Parameter, statistic and inference – Mathematics/Applied Mathematics for NCET

Parameter and statistic

A parameter is a numerical measure describing a population, such as the population mean μ or population standard deviation σ. It is fixed but usually unknown. A statistic is a numerical measure computed from a sample, such as the sample mean x̄ or sample standard deviation s. It varies from sample to sample and is used to estimate the parameter.

Sampling distribution and standard error

If we draw all possible samples of size n and compute a statistic for each, the probability distribution of these values is the sampling distribution of that statistic. The standard deviation of a sampling distribution is called the standard error (SE). For the sample mean, SE(x̄) = σ/√n. Example: σ = 20 and n = 100 give SE = 2. Because of the √n, to halve the standard error you must make the sample four times as large.

Central limit theorem (CLT)

For a population with mean μ and standard deviation σ, when the sample size n is large (often taken as n ≥ 30), the sampling distribution of x̄ is approximately normal with mean μ and standard deviation σ/√n, whatever the shape of the population. This is why normal tables can be used for sample means.

Statistical inference and significance

Statistical inference means drawing conclusions about a population from a sample. In hypothesis testing:

The null hypothesis H₀ states no difference or no effect (for example, μ = 50).
The alternative hypothesis H₁ states what we suspect instead (μ ≠ 50, μ > 50 or μ < 50).
The level of significance α (commonly 5% or 1%) is the probability of rejecting H₀ when it is actually true.
A result is statistically significant if it is unlikely to have happened by chance alone if H₀ were true, so H₀ is rejected.

Key points

Parameter ↔ population (μ, σ); statistic ↔ sample (x̄, s).
SE of the mean = σ/√n.
CLT: x̄ is approximately normal for large n.
α = 0.05 means a 5% risk of rejecting a true H₀.

Common mistakes

Calling the sample mean a parameter.
Thinking the CLT requires the population itself to be normal.

The probability distribution of a statistic, obtained from all possible samples of the same size, is called its:	Easy
A population has standard deviation 20. The standard error of the mean for samples of size 100 is:	Medium
According to the central limit theorem, for a large sample size the sampling distribution of the sample mean is approxi…	Medium
In a hypothesis test, the 5% level of significance means:	Easy
A researcher wants to reduce the standard error of a sample mean to half its present value. The sample size must be:	Hard