20.4 Parameters vs. Test Statistics

This section clarifies the distinction between parameters and test statistics in the context of statistical inference and hypothesis testing.

What is a Parameter?

A parameter is a fixed, numerical value that describes a characteristic of an entire population. These values are typically unknown in practice because it's often impossible or impractical to collect data from every member of a population.

Key Characteristics:

• Represents a population trait.
• Is a constant value for a given population.
• Is generally unknown.

Examples of Population Parameters:

• Population Mean ($\mu$): The average value of a characteristic for all individuals in a population.
• Population Variance ($\sigma^2$): A measure of the spread or dispersion of all values around the population mean.
• Population Proportion ($p$): The proportion of individuals in a population that possess a specific characteristic.

What is a Test Statistic?

A test statistic is a value calculated from sample data that is used in hypothesis testing. Its primary purpose is to quantify how much the observed sample data deviates from what would be expected under the null hypothesis. The test statistic serves as a bridge to make inferences about the unknown population parameter.

Key Characteristics:

• Is calculated from sample data.
• Is a variable value that changes from sample to sample.
• Is known (computed from the available sample).
• Measures the discrepancy between sample results and the null hypothesis.

Examples of Test Statistics:

• t-statistic: Used in t-tests to compare sample means to a hypothesized population mean. $$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$ where:

  • $\bar{x}$ is the sample mean.
  • $\mu_0$ is the hypothesized population mean under the null hypothesis.
  • $s$ is the sample standard deviation.
  • $n$ is the sample size.

• Chi-square ($\chi^2$) statistic: Used in chi-square tests for categorical data, such as goodness-of-fit or independence tests. $$\chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}$$ where:

  • Observed is the observed frequency in a category.
  • Expected is the expected frequency in that category under the null hypothesis.

• F-statistic: Used in ANOVA (Analysis of Variance) to compare means of two or more groups by comparing variances.

Key Differences: Parameter vs. Test Statistic

Example

• Parameter: The true average height ($\mu$) of all adult residents in a specific city. This value is a fixed characteristic of the city's adult population but is likely unknown.

• Test Statistic: If we take a sample of 100 adults from that city and calculate their sample mean height and then compute a t-statistic, this t-value is the test statistic. We use this t-statistic to determine if the sample data provides enough evidence to conclude that the true average height of all adults in the city is different from a specific benchmark value.

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Interview Questions

• What is a parameter in statistics?
• How does a test statistic differ from a parameter?
• Can you give examples of common population parameters?
• What is the role of a test statistic in hypothesis testing?
• Why is a parameter considered unknown in practice?
• How is the t-test statistic calculated and interpreted?
• Explain the difference between population variance and sample variance.
• What is the chi-square test statistic used for?
• How does the F-statistic relate to ANOVA?
• Provide a real-life example of a parameter and a test statistic.