(a) If c(x, t) denote observed proportion of females in the age group (x, x+t) and f(x, t) is the observed proportion of females…

(a) Female birth rate in terms of fertility and correlation

Let age X be uniform on (α, β), with density 1/(β − α). For fixed t write c = c(X, t) and f = f(X, t). The estimated female birth rate is the integral of the product of the age proportion and the age-specific rate:

B̂_f(t) = ∫ c f dx = (β − α)E(cf), and E(cf) = Cov(c, f) + E(c)E(f), with Cov(c, f) = r̂ σ̂_c σ̂_f.

The estimated total fertility is T̂_f(t) = ∫ f dx = (β − α)E(f), so E(f) = T̂_f/(β − α).

The general fertility rate is taken here as the total fertility per unit of the female population in the age range, Ĝ_f = T̂_f/∫c dx. This is the form in which the stated result holds. Hence ∫c dx = T̂_f/Ĝ_f and E(c) = T̂_f/[(β − α)Ĝ_f].

Substituting,

B̂_f(t) = (β − α) r̂ σ̂_c σ̂_f + (β − α) · T̂_f/[(β − α)Ĝ_f] · T̂_f/(β − α)

= r̂_c,f|t σ̂_c σ̂_f (β − α) + [T̂_f(t)]² / [(β − α)Ĝ_f(t)], as required.

(b) Intelligence quotient and reliability of a test

The intelligence quotient measures general mental ability. Originally IQ = (mental age/chronological age) × 100; modern tests use the deviation IQ, a standard score with mean 100 and SD 15.

Procedure: a standardised test (Stanford–Binet, Wechsler or Raven's Progressive Matrices) is given under fixed instructions and time limits, and raw scores are converted using age norms to a mental age or a deviation IQ. Reliability and validity are checked by test–retest, split-half and criterion methods.

An intelligence test measures present general intellectual ability. An aptitude test measures a specific potential, such as mechanical, clerical or musical aptitude, and predicts later training or job success; it is used in guidance and selection.

By the Spearman–Brown formula rₙ = nr/[1 + (n − 1)r], with r = 0.65:

• For rₙ = 0.95: n = 0.95(0.35)/[0.65(0.05)] = 10.23, so the test needs 60 × 10.23 ≈ 614 items, i.e. about 554 more.
• Doubling (n = 2): r = 1.30/1.65 = 0.788. Tripling (n = 3): r = 1.95/2.30 = 0.848.
• For 135 items, n = 2.25 and r = 1.4625/1.8125 = 0.807.

(c) Force of mortality

The force of mortality is the instantaneous death rate at exact age x: μₓ = lim (h→0) of [probability of dying within h after age x]/h = −(1/lₓ)(dlₓ/dx), where lₓ is the number surviving to age x. Hence dlₓ₊ₜ = −μₓ₊ₜ lₓ₊ₜ dt.

First result. The deaths between ages x and x + 1 are ∫₀¹ lₓ₊ₜ μₓ₊ₜ dt, so

qₓ = (1/lₓ) ∫₀¹ μₓ₊ₜ lₓ₊ₜ dt.

Second result. The complete expectation of life is eₓ⁰ = ∫₀^∞ (lₓ₊ₜ/lₓ) dt. Differentiating with respect to x, using ∂lₓ₊ₜ/∂x = −μₓ₊ₜ lₓ₊ₜ and d(1/lₓ)/dx = μₓ/lₓ:

deₓ⁰/dx = −∫₀^∞ μₓ₊ₜ lₓ₊ₜ/lₓ dt + μₓ eₓ⁰.

Since d(lₓ₊ₜ/lₓ)/dt = −μₓ₊ₜ lₓ₊ₜ/lₓ, the integral equals −[lₓ₊ₜ/lₓ] from 0 to ∞ = 1 (as l∞ = 0). Therefore deₓ⁰/dx = μₓeₓ⁰ − 1, and

μₓ = (1/eₓ⁰)[1 + deₓ⁰/dx].