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

2023
Views:

114

Pages:

4
Author:

Sabrina S.

Video 17: testing whether the mean is a specified value Confidence intervals allow us to say things like: Our best estimate of the unknown mean is x̄ and we are 100(1-α)% confident that it lies within the interval x̄ ± zα/2 * σ/√n The Hypothesis Test Statistical inference Significance tests Step 1: Define the null hypothesis (H₀) H₀ is a statement about one of the true, unknown population parameters. H₀ will always state that the true parameter is equal to some (hypothesized) value. H₀ is generally a statement about the status quo. Here, H₀ assumes no effect or difference. In our example of dice → H₀: μ = μ₀ = 3.5 Step 2: Define the alternative hypothesis (H₁) H₁ is still about the true, unknown population parameter. H₁ will always have a ≠, <, or > sign. In our example we have a strong suspicion that the true mean of this die is greater than 3.5 H₁: μ > 3.5 Step 3: Set up a significance level (α) for your test If we set a very tough test, we make it difficult to detect a difference from the hypothesized mean. If α = 1% (tough) Then even if the die is fair (H₀ = 0), we will still erroneously reject H₀ 1% of the time. If α = 10% (less tough) Then even if the die is fair (H₀ = 0), we will still erroneously reject H₀ 10% of the time. In our example: significance level of 5%, α = 0.05 Step 4: Set up a rejection region based on α If α = 5% We will reject H₀ if the sample mean falls within the most extreme 5% values of the Standard Normal distribution. In our example: According to H₁ (H₁: μ > 3.5), we only need to consider rejecting H₀ if the sample mean is in the 5% greatest possible values of the distribution of X. Step 5: Calculate the test statistic We proceed under the assumption that H₀ is true. So, if H₀ is true, then X̄ ~ N(μ₀, σ²/n) In our example: X̄ = 4.7 Z = (X̄ - μ₀) / (σ/√n) = (4.7 - 3.5) / (3.85/√30) = 2.917 Step 6: Draw a conclusion If the test statistic falls within the rejection region, then we reject H₀ in favor of H₁. If the test statistic does not fall within the rejection region, then we do not reject H₀. In our example: Our test statistic of 2.917 falls within the rejection region (values > 1.645). Therefore, we reject H₀ and conclude that the true mean of our friend's die is greater than 3.5. A 2-sided test is the default position unless we have a good reason to suspect a departure in a particular direction. One-sided test Rejection region Type I errors Reject H₀ erroneously This is controlled by α. If α is very small, it will be more difficult to reject H₀, so the chances of making a Type I error will be less. Type II error Accept H₀ erroneously If α is very small: Very small probability of making a Type I error Increasing probability of a Type II error For the purposes of this course, we generally accept that the Pr (Type II error) is acceptable when α = 0.05 or 0.01. Comparing two sample means (with known population variances) X₁ ~ N(μ₁, σ₁²) X₂ ~ N(μ₂, σ₂²) X₁ and X₂ are independent variables Y = X₁ - X₂ Y ~ N(μ₁ - μ₂, σ₁²/n₁ + σ₂²/n₂)