t-Test for Independent Samples

Week 11 Notes: t-Test for Independent Samples (Chapter 12)

Overview of Chapter 12

• Learning Objectives:

  • When the t-test for independent means is appropriate to use.

  • How to compute the observed t value.

  • How to use the T.TEST function.

  • How to use the t-Test Analysis ToolPak tool for computing the t value.

  • Interpreting the t value and understanding its meaning.

  • The difference between significance and effect size.

• Source: Salkind, Statistics for People Who (Think They) Hate Statistics, Excel 2016 4th Edition, SAGE Inc. © 2016.

t-Test for Independent Means

• Definition:

  • A t-test for independent means is used when examining the differences between two groups that are independent from one another.

  • Example: Comparing males (Group 1) and females (Group 2) across a single observation.

  • Note: Each participant is only measured once.

• Significance Level:

  • The test is considered significant if the p-value is less than 0.05 (i.e., p < 0.05). This indicates a significant difference between the two groups.

Example of t-Test Usage

• Research Question:

  • Are blood iron levels (measured in ng/mL) different between males and females?

• Hypotheses:

  • Null Hypothesis (H0): Mean blood iron levels are equal for males and females (μ1 = μ2).

  • Research Hypothesis (H1): Mean blood iron levels are different between males and females (μ1 != μ2).

• Independent Groups:

  • Blood iron levels are measured in two independent samples: one from males (Group 1) and another from females (Group 2).

Computing the Test Statistic

• Components of Test Statistic:

  • Numerator: The difference between the means of the two groups (Mean1 - Mean2).

  • Denominator: The amount of variation within each group, combined with variation between the groups.

Assumptions of the t-Test

• Homogeneity of Variance:

  • Assumes the variability in each of the two groups is equal. This assumption is important, though it is often not violated in practice.

Degrees of Freedom (df)

• Formula for Degrees of Freedom:

  • For the t-test for independent means:
    $df = (n<em>1 - 1) + (n</em>2 - 1)$

  • This can vary based on the test statistic selected, generally approximating sample size.

Steps to Computing the t Statistic

1. State the Hypotheses:

   • Null and research hypotheses need to be clearly defined.

2. Set the Risk Level:

   • Determine the level of risk associated with the null hypothesis, typically set at 0.05.

3. Select Test Statistic:

   • Choose the appropriate test statistic, in this case, t-test for independent means.

4. Compute the Test Statistic Value:

   • Follow the formula and compute to get the observed t value.

5. Determine the Critical Value:

   • Refer to statistical tables (e.g., Table B.2) to find critical values based on df and risk levels.

6. Compare Values:

   • Compare the obtained t value with the critical value to make a decision.

7. Make a Decision:

   • Based on the comparison, accept or reject the null hypothesis.

Example: Alzheimer’s Patients Data

• Data Summary:

  • Group 1 and Group 2 Data on Words Remembered:
    | Group 1 | Group 2 |
    |---------|---------|
    | 7 | 5 |
    | 5 | 5 |
    | and so on… |

Example Computation Steps

• 1. State Null and Research Hypotheses:

  • H0: μ1 = μ2

  • H1: μ1 ≠ μ2

• 2. Set Risk Level:

  • Typically 0.05.

• 3. Select Appropriate Test Statistic:

  • t-Test for independent means.

• 4. Compute Test Statistic Value:

  • Example Result: t = 1.69, with 38 degrees of freedom (df).

Critical Value Determination

• Critical Value Calculation:

  • Use Table B.2 to find critical value, taking into account df and level of risk.

  • For example, if critical value is found to be 2.004, compare it with obtained value.

Decision Making

• Comparison Example:

  • Obtained value: -0.137

  • Critical value: 2.004

  • Conclusion:

  • Since the obtained value is less extreme than the critical value, the null hypothesis is not rejected.

• Interpretation:

  • Despite an 8-point difference in test scores (e.g., 73 vs. 65), there is insufficient evidence to conclude a statistically significant difference.

Interpretation of Results

• Example Result:

  • t (58) = -0.137, p > 0.05

  • Explanation:

  • t indicates the test statistic.

  • 58 represents degrees of freedom.

  • -0.137 is the computed t value.

  • p > 0.05 signifies no significant difference in means between groups.

Using Excel for Calculations

• T.TEST Function:

  • Returns the probability of observing the t value but does not compute the t statistic itself.

• ToolPak Tool:

  • Facilitates t value computation without manual calculations.

Effect Size Understanding

• Effect Size Definition:

  • Measures the magnitude of the difference between two group means, reflecting statistical significance versus meaningfulness.

• Effect Size Calculation Methods:

  • Small: 0.00 – 0.20

  • Medium: 0.20 – 0.50

  • Large: 0.50 and above.

Conclusion: Summary of t-Tests

• t-Test Types:

  • One Sample T-Test: Testing against a known population statistic, df = n-1.

  • Unpaired/Independent T-Test: Comparing two independent sample means, df = n1 + n2 - 2.

  • Paired T-Test: Comparing variables on a subject with two measurements, df = n-1.

• Understanding which t-test to use is crucial depending on the type of study and data collected.