Introduction to T-Tests

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Questions and Answers

Explain in what situations a t-distribution is preferred over a Z-distribution when conducting hypothesis testing. Why is this distinction important in statistical analysis?

A t-distribution is preferred over a Z-distribution when the population standard deviation is unknown. This is important because using the appropriate distribution ensures accurate p-value calculation and valid conclusions.

Describe how degrees of freedom affect the shape of the t-distribution and why this is relevant in the context of the t-test. How does the t-distribution adjust for uncertainty with smaller sample sizes?

Fewer degrees of freedom result in a wider, shorter t-distribution. This is relevant because it reflects greater uncertainty due to smaller sample sizes, making extreme values more probable. This wider distribution necessitates a higher t-value to achieve statistical significance.

Explain why it's important to maintain independence between the two groups when performing an independent samples t-test. Provide an example of what might occur if this assumption is violated.

Independence ensures that observations in one group do not influence observations in the other. Violating this could skew results. For example, testing baseball and basketball players' running speed where some individuals play both sports affects group distinctiveness.

Describe what the null and alternative hypotheses are in the context of an independent samples t-test. Also, why is it important to assume the null hypothesis is true when conducting a t-test?

<p>The null hypothesis (H₀) typically states that there is no difference in the means between the two groups (µG1 = µG2), while the alternative hypothesis (Hᴀ) states that there is a significant difference (µG1 ≠ µG2). We assume the null hypothesis is true because we need a baseline to determine if our observed data is unlikely enough to reject it.</p> Signup and view all the answers

Why is it essential to check for homogeneity of variance between two groups before performing an independent samples t-test? What does this assumption imply about the standard deviations of the two groups?

<p>It's essential because the t-test assumes the variance is approximately equal between the groups. This implies that although the means may differ, the spread of data around those means should be similar. If violated, the t-test may not be valid, and alternative tests might be more appropriate.</p> Signup and view all the answers

Flashcards

What is an Independent Samples T-test?

A test to compare the means of two independent groups to see if they are statistically different.

What is the Null Hypothesis (H₀) in a t-test?

The assumption that there is no difference between the means of the two populations being compared.

What is the t-distribution?

A variation of the normal distribution used when the population standard deviation is unknown.

What are degrees of freedom?

The number of independent pieces of information available to estimate a parameter.

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What is Standard Error?

A measure of the spread of sample means around the population mean.

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Study Notes

Introduction to the T-Test

The t-test is used to compare groups of people in research.
Examples of when to use a T-test: comparing anxiety levels of PhD vs undergrad students, comparing binge drinking in men vs women, comparing performance on standardized tests in students from different school districts
A random sample is split into two groups, G1 and G2, with a numeric variable X measured.
X is assumed to be normally distributed.
The goal is to determine if group membership (G1 vs G2) is associated with different values of X.
The scientific question is to ask if the mean value of X is different between groups G1 and G2
μG1 represents the population-level mean of X for G1, and μG2 represents that for G2
This aims to determine if μG1 = μG2

Statistical Hypotheses of the Independent Samples T-Test

The null and alternate hypotheses are:
- H0: μG1 = μG2
- HA: μG1 ≠ μG2
μG1 = μG2 is the same as μG1 - μG2 = 0
The null and alternate hypotheses can also be:
- H0: μG1 - μG2 = 0
- HA: μG1 - μG2 ≠ 0
The independent sample t-test assesses the probability of the observed data, assuming H0 is true.

Logic of the T-Test

When running a statistical test, assume the null hypothesis is true.
If the null hypothesis is true (μG1 - μG2 = 0 at the population-level), the most likely value is 0 when comparing the group means of X, between samples from G1 and G2.
μG1 and μG2 represent the population-level mean, while 𝑥¯G1 and 𝑥¯G2 represent the mean values of sample groups.
Sampling rarely results in a perfect representation of populations, so the difference in sample group means (𝑥¯G1 - 𝑥¯G2) is unlikely to equal 0.
If the null is true, 𝑥¯G1 - 𝑥¯G2 should be close to 0.
Values of 𝑥¯G1 - 𝑥¯G2 further from 0 are less probable, with negative and positive values being equally likely.
This resembles a normal distribution:
- 0 is the most likely value.
- values closer to 0 are more likely.
- positive and negative values are equally likely.

Introducing the Student's t-Distribution

A normal distribution is defined by a mean value μ and standard deviation σ.
The population-level standard deviation of X might be unknown, even with a normally distributed variable X.
This is common in epidemiologic research due to the lack of population-level data.
The t-distribution, a variation of the standard normal (Z) distribution, is used when the population standard deviation is unknown.
Any normal distribution N(μ, σ) can be transformed to the Z-distribution N(0, 1).
The t-distribution can be understood as a standardized distribution.
The t-distribution is a more conservative version of the Z-distribution, assuming wider variability in observations.
The less information known, the less certain the observations will be near the mean.
The t-distribution is defined as a function of the number of degrees of freedom available to measure data variability.

Degrees of Freedom

Degrees of freedom are the number of parameters able to "vary freely" given an assumed outcome.
In a scenario with 100 participants with a mean age of 60, ages can "vary freely" while maintaining the average.
If the exact age of 99 individuals is known, with a population average of 60, the final person's age cannot "vary freely."
Only one value can achieve the average of 60, therefore, to calculate a mean, one degree of freedom is "spent".
If we have n observations and calculated sample-mean x and standard deviation s, we must "spend" one degree of freedom to calculate x.
This means that we have n - 1 degrees of freedom to calculate s
The fewer observations there are (smaller n), the less information needed to estimate variation of observed variable.
The t-distribution captures uncertainty in measuring the standard deviation from a small sample.
The smaller the sample and therefore having fewer degrees of freedom, the less certain the population-level standard deviation is represented by standard deviation s.
To capture this, the t-distribution is "shorter" and "wider" than the normal distribution.
Under the t-distribution, values further from 0 are more likely than under the Z-distribution.
As sample size n increases, the t-distribution's shape approaches that of the Z-distribution.