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 Mechanics - Question 12 - 2019 - Paper 3

We will use a binomial distribution with parameters:

• $n = 7$ (the number of Christmas holidays without illness since January 2007)
• $p = 0.65$ (the hypothesized probability of falling ill).

We need to find $P(X \leq 2)$ where $X$ is the number of times he falls ill. Using a binomial calculator: $P(X \leq 2) = \sum_{k=0}^{2} {7 \choose k} (0.65)^k (0.35)^{7-k}$ Evaluating this gives approximately 0.0556.

Since $P(X \leq 2) = 0.0556$, which is greater than the significance level of 0.05, we fail to reject the null hypothesis.

Based on our calculations, we accept the null hypothesis, indicating that there is not sufficient evidence to claim that John's rate of illness during the Christmas holidays has decreased since increasing his weekly exercise.

Thus, at the 5% level of significance, we conclude that there is no significant evidence that John's rate of illness during the Christmas holidays has decreased since he increased his weekly exercise.