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
Samples of sizes 13 and 9 give variances 24.5 and 8.2. State the F statistic and its degrees of freedom for the alternative that the first population is the more variable.
Back: The alternative names the first population, so its variance is the numerator: F = 24.5/8.2 = 2.99, on 12 and 8 degrees of freedom, in that order. Larger-over-smaller is the two-tailed rule only; a one-tailed test takes its numerator from the alternative.

Front: RECALL
State the rules for 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 (Spearman assumes no tied ranks):
r = ?, r_s = ?\(r = ?, r_{s} = ?\)r = ?, r_s = ?
Back: S_xy√S_xx S_yy, 1 - 6 Σ d^2n(n^2 - 1)\(\frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}, 1 - \frac{6 Σ d^{2}}{n(n^{2} - 1)}\)(S_xy/√(S_xx S_yy)), 1 - (6 Σ d^2)/(n(n^2 - 1))

Front: STANDARD RESULT — LEARN IT
Pooled variance:
s^2 = ?, degrees of freedom = ?\(s^{2} = ?, \text{degrees of freedom} = ?\)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\(\frac{(n_{1} - 1)s_{1}^{2} + (n_{2} - 1)s_{2}^{2}}{n_{1} + n_{2} - 2}, n_{1} + n_{2} - 2\)(((n_1 - 1)s_1^2 + (n_2 - 1)s_2^2)/(n_1 + n_2 - 2)), n_1 + n_2 - 2

Front: STANDARD RESULT — LEARN IT
Standard error and interval, σ known:
se = ?, interval at 95 per cent = ?\(\text{se} = ?, \text{interval at 95 per cent} = ?\)se = ?, interval at 95 per cent = ?
Back: σ√n, sample mean ± 1.96 × se\(\frac{σ}{\sqrt{n}}, \text{sample mean} ± 1.96 \times \text{se}\)(σ/√(n)), sample mean ± 1.96 × se

Front: STANDARD RESULT — LEARN IT
Difference of two means:
standard error = ?\(\text{standard error} = ?\)standard error = ?
Back: √σ_x^2n_x + σ_y^2n_y. Independent samples: variances add, never standard deviations\(\sqrt{\frac{σ_{x}^{2}}{n_{x}} + \frac{σ_{y}^{2}}{n_{y}}}. \text{ Independent samples: variances add, never standard deviations}\)√(σ_x^2/n_x + σ_y^2/n_y). Independent samples: variances add, never standard deviations

Front: STANDARD RESULT — LEARN IT
Least squares:
b = ?, RSS = ?\(b = ?, \text{RSS} = ?\)b = ?, RSS = ?
Back: S_xyS_xx, S_yy - (S_xy)^2S_xx\(\frac{S_{xy}}{S_{xx}}, S_{yy} - \frac{(S_{xy})^{2}}{S_{xx}}\)(S_xy/S_xx), S_yy - (S_xy)^2/S_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. Once the interval is calculated it either contains that mean or it does not, and no probability is left. What varies from sample to sample is the interval, and 95 per cent of intervals built this way would capture the mean.

Front: STANDARD RESULT — LEARN IT
Continuous uniform:
U(a, b): E(X) = ?, Var(X) = ?\(\text{U}(a, b): \text{E}(X) = ?, \text{Var}(X) = ?\)U(a, b): E(X) = ?, Var(X) = ?
Back: a + b2, (b - a)^212\(\frac{a + b}{2}, \frac{(b - a)^{2}}{12}\)((a + b)/2), (((b - a)^2)/12)

Front: DEFINE
Define: positive skew.
Back: A long tail running to the right, with the bulk of the data to its left. In the single-peaked cases on this paper mode sits below median below mean, and quoting that order with the tail named earns the mark. It is evidence rather than proof: three equal averages do not force symmetry.

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

Front: DEFINE
Define: residual.
Back: The observed y minus the y the regression line predicts at that same x, so a vertical gap and never a perpendicular one. Least squares makes the sum of their squares as small as possible, and the residuals themselves always add to zero.

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

Front: STANDARD RESULT — LEARN IT
Combining independent normals:
X, Y independent: aX ± bY is ?\(X, Y \text{ independent}: aX ± bY \text{ is ?}\)X, Y independent: aX ± bY is ?
Back: N(aμ_x ± bμ_y, a^2σ_x^2 + b^2σ_y^2), so variances always add\(\text{N}(aμ_{x} ± bμ_{y}, \, a^{2}σ_{x}^{2} + b^{2}σ_{y}^{2}), \text{ so variances always add}\)N(aμ_x ± bμ_y, a^2σ_x^2 + b^2σ_y^2), so variances always add

Front: STANDARD RESULT — LEARN IT
Density and distribution:
P(a < X ≤ b) = ?, f(x) = ?\(\text{P}(a < X \le b) = ?, \text{f}(x) = ?\)P(a < X ≤ b) = ?, f(x) = ?
Back: ∫_a^b f(x) dx, dF(x)dx\(\text{∫}_{a}^{b} \text{f}(x) \, dx, \frac{d\text{F}(x)}{dx}\)∫_a^b f(x) dx, (d F(x)/dx)