A-Level Mathematics October/November 2024 Q3(a): A factory owner models the number of employees who use the factory canteen on any day b…

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  Let $X$ be the number of employees who use the factory canteen on a given day. The distribution is given as $X \sim B(25, p)$.

  1. State Hypotheses We are testing if the proportion has increased from the old value of 0.8. Null hypothesis, $H_0: p = 0.8$ Alternative hypothesis, $H_1: p > 0.8$

  Significance level = 10% or 0.1.

  2. Calculate Test Statistic Under $H_0$, the distribution is $X \sim B(25, 0.8)$. The observed value is $x = 23$. For a one-tailed test for an increase, we calculate the probability of observing this value or a more extreme value, which is $P(X \ge 23)$.

  $P(X \ge 23) = P(X=23) + P(X=24) + P(X=25)$

  Using the binomial formula $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$: $P(X=23) = \binom{25}{23}(0.8)^{23}(0.2)^2 = 300 \times 0.0017705... \times 0.04 = 0.07082...$ $P(X=24) = \binom{25}{24}(0.8)^{24}(0.2)^1 = 25 \times 0.0014164... \times 0.2 = 0.02361...$ $P(X=25) = \binom{25}{25}(0.8)^{25}(0.2)^0 = 1 \times 0.003777... \times 1 = 0.00378...$

  $P(X \ge 23) = 0.07082 + 0.02361 + 0.00378 = 0.09821...$ So, the p-value is $0.0982$ (to 4 s.f.).

  3. Compare and Conclude We compare the p-value with the significance level: $0.0982 < 0.1$

  Since the p-value is less than the significance level, we reject the null hypothesis, $H_0$. There is sufficient evidence at the 10% significance level to suggest that the proportion of employees who use the canteen has increased since the new menu was introduced.

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  How the marks are awarded

  • B1 — Correctly stating both the null hypothesis ($H_0: p = 0.8$) and the alternative hypothesis ($H_1: p > 0.8$).
  • M1 — Setting up the correct expression for the tail probability, $P(X \ge 23)$, by summing the probabilities for $X=23, 24,$ and $25$. The full expression must be seen or implied by the individual calculations.
  • A1 — Calculating the correct probability value of 0.0982.
  • M1 — Making a valid comparison between the calculated p-value (0.0982) and the 10% significance level (0.1).
  • A1 — Stating a correct, non-definite conclusion in the context of the problem, consistent with the comparison. This includes rejecting H₀ and mentioning the proportion of employees or the effect of the new menu.

  Common mistakes

  • Using a two-tailed test ($H_1: p \ne 0.8$) instead of a one-tailed test, which is incorrect as the question asks if the value has 'increased'.
  • Calculating the wrong tail probability, such as $P(X > 23)$ or $P(X=23)$ or $P(X \le 23)$, instead of the correct $P(X \ge 23)$.
  • Making an incorrect comparison or conclusion, for example, concluding that $0.0982 > 0.1$ and therefore failing to reject $H_0$.
  • Stating a definite conclusion, such as 'the proportion has increased', instead of a nuanced one like 'there is sufficient evidence to suggest the proportion has increased'.

  Examiner tip: For any hypothesis test, carefully read the question to determine the direction of the test (increase, decrease, or change) to correctly define your alternative hypothesis and identify the correct probability tail to calculate.

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