The frequency density for a class is defined as
$\text{frequency density}=\dfrac{\text{frequency}}{\text{class width}}.$
Show that the frequency density for the $\pounds 10,000$–$\pounds 20,000$ class is $0.0066$.

By reading the histogram, write down the frequency density for the $\pounds 50,000$–$\pounds 90,000$ class.

Hence estimate the number of cars sold in the $\pounds 50,000$–$\pounds 90,000$ class.

Using the axes and scale from the histogram, estimate the frequency in each class and the total number of cars sold. Show your working; you may present your results in a short table.

Using class midpoints together with your class frequencies from part (d), use your GDC to estimate the mean sale price (to the nearest $\pounds 100$).

Comment on the skewness of the distribution, giving a reason from the histogram.

Write a formula for $f(x)$ on $0 \le x \le a$ and for $x$ outside this interval.

Using the fact that the total area under a probability density function is $1$, find $a$. (You may use the area of a quarter circle.)

Use your GDC to evaluate $E(X)=\int_{0}^{a} x f(x),dx$. Give your answer to $3$ significant figures. (Exact working with a correct value of $a$ earns full credit.)

Find $\mathrm{Var}(X)$. You may use technology to compute $E(X^{2})=\int_{0}^{a} x^{2} f(x),dx$. Give an exact simplified form and a decimal value.

State the mode of this distribution and explain briefly whether the median is less than, equal to, or greater than the mean.

Find $AB$, giving your answer to $3$ significant figures.

Find the area of triangle $ABC$.

Determine angle $A$ and the perpendicular height from $C$ to the line segment $AB$.

If $AB$ is increased by $1.2\text{ cm}$ while angle $ABC$ and $BC$ remain unchanged, find the new length of $AC$ (to $3$ significant figures). Comment briefly on the effect on $AC$.

Write down $n(U)$ and the number of parties attended by no child.

Find $n(A)$, $n(B)$, $n(C)$ and $n(D)$.

Compute the following probabilities as simplified fractions:
(i) $P(\text{C only})$ (ii) $P(A \cap D)$ (iii) $P(B \cup C)$ (iv) $P(A' \cap B)$.

Find the probability that a randomly chosen party was attended by exactly one child. Then find the probability that at least three children attended.

Compute $P(A \mid B)$ and $P(B \mid A)$. Which child is more likely to attend given that the other attends? Justify your answer.

Are $A$ and $B$ independent? Justify using $P(A \cap B)$ and $P(A)P(B)$.

Let $D_n$ ($\pounds$) be the debt immediately after the $n$-th payment ($D_0=24,500$). If the monthly payment is $P$ ($\pounds$), write a recurrence for $D_{n+1}$ in terms of $D_n$, $P$ and the monthly interest rate $i=0.0045$.

Use your GDC (TVM/recurrence) to find the monthly payment $P$ that clears the debt in $24$ months. Give $P$ to the nearest pound.

A maintenance fund invests $\pounds 250$ at the end of each month at a monthly rate of $0.35$. Using the geometric-series formula, find the fund value after $24$ months.

Using your answers to parts (b) and (c), decide whether the fund can cover the last two repayments in full at month $23$ (after $24$ deposits). Justify numerically.

Sensitivity analysis. If the loan rate rises to $0.55$ per month, estimate the new monthly payment and the percentage increase from part (b).

Show algebraically that the level-payment formula is
$P=\dfrac{iL}{1-(1+i)^{-n}}$
for principal $L$, monthly rate $i$, and $n$ payments. Outline the geometric-series steps clearly.