MZB127 Chap.9 Normal, t and Sampling Distributions
Normal, t and Sampling Distributions
Normal models describe continuous observations through a mean and standard deviation, while t distributions support mean inference when population spread is unknown. Sampling distributions describe repeated statistics rather than individuals. Their standard error shrinks with sample size.
The central limit theorem explains when a sample mean can be treated as approximately Normal, without claiming that the original data themselves become Normal. Standardising an individual observation uses the population standard deviation, whereas standardising a sample mean uses a standard error that divides spread by the square root of sample size.
Both distributions share the population mean as their centre, but the sample-mean distribution is narrower. Replacing an unknown population spread by the sample standard deviation introduces extra uncertainty, represented by a t distribution with degrees of freedom. Its tails are heavier for small samples and approach the standard Normal as information grows.
Tail direction is determined by the event inequality and should be sketched before a table is read. The approximation still relies on sampling conditions; a large sample does not repair dependence or bias. Clear notation for an observation, a mean and an estimate prevents denominator errors throughout later inference.
What this chapter covers
- 01
Normal location and spread
- 02
Standard Normal conversion
- 03
Student t distributions
- 04
Sampling distributions of means
- 05
Central limit theorem and standard error
Worked example · free
Find a sample-mean upper tail
- 1Identify the statistic as a sample mean.
- 1Calculate standard error 12 divided by square root 36.
- 1Standardise the threshold to z=1.5.
- 1Take the upper tail and interpret repeated samples.
Key terms
- Normal distribution
- A symmetric continuous distribution determined by its mean and standard deviation.
- Standardisation
- The conversion of a value into standard-deviation or standard-error units from its mean.
- t distribution
- A symmetric heavy-tailed family used when population spread is estimated from the sample.
- Sampling distribution
- The distribution of a statistic across repeated samples from the same population.
- Standard error
- The standard deviation of a statistic across repeated samples.
Normal, t and Sampling Distributions FAQ
When is t used instead of z?
For the one-sample mean procedures taught here, t is used when the population standard deviation is unknown and sample s replaces it. The degrees of freedom are n minus one.
What gets approximately Normal under the central limit theorem?
The sampling distribution of the sample mean becomes approximately Normal under the course conditions as sample size grows. The theorem does not say the original observations become Normal.
How does sample size change standard error?
For a mean with known population spread, standard error is sigma divided by square root n. Quadrupling n halves the standard error, improving precision without shifting the target mean.
How can the selected Normal tail be checked?
Sketch the centre and standardised threshold, then shade the event before reading a table. A threshold above the mean must have a lower-tail area above one half and an upper-tail area below one half. This qualitative prediction catches an unintended complement even when the z score itself is correct.
Exam move
Label every variable as one observation or a statistic before standardising. Sketch the requested tail and mark the centre and threshold. Use sigma for a known population spread, s for a sample estimate and the correct square-root n factor for a sample mean. Compare one-observation and sample-mean questions side by side until the denominator difference is automatic.
Return every final statement to the original units and random quantity. Make a three-row comparison sheet for an individual observation, a sample mean with known population spread and a sample mean with estimated spread. Record each centre, denominator, reference distribution and required conditions. For every probability, write the random variable and inequality before inserting numbers.
Draw a bell curve with the centre, threshold and shaded region, then predict whether the answer should be below or above one half. Standardise using the unrounded denominator and carry the unrounded score to the table lookup. Practise complementary and interval probabilities so upper, lower and between areas are selected deliberately.
To understand sample size, compare n, four n and nine n; the corresponding standard errors scale by one, one half and one third. Explain why this narrows the distribution of means without reducing the spread of individual observations. For t work, write degrees of freedom beside the statistic and note why estimated spread produces heavier tails.
State the conditions that support the Normal or t calculation and do not invoke the central limit theorem as a cure for biased sampling. Finish with a sentence about repeated observations or repeated sample means, including the original units. That wording check reveals when an individual probability has been mistakenly interpreted as a sampling probability.
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