University:
McMaster University
Course:
STATS 1LL3 | Introduction to Probability and Statistics
Academic year:

2023
Views:

40

Pages:

3
Author:

Sabrina S.

Problem Statement: Statistically testing whether two dice come from the same or different populations. Testing if they come from the same population with the same underlying true mean. Variables and Notation: Your die: X₁ Your friend's die: X₂ Sample mean: x̄₁ and x̄₂ Population mean: μ₁ and μ₂ Population variance: σ₁² and σ₂² Sample size: n₁ and n₂ Statistical Distributions: X₁ ~ N(μ₁, σ₁²/n₁) X₂ ~ N(μ₂, σ₂²/n₂) X̄₁ - X̄₂ ~ N(μ₁ - μ₂, σ₁²/n₁ + σ₂²/n₂) Hypothesis Testing Steps: Step 1: Ho: μ₁ = μ₂ (null hypothesis) Step 2: Ha: μ₁ ≠ μ₂ (alternative hypothesis) Step 3: Set up significance level (α): α = 0.05 Step 4: Set up rejection region based on α: Reject Ho if absolute value of test statistic > 1.96 Given Values: After 30 rolls of the die: x̄₁ = 3.6, x̄₂ = 4.4 Assume variances of dice are the same as for a fair die: σ₁² = σ₂² = 2.917 Step 5: Calculate test statistic: [Z = ((\overline{X}{1} - \overline{X}{2}) - (\mu_{1} - \mu_{2})) / \sqrt{(\sigma_{1}^{2} / n_{1}) + (\sigma_{2}^{2} / n_{2})}] [= (3.6 - 4.4 - 0) / \sqrt{(2.917 / 30) + (2.917 / 30)} = -1.81] Explanation: This means that if the null hypothesis is true and the two means are actually equal, we have observed a difference that lies 1.8 standard errors away from what we would expect. Step 6: Draw a conclusion: The test statistic does not fall in the rejection region. So we cannot reject Ho and must conclude that there is no difference between the mean value of your die and that of your friend's die. The modified approach: We do not require a significance level to be specified. Rather, we report the observed level of significance or the p-value. Steps: Step 1: Ho: μ₁ = μ₂ Step 2: H₁: μ₁ ≠ μ₂ Step 3: Go straight to test statistic [z = ... = -1.81] P-value: The probability of observing a test statistic at least as extreme as our calculated test statistic. If H₁ is "≠" (two-sided): p-value = Pr(|z| > |test statistic|) × 2 If H₁ is ">" (one-sided): p-value = Pr(z > test statistic) If H₁ is "<" (one-sided): p-value = Pr(z < test statistic) Calculation of p-value: Pr(|z| > |-1.81|) × 2 = (0.5 - 0.4649) × 2 = 0.0351 × 2 = 0.0702 Explanation: What this means is if Ho is true (i.e., there is no difference between the dice), we would observe a difference at least of this size approximately 7% of the time. Commonly used 5% rule: In this course, we commonly use the 5% rule. We can reject Ho when the p-value is lower than 5%. P-value definition: It is the probability, if we assume Ho is true (i.e., there is no difference between the dice), that we would observe a test statistic at least as extreme as the one we are faced with. AKA: Pr(type 1 error)