A-Level Further Maths: Further Statistics Practice

A-Level Further Maths — Further Statistics Practice

16 MCQ practice problems. Select an answer, submit, and review the explanation.

What These Questions Test

These problems test your ability to work with probability distributions, calculate expectations and variances, and apply formal hypothesis testing methods.

Typical question types:

PMF calculations: Given a table of probabilities, find $E(X)$ , $\text{Var}(X)$ , or $P(a \leq X \leq b)$ . Use the definition $E(X) = \sum x \cdot P(X = x)$ .
PDF calculations: Given a probability density function, find probabilities by integrating, calculate $E(X)$ and $\text{Var}(X)$ using integrals, or find the cumulative distribution function.
Coding and transformations: Apply $E(aX + b) = aE(X) + b$ and $\text{Var}(aX + b) = a^{2}\text{Var}(X)$ to transform distributions. Standardise normal variables.
Continuous distributions: Identify and work with uniform, exponential, or other named distributions. Find means, variances, and probabilities.
$t$ -distribution: Determine when to use $t$ instead of $z$ . Calculate $t$ -statistics from sample data. Find critical values or $p$ -values.
$\chi^{2}$ tests: Set up hypotheses for goodness of fit or independence. Calculate expected frequencies. Compute $\chi^{2}$ statistic and compare with critical values. Determine degrees of freedom correctly.

Approach Strategy

Check that probabilities sum to 1. Before doing any calculation, verify that $\sum P(X = x) = 1$ (discrete) or $\int f(x)\,dx = 1$ (continuous). If they don’t, re-read the question.
Draw a table for discrete distributions. Listing $x$ and $P(X = x)$ side by side makes it easier to calculate $E(X)$ and $E(X^{2})$ .
State the hypotheses evidently. For hypothesis testing, write $H_{0}$ and $H_{1}$ explicitly before calculating. This prevents confusion later.
Check degrees of freedom carefully. For $\chi^{2}$ goodness of fit: $df = k - 1 - m$ where $k$ is the number of categories and $m$ is the number of estimated parameters. For contingency tables: $df = (r-1)(c-1)$ .

Intuition

The PDF is like a histogram with infinitely many bins. The height of the curve at any point tells you how dense the probability is in that region. Areas under the curve give probabilities.

The $t$ -distribution reflects real-world uncertainty. When you estimate $\sigma$ from a small sample, you are less confident, so the $t$ -distribution gives wider confidence intervals. This is like adding a “margin of error for your margin of error.”

The $\chi^{2}$ test measures the “distance” between what you observed and what you expected. A large $\chi^{2}$ value means the observed data is far from the expected pattern, which is evidence against the null hypothesis.

Common Mistakes

Forgetting to check that expected frequencies are large enough. For $\chi^{2}$ tests, all expected frequencies should be at least 5. If some are smaller, categories must be combined.
Confusing one-tailed and two-tailed tests. A one-tailed test looks for a difference in one direction; a two-tailed test looks for a difference in either direction. This affects the critical value.
Using $n$ instead of $n-1$ for degrees of freedom in the $t$ -distribution. The degrees of freedom are $n - 1$ , not $n$ . Using $n$ gives slightly incorrect critical values.
Misinterpreting “not significant.” Failing to reject $H_{0}$ does not mean $H_{0}$ is true. It means there is insufficient evidence to reject it. The distinction is important.

A-Level Further Maths: Further Statistics Practice