A-Level Mathematics May/June 2024 Q4(b)(i): The supplier claims that the mean mass of boxes of cereal is 253g. A quality control of…

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  Let $\mu$ be the population mean mass of boxes of cereal in grams.

  1. State Hypotheses:

  • Null hypothesis, $H_0: \mu = 253$
  • Alternative hypothesis, $H_1: \mu > 253$

  This is a one-tailed test at the 5% significance level.

  2. Calculate Sample Mean and Test Statistic:

  • Sample size, $n = 100$
  • Sample mean, $\bar{x} = \frac{25360}{100} = 253.6$ g
  • Population standard deviation, $\sigma = 3.5$ g

  The test statistic is calculated as: $Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ $Z = \frac{253.6 - 253}{3.5 / \sqrt{100}}$ $Z = \frac{0.6}{0.35}$ $Z = 1.71428... \approx 1.714$

  3. Determine Critical Value and Compare: For a one-tailed test at the 5% significance level, the critical value is found from the standard normal distribution tables: $z_{crit} = \Phi^{-1}(1 - 0.05) = \Phi^{-1}(0.95) = 1.645$

  Comparing the test statistic to the critical value: $1.714 > 1.645$

  The test statistic lies in the critical region.

  4. Conclusion: Since the test statistic (1.714) is greater than the critical value (1.645), we reject the null hypothesis $H_0$.

  There is sufficient evidence at the 5% significance level to suggest that the population mean mass of the boxes of cereal is more than 253 g.

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  How the marks are awarded

  • B1 — Correctly stating both the null hypothesis ($H_0: \mu = 253$) and the one-tailed alternative hypothesis ($H_1: \mu > 253$), using the correct population parameter $\mu$.
  • M1 — Correctly applying the standardisation formula for the Z-test statistic, including the sample mean (253.6 or 25360/100) and dividing the population standard deviation (3.5) by the square root of the sample size ($\sqrt{100}$).
  • A1 — Obtaining the correct value for the test statistic, $Z = 1.714$ (or a more accurate value rounding to this).
  • M1 — Making a correct comparison between their calculated test statistic and the correct critical value for a 5% one-tailed test (1.645). For example, stating '1.714 > 1.645'.
  • A1FT — Stating a final conclusion that is consistent with the comparison. The conclusion must reject $H_0$ and be phrased in the context of the problem using non-definite language (e.g., 'sufficient evidence to suggest').

  Common mistakes

  • Using a two-tailed test (H₁: μ ≠ 253) and comparing to the wrong critical value (1.96), which would lead to an incorrect conclusion of not rejecting H₀.
  • Forgetting to divide the standard deviation by the square root of the sample size ($\sqrt{n}$), i.e., calculating $Z = (253.6 - 253) / 3.5$.
  • Using the sample mean $\bar{x}$ instead of the population mean $\mu$ when writing the hypotheses, e.g., $H_0: \bar{x} = 253$.
  • Writing a definite conclusion, such as 'The mean mass is greater than 253 g', which is too strong and lacks the required element of uncertainty.

  Examiner tip: Always carefully interpret the wording of the problem ('more than', 'less than', 'different from') to determine whether a one-tailed or two-tailed test is required, as this dictates your alternative hypothesis and critical region.

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