Hypothesis Testing: The One-, and Two-Sample Case

Hypothesis Testing: The One-, and Two-Sample Case

Introduction to the t-test

The t-test is a statistical test based on the Student’s t-distribution. It is primarily used to compare:

• A group mean against a “test statistic”.

• Means between two different groups.

• Means within one group with two measurements.

Z-test vs. t-test

• Use a z-test when the population standard deviation is known.

• There exists one z-distribution; conversely, there are many t-distributions depending on degrees of freedom.

What is the t-test?

The t-test is characterized by a strict recipe of equations:

• One-sample t-test

• Independent two-sample t-test

• Paired sample t-test

Normal Distribution and Sample Size

The impact of sample size on variability:

• Variability decreases with larger sample sizes.

• The t-distribution curves more sharply compared to the normal distribution as sample sizes vary.

• The t-distribution's shape depends fundamentally on the sample size (denominator involves sample size in variance calculations).

Detailed Comparison: t-test vs. z-test

One-sample z-test

• Purpose: To determine if a sample mean is representative of the population.

• Assumes population standard deviation is known.
  Formula: $z = \frac{X - \mu}{\sigma / \sqrt{N}}$

One-sample t-test

• Purpose: Similar to the z-test, but used when the population standard deviation is unknown.

• Formula: $t = \frac{X - \mu}{s / \sqrt{N}}$ Where:

  • $X$ = sample mean

  • $\mu$ = population mean

  • $s$ = sample standard deviation

  • $N$ = sample size

Steps of Hypothesis Testing

1. State the null hypothesis (H0) and alternative hypothesis (H1).

2. Choose the significance level (α) and determine if the test is one-tailed or two-tailed.

3. State the rejection and acceptance rule based on critical values.

4. Compute the appropriate t or z statistic based on sample data.

5. Make a decision by applying the rejection/acceptance rule.

6. Write a conclusion contextualized to the study.

Application Example: Comparing LSAT Scores

Scenario Overview

Do Ontario university graduates have an average LSAT score that is higher than all Canadian LSAT test takers?

• Population: Entire Canadian population of LSAT test takers

• Sample: LSAT results of n=100 Ontario students

Formulating Hypotheses

• Null Hypothesis (H0): The average LSAT score of Ontario students is equal to the Canadian average.
  $H0: \mu = 153$

• Alternative Hypothesis (H1): The average LSAT score of Ontario students is higher than the Canadian average.
  H1: \mu > 153

• Set significance level (α) as 0.05 for two-tailed tests.

Performing the z-test

• Rejection/accept rule:
  Reject H0 if |zcalculated| > critical z-value (0.05, two-tailed). Accept H0 if |zcalculated| < critical z-value (0.05, two-tailed).

• Given the calculated z-value results, e.g., if $z_{calc} = 3.33$, compare against the critical value.

• If |z_{calc}| > 1.96 , reject the null hypothesis.

Conclusion of LSAT Scores

The conclusion of the test shows that Ontario graduates score significantly (p<0.05) higher than the average e.g., confirming that:

• They have a 95% chance of scoring greater than the Canadian average.