight to the alternative hypothesis.”

Step 4: Conducting Simulations

Outline how to simulate datasets under both H₀ and H₁ scenarios using normal distributions and expected parameters.

Here’s where theory meets computation. The idea is to simulate the distribution of sales revenue under both H₀ and H₁, and then assess the likelihood of observed data under each.

a) Simulating Under Independent Samples

Suppose the assignment provides:

• Log sales are normally distributed with μ = 13.7, σ = 1.11 for PS3,
• Effect size under H₁ is 0.6 (i.e., μ_PS3 − μ_Wii = 0.6),
• Sample size = 54 for each platform.

The simulation involves:

• Drawing 10,000 samples of size 54 from N(13.7, 1.11) for PS3,
• Drawing 10,000 samples for Wii from either N(13.1, 1.11) or N(13.7, 1.11) depending on whether H₁ or H₀ holds,
• Computing test statistics (e.g., difference in means),
• Estimating the posterior probability of H₀ and H₁ given the simulated data.

Use Bayes’ Rule:

Posterior H₁ = [Likelihood of data under H₁ × Prior H₁] / [Total probability of data]

b) Simulating Under Paired Samples

• Assume each difference D_i is drawn from N(0.6, 1.0) under H₁ and N(0, 1.0) under H₀,
• Sample 54 such differences,
• Compute the mean of the differences,
• Repeat 10,000 times,
• Again apply Bayes’ rule to estimate posteriors.

Assumptions to document:

• Normality of data,
• Independence of observations (or lack thereof),
• Equal variances if assumed.

Step 5: Interpreting Simulation Outcomes

Guide students on how to analyze posterior probabilities from the simulations and what they imply about the hypotheses.

Once you compute the posterior probabilities of H₀ and H₁, your task becomes interpretive.

Example phrasing:

“Based on our simulation, the posterior probability of H₁ is 0.82, suggesting strong evidence that PlayStation 3 had higher average game revenue.”

Explain:

• Why your posterior makes sense in the context,
• Whether it supports decision-making,
• And the risk of Type I and Type II errors.

Step 6: Evaluating Sample Size Adequacy

Explain how to assess whether the current sample size is statistically sufficient, and how to simulate different sizes to evaluate power.

Assignments often ask:

“Is the current sample size enough?”

You answer this by:

• Simulating different sample sizes (e.g., n = 30, 54, 80, 100),
• Measuring power (i.e., proportion of times H₁ is correctly favored),
• Plotting power curves (theoretical),
• Recommending the smallest sample size that achieves at least 80% power.

“Our simulations show that with 54 observations, we achieve 70% power; increasing to 80 games per platform raises this to 85%, making it a preferable choice.”

Step 7: Comparing Study Designs

Compare the strengths and weaknesses of the independent and paired approaches in terms of power, generalizability, and bias.

Theoretical justification for paired vs. independent design is essential. Here's how to argue this:

• Paired design reduces variability because it controls for game-specific factors.
• Independent design is more generalizable but less sensitive.

A good theoretical explanation:

“Paired designs inherently reduce within-group variance by comparing the same titles across platforms. This leads to more powerful tests, especially when games are not perfectly matched across groups in the independent design.”

Step 8: Communicating to General Audiences

Offer tips on explaining technical results in accessible language for decision-makers, stakeholders, or non-statisticians.

Many assignments ask for explanations accessible to non-technical readers. Keep it simple and persuasive:

“By comparing the same game released on both platforms, we can control for a lot of other factors that might influence sales. This gives us a clearer picture of which console was better for game developers.”

This shows statistical storytelling—an essential academic skill.

Step 9: Formal Statistical Testing

Review how to perform t-tests (independent and paired) and interpret p-values in the context of hypothesis testing.

Most assignments also request a frequentist test (e.g., t-test).

• For Approach A: use a two-sample t-test assuming equal/unequal variance,
• For Approach B: use a paired t-test.

Even in a theoretical blog, mention the test structure:

“We tested the null hypothesis of equal mean log sales using a two-sided paired t-test. The p-value of 0.03 suggests a statistically significant difference in favor of PS3.”

Step 10: Addressing Type I and Type II Errors

Discuss the implications of false positives and false negatives, and how your simulation results relate to the risk of such errors.

Finally, always include a discussion on potential decision errors:

• Type I: False positive—rejecting H₀ when it’s true,
• Type II: False negative—not rejecting H₀ when H₁ is true.

Link this back to your simulation:

“Given the posterior probabilities and observed p-values, it’s unlikely that our conclusion is due to a Type I error. However, if our sample size were smaller, we would face an increased risk of missing a real effect.”

Conclusion

Assignments like these test your ability to:

1. Translate real-world problems into statistical frameworks,
2. Choose appropriate inferential methods (paired vs. unpaired),
3. Simulate scenarios under different assumptions,
4. Balance technical rigor with interpretive clarity.

By carefully setting up your hypotheses, selecting priors, executing simulations, and justifying your recommendations, you can successfully navigate the complexities of comparative simulation-based statistical assignments.

Whether you're comparing gaming consoles, smartphone apps, or e-commerce platforms, this structure helps you produce coherent, theoretically grounded, and statistically defensible work—precisely what your professors (and future employers) look for.