9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

2:14 minutes

Problem 9.5.4

Textbook Question

Randomization vs t Test Two samples of commute times from Boston and New York are randomly selected and it is found that the samples sizes are n1 = 18 and n2 = 12 and each of the two samples appears to be from a population with a distribution that is dramatically far from normal. Which method is more likely to yield better results for testing Mu1 is not equals to Mu2. Hypothesis test using the t distribution (as in Section 9-2) or the resampling method?

Verified step by step guidance

Step 1: Understand the problem. The goal is to determine which method is more appropriate for testing the hypothesis that the population means (μ₁ and μ₂) of commute times for Boston and New York are not equal, given that the sample sizes are n₁ = 18 and n₂ = 12, and the population distributions are far from normal.

Step 2: Recall the assumptions of the t-test. The t-test assumes that the data comes from a normal distribution or that the sample sizes are sufficiently large for the Central Limit Theorem to apply. Since the population distributions are dramatically far from normal and the sample sizes are relatively small (n₁ = 18 and n₂ = 12), the t-test may not yield reliable results.

Step 3: Consider the resampling method. Resampling methods, such as bootstrapping or permutation tests, do not rely on the assumption of normality. These methods involve repeatedly sampling from the observed data to create a sampling distribution, making them more robust for non-normal data.

Step 4: Compare the two methods. Given the small sample sizes and the non-normality of the population distributions, the resampling method is likely to yield better results because it does not depend on the normality assumption and can handle small sample sizes effectively.

Step 5: Conclude the reasoning. Based on the information provided, the resampling method is more appropriate for testing the hypothesis that μ₁ ≠ μ₂ under these conditions. The t-test is less suitable due to the violation of its assumptions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Randomization

Randomization is a fundamental principle in statistics that involves randomly assigning subjects to different groups to eliminate bias and ensure that the groups are comparable. This method helps in making valid inferences about the population from which the samples are drawn. In the context of hypothesis testing, randomization can enhance the reliability of results, especially when dealing with non-normal distributions.

t Test

The t Test is a statistical method used to determine if there is a significant difference between the means of two groups. It assumes that the data is normally distributed and is particularly useful when sample sizes are small. However, when the underlying population distributions are not normal, the t Test may yield unreliable results, making it essential to consider alternative methods.

Resampling Method

The resampling method, including techniques like bootstrapping, involves repeatedly drawing samples from the observed data to estimate the sampling distribution of a statistic. This approach is particularly useful when the assumptions of traditional parametric tests, such as normality, are violated. Resampling can provide more accurate confidence intervals and hypothesis tests, especially in cases with small sample sizes or non-normal distributions.

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