Population, parameter, statistic; samples of 3,7,11,15

Definitions

• Population: the complete collection of all individuals or items under study, for example all 2000 IOE students of a campus.
• Sample: a part of the population selected for study, for example 100 of those students.
• Parameter: a numerical measure of a population, such as the population mean μ, standard deviation σ or proportion P. It is fixed (usually unknown).
• Statistic: a numerical measure computed from a sample, such as the sample mean x̄, standard deviation s or proportion p̂. It varies from sample to sample.

Example: the average marks of all 2000 students is a parameter (μ); the average of the 100 sampled students is a statistic (x̄).

Numerical

Population: 3, 7, 11, 15 (N = 4), n = 2 without replacement, 6 samples.

Population mean and standard deviation

μ = Σx/N = 36/4 = 9

σ² = Σ(x − μ)²/N = (36 + 4 + 4 + 36)/4 = 80/4 = 20

σ = √20 = 4.4721

All samples

Sum of sample means = 5 + 7 + 9 + 9 + 11 + 13 = 54

Mean of sample means = 54/6 = 9

(equal to μ = 9).

Standard error of the mean

With sampling without replacement from a finite population:

SE = (σ/√n) × √((N − n)/(N − 1)) = (4.4721/√2) × √((4 − 2)/(4 − 1))

= 3.1623 × 0.8165 = 2.5820

(Direct check: the standard deviation of the 6 sample means is 2.5820.)

Answer: population mean = 9; standard error of the mean = 2.5820.