A-Level Mathematics May/June 2024 Q7(a): Every July, as part of a research project, Rita collects data about sightings of a part…

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  Let $p$ be the probability of seeing the bird on a randomly chosen day in July. The number of days the bird is seen, $X$, is modelled by $X \sim B(31, p)$.

  1. State Hypotheses We are testing the claim that the probability has been reduced from the previous value of 0.3. This is a one-tailed test. Null hypothesis, $H_0: p = 0.3$ Alternative hypothesis, $H_1: p < 0.3$

  Significance level, $\alpha = 0.05$

  2. Calculate Test Statistic Probability (p-value) Under $H_0$, the distribution is $X \sim B(31, 0.3)$. The observed value is $x=4$. We need to find the probability of observing a result as extreme or more extreme than 4, in the direction of the alternative hypothesis. This is $P(X \le 4)$.

  $P(X \le 4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)$

  Using a calculator or the binomial formula: $P(X \le 4) = \binom{31}{0}(0.3)^0(0.7)^{31} + \binom{31}{1}(0.3)^1(0.7)^{30} + \binom{31}{2}(0.3)^2(0.7)^{29} + \binom{31}{3}(0.3)^3(0.7)^{28} + \binom{31}{4}(0.3)^4(0.7)^{27}$

  $P(X \le 4) = 0.02391...$ $p\text{-value} = 0.0239$ (to 3 s.f.)

  3. Compare and Conclude We compare the p-value with the significance level: $0.0239 < 0.05$

  Since the p-value is less than the 5% significance level, we reject the null hypothesis, $H_0$.

  There is sufficient evidence at the 5% significance level to conclude that Rita's suspicion is justified. The probability of seeing this kind of bird appears to have been reduced.

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  How the marks are awarded

  • B1 — Awarded for correctly stating both the null hypothesis ($H_0: p = 0.3$) and the one-tailed alternative hypothesis ($H_1: p < 0.3$) based on the context of a suspected 'reduction'.
  • M1 — Awarded for identifying the correct binomial distribution $B(31, 0.3)$ and attempting to find the correct tail probability, $P(X \le 4)$, by summing the individual probabilities from $X=0$ to $X=4$.
  • A1 — Awarded for the correct p-value of $0.0239$ (or more accurately, 0.02391...), correctly rounded to at least 3 significant figures.
  • M1 — Awarded for making a valid comparison between the calculated p-value (0.0239) and the significance level (0.05), as shown by the inequality '$0.0239 < 0.05$'.
  • A1FT — Awarded for a correct conclusion in context, which follows from the comparison. The conclusion correctly states there is sufficient evidence to support Rita's suspicion, is not definite, and references the 5% significance level.

  Common mistakes

  • Using a two-tailed alternative hypothesis ($H_1: p \neq 0.3$) instead of a one-tailed one ($H_1: p < 0.3$), as the question specifically mentions a suspected 'reduction'.
  • Calculating the wrong tail probability, such as $P(X \ge 4)$ or just $P(X=4)$, instead of $P(X \le 4)$. The alternative hypothesis $p < 0.3$ indicates the lower tail of the distribution is the critical region.
  • Writing a definite conclusion, such as 'the probability has been reduced' or 'Rita's suspicion is correct'. Hypothesis tests only provide evidence for or against a claim; they do not prove it.
  • Forgetting to state the conclusion in the context of the original problem, for example by just writing 'Reject H₀'.

  Examiner tip: Always carefully read the question to determine whether a hypothesis test is one-tailed (e.g., 'increased', 'reduced', 'less than') or two-tailed (e.g., 'changed', 'different from') as this dictates which probability you must calculate.

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