Further Math A2s 4.3.2 Goodness-of-fit tests

[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 CAIE A-Level Further Math A2 4.3.2 Goodness of Fit Test Questions question 1 — CAIE A-Level Further Math A2

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.

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

Combine first two columns: 6, 9.0278
Must be seen, or implied by 1.0155 .

Calculate value of chi-squared: 1.0155+0.5715+0.7661+0.2183+0.5885
At least 2 correct values (at least 3sf) or
expressions seen.
Allow columns not combined or three columns
combined.

3.16
CWO.

Dependent on M1M1 scored. SCB1 for 3.16 with no working

$\mathrm{H}_{0}: \mathrm{f}$ is a good fit for the data

$\mathrm{H}_{1}$ : f is not a good fit for the data
Tabular value of chi-squared: 7.78

' 3.16 ' < 7.78 and accept $\mathrm{H}_{0}$
Correct tabular value: allow correct FT value if
columns not combined (9.236) or three columns
combined (6.251).

Insufficient evidence to suggest that f is not a good fit for the data

Condone: sufficient evidence to suggest that f is a good fit for the data.
Correct conclusion in context, following correct
work, level of uncertainty in language (not 'prove',
not 'there is no evidence'), no contradictions.
A0 if hypotheses reversed

6

CAIE A-Level Further Math A2 4.3.2 Goodness of Fit Test Questions question 1