4.2 Inference using normal and t-distributions

2026 Syllabus Objectives

  1. Formulate hypotheses and apply a hypothesis test concerning the population mean using a small sample drawn from a normal population of unknown variance, using a t-test
  2. Calculate a pooled estimate of a population variance from two samples (calculations based on either raw or summarised data may be required)
  3. Formulate hypotheses concerning the difference of population means, and apply, as appropriate: a 2-sample t-test; a paired sample t-test; a test using a normal distribution (the ability to select the test appropriate to the circumstances of a problem is expected)
  4. Determine a confidence interval for a population mean, based on a small sample from a normal population with unknown variance, using a t-distribution
  5. Determine a confidence interval for a difference of population means, using a t-distribution or a normal distribution, as appropriate

🔑 The t-distribution

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When to use the t-distribution

When collecting a sample from a normal distribution to carry out a hypothesis test concerning the mean, we need to make certain assumptions to use the normal distribution. Specifically, we assume that:

  • The population variance is known, OR
  • The sample size is sufficiently large (typically n≥30n \geq 30) that we can use s2s^2, the unbiased estimator of the variance, instead of the population variance σ2\sigma^2

The problem with small samples: If the sample size is small and we do not know what the variance is, then it is no longer appropriate to use the unbiased estimator with the normal distribution.

The solution: The t-distribution was developed so that the unbiased estimator could be used. It is a better model in this situation.

Degrees of freedom

The t-distribution is a family of distributions with (n−1)(n - 1) degrees of freedom.

Degrees of freedom: The number of independent observations in a set of data, typically (n−1)(n - 1) for a sample of size nn.

Shape of the t-distribution

As the sample size increases, the t-distribution looks more like the standard normal distribution:

  • The t-distribution is symmetric and bell-shaped, centered at 0
  • It has heavier tails than the normal distribution (more probability in the extremes)
  • With fewer degrees of freedom, the distribution is wider and flatter
  • As degrees of freedom increase, it approaches the standard normal distribution

📊 Visual comparison:

  • Standard normal curve: tallest and narrowest
  • t-distribution with 9 degrees of freedom: slightly lower and wider
  • t-distribution with 2 degrees of freedom: lowest at center with the heaviest tails

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