ECMT1020 Chap.8 Distributions Used for Inference
Distributions Used for Inference
Reference distributions arise from how statistics are constructed. This chapter links standardised averages to the normal distribution, estimated scale to the t distribution, sums of squared standard normals to chi-squared laws and scaled variance ratios to F laws. It uses support, symmetry, tail weight and degrees of freedom to check a distribution choice before applying critical values.
The t distribution is symmetric like the normal but pays for estimating scale with heavier tails at low degrees of freedom. Chi-squared and F distributions are nonnegative and typically asymmetric because squares and variance ratios generate them. These shapes affect where critical regions lie. A distribution name is therefore the end of a derivation, not a guess based on whether the original variable looks bell-shaped.
The statistic's numerator, denominator and assumptions determine the reference law.
What this chapter covers
- 01
Standard normal distribution
- 02
Student t distribution
- 03
Chi-squared distribution
- 04
F distribution
- 05
Degrees of freedom
- 06
Critical regions and tails
Worked example · free
Choose a t reference
- +1Population scale is unknown, so the sample standard deviation is used.
- +1Degrees of freedom are 16-1=15.
- +1The estimated standard error is 8/√16=2.
- +1The null gap is 74-70=4.
- +1The statistic is 4/2=2.
- +1Its reference law is t with 15 degrees of freedom.
Key terms
- Standard normal
- A normal distribution with mean zero and variance one.
- Student t
- A symmetric reference distribution with heavier tails than the normal.
- Chi-squared distribution
- A nonnegative distribution generated by sums of squared standard normals.
- F distribution
- A nonnegative distribution generated by a ratio of scaled variances.
- Degrees of freedom
- The number of independent pieces of information remaining after restrictions.
- Tail weight
- The amount of probability a distribution places far from its centre. At low degrees of freedom, a t distribution has heavier tails than a standard normal because its denominator contains an estimated scale.
- Critical region
- The set of reference-distribution values that lead to rejection under a preselected rule. Its location depends on the alternative direction, while its probability under the null is controlled by the significance level.
- Reference distribution
- The probability law used to judge a statistic under stated assumptions. It follows from the statistic's construction rather than from the appearance of the raw variable.
Distributions Used for Inference FAQ
Why does t have heavier tails?
Estimating the denominator adds uncertainty. The t distribution assigns more probability to large standardised values, especially when degrees of freedom are low and scale estimation is imprecise.
When does t approach normal?
As degrees of freedom grow, the uncertainty introduced by estimating scale diminishes and the t curve approaches the standard normal curve in centre, spread and tail behaviour.
How does support check a distribution?
Chi-squared and F variables cannot be negative because of their squared constructions. Normal and t variables can take values on either side of zero, matching their full real-line support.
Exam move
Learn each distribution through its construction, not just its curve. For a proposed statistic, trace the numerator, denominator and estimated quantities, then name degrees of freedom. Check whether the resulting value fits the distribution's support and whether a one-sided or two-sided rule matches the question before reading a critical value. Build a reference-law map with four branches.
Put standardised averages with known scale under normal, estimated scale under t for the stated normal-sample result, sums of squared standard normals under chi-squared, and ratios of scaled chi-squared quantities under F. Beside every branch write its support, symmetry and degrees-of-freedom requirement. Reconstruct the map from memory rather than memorising a table of names.
For t problems, underline whether the population standard deviation is known or replaced by the sample standard deviation. Calculate the standard error and degrees of freedom on different lines. Compare t curves at small and large degrees of freedom and explain the tail change as uncertainty about scale. For chi-squared and F problems, use nonnegative support as an immediate check.
A negative calculated value signals a transformation or arithmetic error before any table lookup. Practise matching tail instructions to a diagram. Mark the significance probability in the requested upper, lower or two-sided regions, and write how it is divided before reading a critical value. This stops a correct statistic from being compared with the wrong cutoff. Separate exact from approximate reasoning.
If a result depends on normal sampling in a small sample, state that condition. If a large-sample argument is being used, identify it as an approximation rather than silently changing the theorem. Your final audit should answer five questions: what was standardised, what scale was known or estimated, which distribution follows, what degrees of freedom apply, and where the relevant tail area lies.
When all five agree, the inference calculation has a coherent reference model.
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