Statistics, Probability and Discrete Mathematics — BMATB301 (Modules 1–5): 19 MCQs with answers

$\sum (y_i - \hat{y}_i)^2$

Least squares minimises the sum of squared vertical deviations between observed and fitted values.

Three

There are three unknown coefficients $a_0, a_1, a_2$, so three normal equations are needed.

$(\bar{x}, \bar{y})$

The two regression lines intersect at the point of means $(\bar{x}, \bar{y})$.

$r = \pm\sqrt{b_{yx} \cdot b_{xy}}$

The product of the two regression coefficients equals $r^2$, so $r = \pm\sqrt{b_{yx} \cdot b_{xy}}$.

$\sum P(x) = 1$

The total probability over all outcomes must equal 1, which fixes $k$.

$z = (X - \mu)/\sigma$

Standardisation subtracts the mean and divides by the standard deviation.

The conclusion must be true whenever all premises are true

Validity is about the logical link: true premises force a true conclusion.

True

The method of least squares is the standard technique for fitting curves and regression lines because it minimises the sum of squared vertical deviations.

Take logarithms of both sides

Taking logs gives $\log y = \log a + b\log x$, a linear form in $\log x$ and $\log y$.

$\rho = 1 - \frac{6\sum d_i^2}{n(n^2-1)}$

Spearman's formula uses the sum of squared rank differences $d_i$ over $n$ items.

$n$ is large and $p$ is small

Poisson is the limiting case of binomial for large $n$ and small $p$, with $\lambda = np$.

$f(t) = \frac{1}{\theta}e^{-t/\theta}$

The exponential density with mean $\theta$ is $f(t) = \frac{1}{\theta}e^{-t/\theta}$ for $t \ge 0$.

t-test

The t-test handles small samples with unknown $\sigma$, using degrees of freedom.

Observed and expected frequencies

It measures how far observed frequencies deviate from those expected under a theoretical model.

$1 - e^{-0.8}$

$P(T < t) = 1 - e^{-t/\theta}$; with $t = 8$ and $\theta = 10$, this is $1 - e^{-0.8}$.

$\{(p \to r) \land (q \to r)\} \to \{(p \lor q) \to r\}$

This compound proposition is true for every truth assignment, making it a tautology.

$\sum y_i = 5a_2 + a_1 \sum x_i + a_0 \sum x_i^2$ and $\sum x_i y_i = a_0 \sum x_i^3 + a_1 \sum x_i^2 + 5a_2$

The correct normal equations for fitting a quadratic polynomial are derived from minimizing the sum of squared differences. The first equation relates the total $y$ values to the coefficients and sums of $x$ values, while the second relates the product of $x$ and $y$ values to the coefficients.

$b_{yx} \cdot b_{xy} = r^2$

The identity $r^2 = b_{yx} \cdot b_{xy}$ holds true, where $r$ is the correlation coefficient. This relationship is fundamental in regression analysis, confirming that the product of the slopes of the two regression lines equals the square of the correlation coefficient.

The intercept of the new regression line represents the logarithm of the original intercept.

When you apply a logarithmic transformation to both variables, the resulting regression line will be linear, and the intercept of the new line will represent the logarithm of the original intercept. This transformation is a common technique to linearize power-law relationships.