HW5: Pixel Quest Performance Analytics

A Normal Distribution Investigation

Author

Dr. Marcela Alfaro Córdoba

🎮 The Case

Welcome back, Statistical Detective!

Your discrete probability work impressed GameZone so much that Alex Rivera has a new challenge for you. The analytics team has been collecting continuous data—player reaction times, session lengths, loading speeds, and more—and they need help understanding these measurements.

“Detective, we’ve moved beyond counting events. Now we’re measuring performance metrics that can take any value within a range. Our game designers need to know what ‘normal’ looks like, how to spot outliers, and how to make predictions about player averages. We need someone who understands the normal distribution and sampling to help us optimize the player experience. Ready for the next level?”

Your mission: Use the normal distribution and Central Limit Theorem to help GameZone analyze continuous player data and make data-driven decisions!

Question 1: Normal Distribution Foundations

a. Define these terms in the context of GameZone:

  • Normal Distribution:

  • Z-score:

  • Empirical Rule (68-95-99.7 Rule):

b. Sketch and label:

Draw a normal curve and mark the approximate locations of the mean (μ) and one, two, and three standard deviations from the mean, including the percentages in each region.

Question 2: Z-scores and Normal Probabilities

Player reaction times in “Pixel Quest” follow a normal distribution with μ = 285 milliseconds and σ = 42 milliseconds.

a. Calculate z-scores:

Find the z-score for:

  1. A player with reaction time of 320 ms
  2. A player with reaction time of 250 ms

Show your work using: \(z = \frac{x - \mu}{\sigma}\)

b. Calculate probabilities:

  1. P(X < 300) = probability a randomly selected player has reaction time under 300 ms
  2. P(260 < X < 310) = probability reaction time is between 260 and 310 ms

c. Percentile:

What reaction time represents the 90th percentile? (90% of players are faster)

d. Business application:

GameZone wants to identify the top 10% of players for a tournament. What reaction time threshold should they use?

Question 3: Empirical Rule Application

Daily session length for “Pixel Quest” is normally distributed with μ = 47 minutes and σ = 12 minutes.

a. Using the Empirical Rule:

  1. What percentage of sessions last between 35 and 59 minutes?
  2. What percentage of sessions last between 23 and 71 minutes?
  3. What percentage of sessions last more than 71 minutes?

b. Design decision:

Alex wants to send a “take a break” notification to players in the longest 2.5% of sessions. After approximately how many minutes should this trigger?

Question 4: Introduction to Sampling Distributions

GameZone collects data on player skill ratings, which are normally distributed with μ = 1,500 and σ = 200.

a. Fill in this comparison table:

Individual Players Sample Mean (n=25) Sample Mean (n=100)
Mean (μ or μₓ̄)
Std Dev (σ or σₓ̄)

Use: \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\)

b. Interpret:

How does the standard deviation of sample means change as sample size increases? What does this mean for GameZone’s data collection?

c. Calculate:

If we take a random sample of 36 players:

  1. What is the probability their mean skill rating is above 1,550?
  2. What is the probability their mean skill rating is between 1,480 and 1,520?

Question 5: Central Limit Theorem - The Foundation

a. State the Central Limit Theorem:

In your own words, explain what the CLT tells us about sampling distributions.

b. Conditions:

List and explain the THREE conditions/guidelines for applying the CLT.

c. Identify:

For each scenario, state whether CLT can be applied and why:

Scenario Can CLT Apply? Explanation
Population is normal, n = 15
Population is skewed right, n = 45
Population shape unknown, n = 8
Population is uniform, n = 100

Question 6: CLT with Non-Normal Populations

Loading times for “Pixel Quest” are right-skewed with μ = 3.8 seconds and σ = 1.6 seconds.

a. Individual vs. Sample:

  1. Can we use normal distribution methods for a single randomly selected loading time? Why or why not?
  2. Can we use normal distribution methods for the mean loading time of 50 randomly selected loads? Why or why not?

b. Calculations (n = 50):

  1. What is the probability the mean loading time for 50 loads exceeds 4.0 seconds?
  2. What is the probability the mean loading time is between 3.6 and 4.0 seconds?

c. Quality assurance:

GameZone considers loading performance “acceptable” if the mean of 50 tests is below 4.2 seconds. What’s the probability they’ll meet this standard?

Question 7: Putting It All Together

Bug fix times (in minutes) have μ = 85 and σ = 28 (shape unknown).

a. Multiple scenarios:

For EACH scenario below, state whether you can calculate the probability using normal methods, and if so, calculate it:

  1. A single bug fix takes less than 70 minutes: P(X < 70) = ?
  2. Mean of 5 bug fixes is less than 70 minutes: P(X̄ < 70) = ?
  3. Mean of 50 bug fixes is less than 70 minutes: P(X̄ < 70) = ?

b. Quality metrics:

If GameZone measures 36 bug fixes, between what two values will the middle 95% of sample means fall?

💭 Question 8: Detective’s Reflection

Reflect on your normal distribution investigation (5-7 sentences):

  • How does the Central Limit Theorem change what we can do with non-normal data?
  • Why is the normal distribution so powerful for business analytics?
  • What’s the key difference between analyzing individual values vs. sample means?
  • How does sample size affect our ability to make predictions?
  • What was most challenging or what clicked for you?
  • Name one scenario from gaming, business, or daily life where CLT would be crucial for decision-making.

🎉 Excellent work, Statistical Detective! GameZone now has the tools to analyze continuous data, understand sampling variability, and make confident predictions about player behavior. Your mastery of the normal distribution and CLT has leveled up their analytics game! ```