Solved: On 90 journeys to his office, Isaac noted whether or not it rained. He also... [IB Mathematics Applications & Interpretation (Math AI)]

On 90 journeys to his office, Isaac noted whether or not it rained. He also recorded hisjourney time to the office, and classified each journey as short, medium or long.Of the 90 journeys to the office, there were3 short journeys when it rained, 22 mediumjourneys when it rained, and 15 long journeys when it rained. There were also 14 shortjourneys when it did not rain.Isaac carried out a χ2 test at the 5% level of significance on these data, looking at theweather and the types of journeys.

Question

SLPaper 1

On $90$ journeys to his office, Isaac noted whether or not it rained. He also recorded hisjourney time to the office, and classified each journey as short, medium or long.

Of the $90$ journeys to the office, there were $3$ short journeys when it rained, $22$ mediumjourneys when it rained, and $15$ long journeys when it rained. There were also $14$ shortjourneys when it did not rain.

Isaac carried out a $χ^{2}$ test at the $5 %$ level of significance on these data, looking at theweather and the types of journeys.

Write down $H_{0}$ , the null hypothesis for this test.

[1]

Verified

Solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

type of journey and whether it rained are independent (A1) (C1)

Note: Accept “there is no association” or “not dependent”. Do not accept “not related” or “not correlated”. Accept equivalent terms for ‘type of journey’.

[1 mark]

Find the expected number of short trips when it rained.

[3]

Verified

Solution

$\frac{17}{90} \times \frac{40}{90} \times 90$ OR $\frac{17 \times 40}{90}$ (A1)(M1)

Note: Award (A1) for $17$ or $40$ seen. Award (M1) for $\frac{17}{90} \times \frac{40}{90} \times 90$ OR $\frac{17 \times 40}{90}$ seen.

$7.56 (7.55555 \dots, \frac{68}{9})$ (A1) (C3)

[3 marks]

The $p$ -value for this test is $0.0206$ .

State the conclusion to Isaac’s test. Justify your reasoning.

[2]

Verified

Solution

reject (do not accept) $H_{0}$ (A1)

type of journey and whether it rained are not independent (A1)

Note: Follow through from part (a) for their phrasing of the null hypothesis.

$0.0206 < 0.05$ (R1) (C2)

Note: A comparison must be seen, either numerically or in words (e.g. $p$ -value < significance level). Do not award (R0)(A1).

[2 marks]

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