Chapter 1: Tests for One Continuous Population Mean

This chapter introduces statistical tests designed to analyze a single sample, which is a fundamental task in data analysis across many disciplines. Whether you’re evaluating whether the average recovery time from a treatment differs from a known standard, assessing whether student test scores exceed a benchmark, or testing if the proportion of success in a group differs from an expected rate, these methods help determine whether the observed values are statistically significant or simply due to chance.

mindmap
  root((Frequentist
  Hypothesis 
  Testings
  ))
    Simulation Based<br/>Tests
    Classical<br/>Tests
      (Chapter 1: <br/>Tests for One<br/>Continuous<br/>Population Mean)
        {{Unbounded<br/>Response}}
          Known<br/>Population<br/>Variance
            )One sample<br/>t test(
          Unknown<br/>Population<br/>Variance
            )One sample<br/>z test(
        {{Proportion between<br/>0 and 1<br/>obtained from a <br/>Binary Response}}
          )One sample<br/>z test(
      (Chapter 2: <br/>Tests for Two<br/>Continuous<br/>Population Means)
      (Chapter 3: <br/>ANOVA-related <br/>Tests for<br/>k Continuous<br/>Population Means)

Figure 1: A specific hypothesis testing mind map outlining the techniques explored in this chapter, which are classical tests for one continuous population mean.

There are several statistical tests used to evaluate hypotheses about a single sample. The appropriate test depends on the type of variable (mean or proportion), sample size, and whether population parameters like variance are known.

We test whether a population mean equals a specific value. The right test depends on:

In this chapter, we focus on statistical tests used to evaluate hypotheses about a single population mean or proportion, based on sample data. These tests help determine whether a sample provides sufficient evidence to conclude that the population mean (or proportion) differs from a specified value.

We cover two cases for the mean — depending on whether the population variance is known or unknown — and one test for binary outcomes where we’re testing a population proportion.

Key tests include:

One-sample z-test for the mean

Use this test when: - The population variance σ² is known, and - The sample comes from a normally distributed population, or the sample size is large (typically ( n )).

The test statistic is:

\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]

Where: - ( \(\bar{x}\) ) is the sample mean
- ( \(\mu_0\) ) is the hypothesized population mean
- ( \(\sigma\) ) is the known population standard deviation
- ( n ) is the sample size

We compare the calculated ( z )-value to a standard normal distribution to compute a p-value or make a decision based on a critical value.

One-sample t-test for the mean

Use this test when: - The population variance is unknown, and - The sample is either normally distributed or large enough to rely on the central limit theorem.

The test statistic is:

\[t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\]

Where: - ( s ) is the sample standard deviation (used instead of ( \(\sigma\) ))

This statistic follows a t-distribution with ( n - 1 ) degrees of freedom. It is more appropriate for real-world scenarios where we rarely know the true population variance.

One-sample z-test for proportions

Use this test when: - The variable is binary (success/failure, yes/no, etc.), and - You want to test a population proportion ( p ), using a large enough sample.

The test statistic is:

\[z = \frac{\hat{p} - p_0}{\sqrt{p_0(1 - p_0)/n}}\]

Where: - ( \(\hat{p}\) ) is the sample proportion
- ( \(p_0\) ) is the hypothesized population proportion
- ( \(n\) ) is the sample size

This test assumes ( \(np_0 \geq 5\) ) and ( \(n(1 - p_0)\) \(\geq\) 5 ) to justify the normal approximation to the binomial distribution.