Selecting the Right Test

Statistical Hypothesis Testing Framework

Understanding Major Concepts:

  • Null Hypothesis (H0): A statement that there is no effect or no difference, which researchers aim to test against.

  • Alternate Hypothesis (HA): Contradicts the null hypothesis; suggests there is an effect or a difference.

  • P value: Probability that measures the strength of evidence against the null hypothesis.

  • Decision Rule: Reject H0 if p < 0.05.

Choosing Statistical Tests

  • Importance of Fit: Statistical tests serve as tools for hypothesis testing, necessitating the correct selection for the data at hand.

  • Correct Application: It’s critical to understand not only the selection processes for tests but also the appropriate operational specifics.

Types of Data & Graphing

Variables in Analysis

  • Independent Variable: Considered the cause; plotted on the x-axis.

  • Dependent Variable: Considered the effect; plotted on the y-axis.

  • Graph Selection: Understanding the nature of variables aids in deciding which statistical tool is appropriate.

Graph Types

  • Bar Graphs:

    • Suited for tests of differences among two or more groups.

  • Scatter Plots

    • Proper for examining relationships between two variables.

Normal Distribution Definition

  • Defined by mean value (µ) and standard deviation (σ).

  • Statistical Ranges:

  • 68.2% of observations lie within µ±1σµ ± 1σ.

  • 95.5% lie within µ±2σµ ± 2σ.

Assumptions of Parametric Statistical Methods

  • Parametric vs Non-Parametric: Focus on methods suitable for continuous data predominantly following a normal distribution.

  • Parametric tests assume normal distribution and homogeneity of variances.

  • The Central Limit Theorem: Suggests that with larger sample sizes, distributions will tend more closely toward normal.

Measuring Sample Characteristics

  • Sample Mean (x): The average of the sample data.

  • Sample Standard Deviation (s): Measure of dispersion within the sample data.

  • Skew: deviation from symmetry caused by a long tail

  • Kurtosis: Measures the peak and tail thickness of the distribution.

9. Normality Tests

  • Formulation of Null and Alternate Hypotheses for Normality:

    • Null hypothesis (H0): Data follows a normal distribution.

    • Alternate hypothesis (HA): Data does not follow a normal distribution.

  • Example Test: Shapiro-Wilk test yields a p value for evaluation.

Q-Q Plots

  • Q-Q plots compare quantiles of observed data against theoretical quantiles of a normal distribution.

  • Points close to the reference line indicate normality; curves suggest departures from normality.

  • Caution: Normality tests can yield poor results with small sample sizes, making Q-Q plots more useful.

  • Parametric tests assume homogeneity of variances, which is crucial for analytical validity.

Error Measurements

Standard Error of the Mean (SEM)

  • Definition: SEM is the standard deviation of sample means.

  • Calculation: SEM=snSEM = \frac{s}{\sqrt{n}} where n is the sample size.

95% Confidence Interval (CI)

  • CI indicates the range surrounding the sample mean where the true population mean is expected to fall in 95% of samples.

Representing Data Variability

  • Always indicate variability around means through:

    • Sample Standard Deviation.

    • Standard Error of the Mean.

    • 95% Confidence Intervals.

  • Clearly state choices in legends upon plotting.

Key Takeaways

  • Understand key concepts:

    • Difference between independent & dependent variables.

    • Recognizing various types of data.

    • Application and implications of mean, standard deviation, and standard error in context.

  • Selected parametric tests are reserved for continuous, normally distributed variables.

  • Utilize normality tests and Q-Q plots to ensure analytical assumptions are met for valid statistical inference.