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CAIE A-Level Further Mathematics 4.3.2 Goodness-of-Fit Tests

Practise testing a named probability model by calculating expected frequencies, combining low-count classes and comparing the chi-squared statistic correctly.

Syllabus
2028–2030
Course
Further Mathematics 9231
Level
AS

Exam points

  • state hypotheses that name the proposed distribution or density model and the observed data
  • combine adjacent classes until every expected frequency used in the test is at least five
  • adjust degrees of freedom for estimated parameters and compare the statistic with the critical value

4.3.2—Goodness-of-fit test question 1

[Maximum number: 6]

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 4.3.2—Goodness-of-fit test 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)={124(4x2+x2)1⩽x⩽40 otherwise 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.

Table for Question 4.3.2—Goodness-of-fit test question 1 — CAIE A-Level Further Math AS

Carry out a goodness of fit test, at the 10% significance level, to test whether f is a satisfactory model for the data.

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