ASP104: Hypothesis Testing with t-test

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Questions and Answers

What is the purpose of a paired t-test?

To evaluate the mean difference from same subjects before and after a treatment.

What are the null and alternative hypotheses in a paired t-test?

H0: Physiotherapy did not produce a difference in recovery (ubefore = uafter), H1: Physiotherapy produced a difference in recovery (ubefore ≠ uafter).

What does the t-value represent in a paired t-test?

The t-value represents the test statistic, which indicates the difference from the mean due to chance for the null hypothesis to be true.

How is the critical t-value determined in a paired t-test?

<p>The critical t-value is determined based on the degrees of freedom (df) and the significance level (α), typically at α = 0.05 for a two-tailed test.</p>
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What can be concluded if the t-calculated value is greater than the critical t-value in a paired t-test?

<p>Reject the null hypothesis (D)</p>
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What does a small p-value in a paired t-test indicate?

<p>A small p-value (typically &lt; 0.05) indicates that the observed difference is statistically significant.</p>
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What is the purpose of a t-test?

<p>The purpose of a t-test is to compare if there is a significant difference between the means of two groups or samples.</p>
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What are the assumptions of a t-test? (Select all that apply)

<p>Independence (A), Normality (B), Equal variances (D)</p>
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For a t-test, the degrees of freedom (df) for one sample is ___ and for two samples is ___.

<p>n-1, n1+n2-2</p>
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In a t-test, the null hypothesis assumes no relationship exists between two different groups.

<p>True (A)</p>
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Match the t-test type with its description:

<p>Paired t-test = Compare related groups (same subjects at 2 different times or conditions) Independent sample t-test = Compare the means of 2 independent groups One sample t-test = Compare the mean of a single sample to a known value or hypothesized mean of a population</p>
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Study Notes

Introduction to t-test

  • A powerful inferential statistical tool used to compare if there is a significant difference between the means of two groups or samples
  • T-distribution is a continuous probability distribution that arises from an estimation of the mean of a normally distributed population using a small sample size and an unknown standard deviation for the population

Assumptions of t-test

  • Normality: The data should follow a normal distribution, i.e., a bell-shaped curve
  • Independence: The observations within each group should be independent of each other
  • Equal variance: When the standard deviations of the samples are approximately equal

t-distribution

  • Used to determine statistical significance of the differences between groups
  • A t-distribution is a distribution of the t-statistic, a measure of the difference between the sample mean and the unknown population mean

t-test Procedure

  1. Formulate hypotheses (H0 & H1)
  2. Choose the appropriate t-test based on study design and assumptions
  3. Collect and organize data
  4. Calculate the t-statistic
  5. Determine the degrees of freedom (df)
  6. Find the critical value or p-value
  7. Make a decision: Reject or fail to reject the null hypothesis

Types of t-tests

  • 1-sample t-test: Compares the mean of a single sample to a known population mean
  • Independent samples t-test (2-sample unpaired t-test): Compares the means of two independent samples
  • Paired t-test (2-sample paired t-test): Compares the means of two related samples or matched pairs

Paired t-test

  • Used to evaluate the mean difference from the same subjects before and after a treatment
  • Accounts for the correlation between the two samples, making it more powerful than an unpaired t-test in certain scenarios

1-sample t-test Example

  • Testing if the mean time for a student athletics team 2 km run is significantly different from a known population mean
  • Calculating the t-statistic and comparing it to the critical value

2-sample Unpaired t-test Example

  • Comparing the means of two milk formulas on the growth of infants
  • Calculating the t-statistic and comparing it to the critical value

Paired t-test Example

  • Determining if physiotherapy made a difference in recovery at a significance level of 0.05
  • Calculating the t-statistic and comparing it to the critical value

Interpreting t-test Results

  • If p-value is less than the chosen significance level (typically 0.05), the difference observed in the data is statistically significant
  • Considering the effect size and practical significance is crucial

Limitations of t-test

  • Sensitivity to sample size
  • Normality assumption
  • Homogeneity of variance
  • Limited to two groups

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