Last year the mean time for pizza deliveries from Pete's Pizza Pit was 32.4 minutes. This year the time, t minutes, for pizza deliveries from Pete's Pizza Pit was recorded for a random sample of 50 deliveries. The results were as follows. n = 50 Σt = 1700 Σt² = 59050

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [1] (a) Unbiased estimate of population mean, x̄ = 1700/50 = 34.
2. [1] (a) Correct method for unbiased estimate of population variance, s² = (n/(n-1)) * (Σt²/n - x̄²) OR (1/(n-1)) * (Σt² - (Σt)²/n).
3. [1] (a) s² = 1250/49 ≈ 25.5 (3 s.f.).
4. [1] (b) Hypotheses: H₀: μ = 32.4, H₁: μ ≠ 32.4 (where μ is population mean time).
5. [1] (b) Calculation of test statistic z = (x̄ - μ) / sqrt(s²/n). Substitute values (34 - 32.4) / sqrt(25.5/50). Must have sqrt(n) (or 50) and not n (or 50²).
6. [1] (b) Calculated z-value = 1.6 / sqrt(0.51) ≈ 2.24 (3 s.f.).
7. [1] (b) Valid comparison of z-value to critical value (±2.326 for 2% two-tailed) OR p-value (0.025) to significance level (0.02). Since |2.24| < 2.326 (or p-value 0.025 > 0.02).
8. [1] (b) Conclusion: Do not reject H₀. Insufficient evidence that (mean) time has changed. Conclusion must be in context and not definite.
9. [1] (c) The distribution of times in the population is normal (Accept underlying distribution for population).