Topic 8

Testing the Null Hypothesis

Researchers test the null hypothesis using various statistical tests to validate their hypotheses against observed data. The most common statistical test employed in this context is the t-test, renowned for its effectiveness in comparing means from different groups or samples.

t-Test Basics

  • t statistic formula: The fundamental formula for calculating the t statistic is given ast = (X̄ - µ) / (SD / √N)where:

    • X̄ = sample mean

    • µ = population mean

    • SD = standard deviation

    • N = sample size

  • Degrees of Freedom (df): Calculated as df = N - 1, where N represents the total sample size. This parameter is crucial for determining the t-distribution used in hypothesis testing.

Example Question

  • Research Question: Do KINE students' heights differ significantly from those of YorkU students (with a known population mean µ = 169 cm)?

  • Sample Data:

    • N = 15 (KINE students)

    • Sample Mean (KINE) = 178 cm

    • Standard Deviation (SD) = 20 cm

    • Standard Error of Mean (SEM) = 5.16

  • Calculated t value: Upon applying the t statistic formula, we arrive at:t = 1.74

Testing Between Population and Sample Means

  • Null Hypothesis (H0): In this scenario, H0 asserts that µYORKU = µKINE, specifically that the average height of KINE students is equal to that of YorkU students, where µ = 169 cm.

  • Under actual conditions, we anticipate t = 0; however, sampling errors may lead to variations in results. This particular test is classified as a one-sample t-test, allowing for the analysis of one sample against a known population mean.

Understanding t-values

  • Key Question: Does the observed t value arise from mere sampling error, or is there a substantial difference between groups? Utilize the t-distribution table, a critical tool for statistical inference, to make judgments regarding significance levels.

  • Degrees of Freedom Calculation: For our example, we compute df = 15 - 1 = 14.

Critical t-value

At the commonly used significance level of 0.05, the critical value for df = 14 is determined to be 2.145.

  • Evaluation:

    • Calculated t-value = 1.74

    • Range of acceptance: -2.145 to +2.145

Decision Making

Based on the calculated t-value and the critical t-value:

  • If calculated t > critical value, we reject the null hypothesis, indicating a less than 5% chance of the observed difference being due to sampling error.

  • Conversely, if calculated t < critical value, we