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

2023
Views:

343

Pages:

2
Author:

Sabrina S.

Video 20: Comparing two means Rerun experiment of rolling die with friend, potentially biased die. Your die x̄₁ = 3.9 s₁ = 2.00 Their die x̄₂ = 4.3 s₂ = 1.95 Step 1: define the null hypothesis (H₀) H₀: μ₁ = μ₂ Step 2: define the alternative hypothesis (H₁) H₁: μ₁ ≠ μ₂ Step 3: Significance level α = 0.05 Step 4: calculate rejection region remember! t = (x̄₁ - x̄₂) - (μ₁ - μ₂) / √(s₁²/n₁ + s₂²/n₂) we assume that the population variances are equal Therefore s₁ and s₂ can both be viewed as estimates of the same true variance s_p: pooled estimate s_p = √((n₂ - 1)s₁² + (n₁ - 1)s₂² / (n₁ + n₂ - 2)) test statistic → t = (x̄₁ - x̄₂) - (μ₁ - μ₂) / (s_p √(1/n₁ + 1/n₂)) t = (3.9 - 4.3) - 0 / (√(1.95² / 10 + 2.00² / 10)) t = -0.78 this test statistic will follow a t-distribution with n₁ + n₂ - 2 degrees of freedom Rejection region → |test statistic| > t_α/2,n₁+n₂-2 (reject H₀ if) critical value-1.000 test statistic t -0.78 20 1.0 t distribution with 60 degrees of freedom at 5% level we do not reject H₀ this means we conclude that the two means are equal What is the probability that our test statistic is (using the modified approach) 0.78 or <-0.78? use the table : 60 degrees of freedom find highest level at which test statistic would be significant 0.3 p-value is < 0.6 two-sided, we double We need the t-distribution because we no longer assume we know the population variances (estimated from two sample variances) we assume that the two populations have equal variances so we can combine the two sample statistics and a pooled standard deviation estimate