During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching. In January 2007, he increased his weekly exercise to try to improve his health. For the next 7 years, he only fell ill during 2 Christmas holidays. 12 (a) Using a binomial distribution, investigate, at the 5% level of significance, whether there is evidence that John's rate of illness during the Christmas holidays had decreased since increasing his weekly exercise.

A-Level AQA Maths Pure Arithmetic Sequences & Series: During the 2006 Christmas holida

During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching - AQA - A-Level Maths Pure - Question 12 - 2019 - Paper 3

Question 12

During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching. In Jan... show full transcript

Worked Solution & Example Answer:During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching - AQA - A-Level Maths Pure - Question 12 - 2019 - Paper 3

Step 1

State both hypotheses correctly for one-tailed test

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Answer

Let $X$ be the number of Christmas holidays without illness since January 2007.

Null Hypothesis ( $H_0$ ): $p = 0.65$
Alternative Hypothesis ( $H_1$ ): $p < 0.65$

Step 2

States model used (condone 0.09 rather than 0.05 PI)

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Answer

Using a binomial model with parameters $n = 7$ (the number of Christmas holidays) and $p = 0.65$ , we can evaluate the probability of falling ill during two or fewer holidays.

Step 3

Using calculator, 0.056 or better

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Answer

Calculate $P(X imes ext{or fewer})$ using a binomial distribution calculator or software:

$P(X ext{ ≤ 2}) = inom{7}{0}(0.35)^7 + inom{7}{1}(0.35)^6(0.65) + inom{7}{2}(0.35)^5(0.65)^2$

After calculation, we find that the probability is approximately $0.0556$ .

Step 4

Evaluate binomial model by comparing $P(X ≤ 2)$ with 0.05 PI

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Answer

Since $P(X ≤ 2) = 0.0556 > 0.05$ , we do not reject the null hypothesis.

Step 5

Conclude correctly in context

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Answer

There is not sufficient evidence to support the claim that John's rate of illness during the Christmas holidays has decreased since he increased his weekly exercise.

During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching - AQA - A-Level Maths Pure - Question 12 - 2019 - Paper 3

Worked Solution & Example Answer:During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching - AQA - A-Level Maths Pure - Question 12 - 2019 - Paper 3

State both hypotheses correctly for one-tailed test

States model used (condone 0.09 rather than 0.05 PI)

Using calculator, 0.056 or better

Evaluate binomial model by comparing $P(X ≤ 2)$ with 0.05 PI

Conclude correctly in context

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