Title: InkMaths / Further Statistics 2: regression, densities and tests
On every note of this deck, unless a note says otherwise:
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Front: CHECK YOURSELF
Two samples of sizes 13 and 9 give variances 24.5 and 8.2. State the F statistic and its degrees of freedom.
Back: Larger variance on top: F = 24.5/8.2 = 2.99, on 12 and 8 degrees of freedom in that order.

Front: DEFINE
Define: the correlation test.
Back: Hypotheses are about the population value ρ or ρs, never the sample r. Critical values fall as n grows, and the product moment version assumes a bivariate normal population.

Front: STANDARD RESULT — LEARN IT
Correlation coefficients:
r = ?, r_s = ?
Back: S_xy√S_xx S_yy, 1 - 6 Σ d^2n(n^2 - 1)

Front: STANDARD RESULT — LEARN IT
Pooled variance:
s^2 = ?, degrees of freedom = ?
Back: (n_1 - 1)s_1^2 + (n_2 - 1)s_2^2n_1 + n_2 - 2, n_1 + n_2 - 2

Front: STANDARD RESULT — LEARN IT
Standard error and interval:
se = ?, interval at 95 per cent = ?
Back: σ√n, sample mean ± 1.96 × se

Front: STANDARD RESULT — LEARN IT
Difference of two means:
standard error = ?
Back: √σ_x^2n_x + σ_y^2n_y: the variances add

Front: STANDARD RESULT — LEARN IT
Least squares:
b = ?, RSS = ?
Back: S_xyS_xx, S_yy - (S_xy)^2S_xx

Front: SPOT THE ERROR
Say what is wrong with this claim:
A 95% confidence interval has a 95% chance of containing the population mean.
Back: The population mean is fixed, so once the interval is calculated it either contains it or does not. What varies is the interval: 95% of intervals built this way from repeated samples would capture the mean.

Front: STANDARD RESULT — LEARN IT
Continuous uniform:
U(a, b): E(X) = ?, Var(X) = ?
Back: a + b2, (b - a)^212

Front: DEFINE
Define: positive skew.
Back: Mode below median below mean, with the long tail running to the right. Reversing the order gives negative skew; all three equal means symmetric.

Front: STANDARD RESULT — LEARN IT
Testing a variance:
statistic = ?, degrees of freedom = ?
Back: (n - 1)S^2σ^2 is χ^2, n - 1

Front: DEFINE
Define: residual.
Back: The observed y minus the value the regression line predicts at the same x: a vertical gap. Least squares minimises the sum of their squares, and they always add to zero.

Front: STANDARD RESULT — LEARN IT
Continuous mean and variance:
E(X) = ?, Var(X) = ?
Back: ∫ x f(x) dx, E(X^2) - [E(X)]^2

Front: STANDARD RESULT — LEARN IT
Combining normals:
aX ± bY is ?
Back: N(aμ_x ± bμ_y, a^2σ_x^2 + b^2σ_y^2)

Front: STANDARD RESULT — LEARN IT
Density and distribution:
P(a < X ≤ b) = ?, f(x) = ?
Back: ∫_a^b f(x) dx, dF(x)dx