HW6: Pixel Quest Confidence & Decisions

A Journey Through Statistical Inference

🎮 The Case

Welcome back, Statistical Detective!

Your work with the normal distribution and sampling has been invaluable to GameZone. Now Alex Rivera needs you to tackle their biggest challenge yet: making confident decisions with incomplete information.

“Detective, we’ve learned to describe what we see in our data. But now we need to make claims about ALL our players based on samples, and we need to test whether our game updates actually improve performance. We’re talking confidence intervals, hypothesis testing, and making decisions that affect millions of players. The stakes are high—we can’t afford to be wrong, but we also can’t collect data forever. Ready to help us make confident, data-driven decisions?”

Your mission: Master confidence intervals and hypothesis testing to help GameZone make reliable inferences and test their hypotheses about player behavior!

Question 1: Foundations of Statistical Inference

1. Define these terms in the context of GameZone:

• Point Estimate:
• Confidence Interval:
• Confidence Level:
• Margin of Error:

2. Interpretation challenge:

GameZone calculates a 95% confidence interval for mean daily playtime as (38.5, 43.7) minutes. Write THREE statements:

• A CORRECT interpretation of this interval
• An INCORRECT interpretation (explain why it’s wrong)
• What the 95% confidence level really means

Question 2: Confidence Intervals (σ Known)

GameZone knows from years of data that player spending has σ = $15.80. A random sample of n = 100 players from a new region shows x̄ = $42.30.

1. Construct confidence intervals:

Calculate and interpret a confidence interval for the population mean at:

• 90% confidence level (z* = 1.645)

• 95% confidence level (z* = 1.96)

• 99% confidence level (z* = 2.576)

Use: \(\text{CI} = \bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}}\)

2. Compare:

How does the width of the interval change as confidence level increases? Why does this make sense?

3. Business decision:

GameZone wants to know if this new region’s mean spending differs from their global average of $45.00. Based on your 95% CI, what can you conclude? Explain your reasoning.

Question 3: Confidence Intervals (σ Unknown, Large Sample)

A random sample of n = 64 “Pixel Quest” sessions yields:

• x̄ = 52.4 minutes
• s = 14.8 minutes

1. Construct a 95% confidence interval:

Use: \(\text{CI} = \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}\)

(For large samples, you can use z* ≈ t, so use z = 1.96)

2. Margin of error:

What is the margin of error for this interval? What would it take to cut this margin of error in half?

3. Sample size calculation (use Google Sheets):

GameZone wants to estimate mean session length with a margin of error of ±2 minutes at 95% confidence. Assuming σ ≈ 14.8, what sample size do they need?

Use: \(n = \left(\frac{z^* \cdot \sigma}{E}\right)^2\) where E is the desired margin of error

Question 4: Confidence Intervals (σ Unknown, Small Sample)

GameZone tests a new tutorial on n = 12 randomly selected new players. Completion times (minutes) are:

18, 22, 19, 25, 21, 20, 23, 19, 24, 21, 20, 22

1. Calculate summary statistics:

Find x̄ and s for this sample (show your work or describe your process).

2. Check conditions:

List the conditions needed for a t-interval. Assess whether they’re met (you may assume the population is approximately normal).

3. Construct a 90% confidence interval:

Use the t-distribution with df = 11 and t* = 1.796

\(\text{CI} = \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}\)

4. Application:

If the old tutorial had a mean completion time of 25 minutes, what can GameZone conclude about the new tutorial based on this interval?

Question 5: Confidence Intervals for Proportions

GameZone surveys n = 400 randomly selected players. 312 report they would recommend “Pixel Quest” to friends.

1. Calculate the sample proportion:

\(\hat{p} = \frac{x}{n}\)

2. Check conditions:

Verify that conditions for a proportion confidence interval are met:

• Random sample?
• np̂ ≥ 10 and n(1 - p̂) ≥ 10?

3. Construct a 95% confidence interval:

Use: \(\text{CI} = \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)

(Use z* = 1.96)

4. Marketing claim:

Can GameZone confidently claim that “more than 75% of players would recommend our game”? Explain using your confidence interval.

5. Sample size for proportions:

GameZone wants to estimate the proportion within ±0.03 at 95% confidence. If they have no prior estimate, what sample size should they use?

Use: \(n = \left(\frac{z^*}{E}\right)^2 \cdot 0.25\) (worst case scenario)

Question 6: Hypothesis Testing Foundations

1. Key terms. Define in the context of GameZone:

• Null Hypothesis (H₀):

• Alternative Hypothesis (Hₐ):

• Test Statistic:

• P-value:

• Significance Level (α):

2. Formulate hypotheses:

For each research question, write appropriate null and alternative hypotheses:

• GameZone wants to test if a new matchmaking algorithm changes mean wait time (currently 45 seconds)
• They want to test if a new difficulty setting increases player satisfaction scores (currently μ = 7.2 out of 10)
• They want to test if more than 60% of players complete the tutorial

Question 7: Type I and Type II Errors

GameZone is testing whether a new anti-cheat system reduces cheating below the current rate of 8%.

• H₀: p = 0.08 (cheating rate is still 8%)
• Hₐ: p < 0.08 (cheating rate has decreased)

1. Define errors in context:

• Describe a Type I error for this test. What’s the consequence?
• Describe a Type II error for this test. What’s the consequence?

2. Error probabilities:

If α = 0.05, what is the probability of Type I error?

If the power of the test is 0.85, what is the probability of Type II error?

3. Trade-offs:

GameZone is considering using α = 0.01 instead of α = 0.05. How would this affect:

• The probability of Type I error?
• The probability of Type II error?

Which error is GameZone more concerned about avoiding in this scenario?

Question 8: Complete Hypothesis Test (σ Known)

GameZone claims their new server upgrade reduces mean loading time from the previous μ = 3.8 seconds. They know from extensive data that σ = 1.2 seconds. A random sample of n = 50 loads shows x̄ = 3.52 seconds.

Test at α = 0.05.

1. Set up:

• State H₀ and Hₐ
• Check conditions for using the normal distribution
• What significance level are you using?

2. Calculate test statistic:

Use: \(z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\)

3. Find p-value:

Sketch a normal curve, shade the relevant area, and find the p-value.

4. Make decision:

Compare p-value to α and state your decision (reject H₀ or fail to reject H₀).

5. Conclusion:

Write a complete conclusion in context that GameZone’s management can understand.

6. What if:

If the sample mean had been 3.65 seconds instead, would your conclusion change? Explain without doing the full test.

Question 9: Complete Hypothesis Test (σ Unknown)

GameZone believes a new tutorial has decreased completion time from the old mean of μ = 25 minutes. They test n = 16 new users and find:

x̄ = 22.3 minutes

s = 4.8 minutes

Test at α = 0.05 (assume population is approximately normal).

1. Set up:

• State H₀ and Hₐ
• What test statistic distribution should you use and why?
• What are the degrees of freedom?

2. Calculate test statistic:

Use: \(t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\)

3. Critical value approach:

Find the critical value for this one-tailed test (t* = -1.753 for df = 15)

Make your decision based on comparing your test statistic to the critical value

4. P-value approach:

If the p-value is approximately 0.015, make your decision and state your conclusion.

5. Connect to confidence intervals:

Construct a 90% confidence interval for the true mean completion time (t* = 1.753). Does this interval support your hypothesis test conclusion? Explain.

Question 10: Hypothesis Test for Proportions

GameZone’s competitor claims that 40% of mobile gamers prefer their game. GameZone surveys n = 500 random mobile gamers and finds 175 prefer the competitor’s game.

Test at α = 0.10 whether the competitor’s claim is accurate.

1. Set up:

• State H₀ and Hₐ (two-tailed test)
• Check conditions for proportion test
• Calculate the sample proportion

2. Calculate test statistic:

Use: \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\)

3. P-value:

This is a two-tailed test. Find the p-value.

4. Decision and conclusion:

Make your statistical decision and write a conclusion in context.

5. Practical significance:

Even if the result is statistically significant, discuss whether the difference is practically significant for business decisions.

Question 11: Choosing the Right Test

• For each scenario, identify:
• Parameter of interest (μ or p)
• Known or unknown population standard deviation
• Sample size
• Which distribution to use (z or t)
• Whether conditions are met

Scenarios:

• Testing if mean score differs from 500; n=35, s=45
• Testing if proportion completing tutorial > 0.80; n=200, x=165
• Testing mean time; n=9, s=2.3, population normal
• Estimating mean revenue; n=100, σ=250 from historical data

Question 12: P-value Interpretation

1. True or False (explain each):

• A p-value of 0.03 means there’s a 3% chance the null hypothesis is true.
• A smaller p-value provides stronger evidence against H₀.
• If p-value = 0.06 and α = 0.05, we accept the null hypothesis.
• A p-value tells us the probability of getting our sample result (or more extreme) if H₀ is true.

2. Critical thinking:

GameZone runs 20 different A/B tests on game features, each at α = 0.05. If none of the features actually have an effect, approximately how many tests would you expect to show “statistically significant” results just by chance? What does this suggest about multiple testing?

💭 Question 13: Detective’s Final Reflection

Reflect on your statistical inference journey (6-8 sentences):

• How do confidence intervals and hypothesis tests help GameZone make decisions with limited data?
• What’s the relationship between confidence intervals and hypothesis testing?
• Why is it important to understand Type I and Type II errors?
• How does sample size affect both confidence intervals and hypothesis tests?
• What surprised you most about statistical inference?
• Describe a real-world scenario (gaming, business, or daily life) where you’d need to construct a confidence interval or test a hypothesis.
• If you were advising GameZone’s analytics team, what’s the most important concept from this unit they should remember?

🎉 Outstanding work, Statistical Detective!

You’ve completed your training in statistical inference! GameZone now has the tools to make confident claims about their player population and test whether changes to their game actually make a difference. Your mastery of confidence intervals, hypothesis testing, and statistical decision-making has transformed their analytics from descriptive to inferential. You’re ready to tackle any data challenge that comes your way!

Remember: In statistics, we never prove anything with absolute certainty—we build confidence and make informed decisions based on evidence. That’s the power and the humility of statistical thinking.