CAIE A-Level Further Mathematics 4.3.1 Fitted Distributions

Practise estimating distribution parameters from frequency data and calculating expected frequencies for goodness-of-fit tables.

Exam points

estimate a binomial or Poisson parameter from totals, frequencies or the sample mean
calculate expected frequencies by multiplying model probabilities by sample size
combine low expected-frequency classes before completing χ² contributions where required

[Maximum number: 3]

A random sample of 50 values of the continuous random variable X was taken. These values are summarised in the following table.

Table for Question 4.3.1—Fitted distributions question 1 — CAIE A-Level Further Math AS

It is required to test the goodness of fit of the distribution with probability density function f given by

f(x)= \begin{cases}\frac{1}{24}\left(\frac{4}{x^{2}}+x^{2}\right) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise }\end{cases}

The expected frequencies, correct to 4 decimal places, are given in the following table.

Show that a=4.6007 and find the value of b.

$\int_{1.5}^{2} \frac{1}{24}\left(\frac{4}{x^{2}}+x^{2}\right) \mathrm{d} x=\frac{1}{24}\left[-\frac{4}{x}+\frac{x^{3}}{3}\right]$

M1

Integration with correct powers and correct limits
seen.

$a=50 \times 0.092014=4.6007$

A1

AG

$\frac{53}{576}$ or $\frac{1325}{288}$ or 0.092014 or 4.60069 seen

b=11.4211

B1

Or 11.421(0)

3

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4.3.1—Fitted distributions question 1