Parametric tests of difference: unrelated and related t-tests

Question 1

Overview of Parametric Tests of Difference

Answer 1

• Purpose: To assess whether there are significant differences in the means of two groups.
• Types: Unrelated t-test (independent samples) and related t-test (dependent samples).

Question 2

Unrelated T-Test (Independent T-Test)

Answer 2

• Definition: A parametric test used to compare the means of two independent groups to determine if there is a statistically significant difference between them.
• When to Use:
  o Two independent groups (e.g., treatment group vs. control group).
  o Data is normally distributed and measured on an interval or ratio scale.

Question 3

Steps for Conducting an Unrelated T-Test

Answer 3

1. State the Hypotheses:
   o Null hypothesis (H0): There is no difference between the group means (e.g., μ1=μ2\mu_1 = \mu_2μ1=μ2).
   o Alternative hypothesis (H1): There is a difference (e.g., μ1≠μ2\mu_1 \neq \mu_2μ1=μ2).
2. Calculate the Means and Standard Deviations:
   o Compute the mean and standard deviation for each group.
3. Compute the T-Statistic:
   o Use the formula:
   t=Xˉ1−Xˉ2spooled2(1n1+1n2)t = \frac{\bar{X}1 - \bar{X}2}{\sqrt{s^2{pooled}\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}t=spooled2(n11+n21)Xˉ1−Xˉ2
   where spooled2=(n1−1)s12+(n2−1)s22n1+n2−2s^2{pooled} = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}spooled2=n1+n2−2(n1−1)s12+(n2−1)s22.
4. Determine Degrees of Freedom (df):
   o df=n1+n2−2df = n_1 + n_2 - 2df=n1+n2−2.
5. Compare to Critical Value or Calculate P-Value:
   o Use a t-table or statistical software to find the critical value or p-value.

Question 4

Example of Unrelated T-Test

Answer 4

• Scenario: Testing the effectiveness of two different diets on weight loss.
• Data Collection: Group A (diet 1) and Group B (diet 2) weigh-ins after 4 weeks.
• Analysis: Conduct an independent t-test to see if the mean weight loss differs significantly between the two diets.

Question 5

Related T-Test (Dependent T-Test)

Answer 5

• Definition: A parametric test used to compare the means of two related groups or matched samples to determine if there is a statistically significant difference.
• When to Use:
  o Two related samples (e.g., pre-test vs. post-test scores).
  o Data is normally distributed and measured on an interval or ratio scale.

Question 6

Steps for Conducting a Related T-Test

Answer 6

1. State the Hypotheses:
   o Null hypothesis (H0): There is no difference in means (e.g., μpre=μpost\mu_{pre} = \mu_{post}μpre=μpost).
   o Alternative hypothesis (H1): There is a difference (e.g., μpre≠μpost\mu_{pre} \neq \mu_{post}μpre=μpost).
2. Calculate the Differences:
   o For each pair, find the difference (D) between scores.
3. Calculate the Mean and Standard Deviation of Differences:
   o Compute the mean (Dˉ\bar{D}Dˉ) and standard deviation (SD) of the differences.
4. Compute the T-Statistic:
   o Use the formula:
   t=DˉSDnt = \frac{\bar{D}}{\frac{SD}{\sqrt{n}}}t=nSDDˉ
   where nnn is the number of pairs.
5. Determine Degrees of Freedom (df):
   o df=n−1df = n - 1df=n−1.
6. Compare to Critical Value or Calculate P-Value:
   o Use a t-table or statistical software to find the critical value or p-value.

Question 7

Example of Related T-Test

Answer 7

• Scenario: Measuring the impact of a study skills program on student performance.
• Data Collection: Collect test scores of students before and after the program.
• Analysis: Conduct a dependent t-test to see if there is a significant difference in scores before and after the program.

Question 8

Reporting Results

Answer 8

• Format: When reporting results, include the test statistic, degrees of freedom, and p-value.
  o Example for unrelated t-test: “An independent t-test revealed a significant difference in weight loss between diet groups, t(28) = 3.21, p < 0.01.”
  o Example for related t-test: “A dependent t-test indicated a significant increase in scores after the program, t(15) = 5.42, p < 0.001.”

Question 9

Assumptions of T-Tests

Answer 9

• Normality: Data should be normally distributed (especially important for smaller sample sizes).
• Homogeneity of Variance: For unrelated t-tests, variances in both groups should be approximately equal. Use Levene’s test to check this assumption.

Question 10

Advantages and Limitations

Answer 10

• Advantages:
  o Powerful tests if assumptions are met.
  o Provides information on the size and direction of the difference.
• Limitations:
  o Sensitive to outliers, which can distort results.
  o Assumes normality and homogeneity of variance.