| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | One-tail z-test (upper tail) |
| Difficulty | Standard +0.3 This is a straightforward one-sample z-test application with all values provided explicitly. Students must identify it as a one-tailed test, calculate the test statistic z = (3360-3300)/(450/√200) ≈ 1.886, compare to critical value 1.96, and conclude. While it requires correct identification of the test type and careful calculation, it follows a standard procedure with no conceptual subtleties or multi-step reasoning, making it slightly easier than average. |
| Spec | 2.05e Hypothesis test for normal mean: known variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| A: \(H_0: \mu = 3300\); \(H_1: \mu > 3300\) where \(\mu\) = (pop) mean mass | B1B1 | |
| B: \(H_0: \mu = 3300\); \(H_1: \mu > 3300\) | B1B0 | |
| C: \(H_0\): The (pop) mean mass is 3300; \(H_1\): The (pop) mean mass is greater than 3300 | B1B0 | Must be in terms of parameter values |
| D: \(H_0 = 3300\); \(H_0 > 3300\) | B0B0 | |
| E: \(H_0: \mu = 3300\); \(H_1: \mu \neq 3300\) where \(\mu\) = (pop) mean mass | B1B0 | |
| F: \(H_0: \mu = 3300\); \(H_1: \mu \neq 3300\) | B0B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| G: No statement of distribution; \(P(\bar{X} = 3360) = 0.0297\); \(0.0297 > 0.025\); Don't reject \(H_0\); There is no evidence that mean mass has increased | M1A1, A1, M1, A1 | |
| H: \(P(\bar{X} = 3360.5) = 0.0286\); \(0.0286 > 0.025\); Accept \(H_0\); There is evidence that mean mass hasn't increased | M1A0, A1, M1, A0 | |
| I: \(P(\bar{X} > 3360.5) = 0.0286\); Accept \(H_0\); There is evidence that mean mass hasn't increased | M1A0, A0, M1, A0 | |
| J: \(P(\bar{X} = 3359.5) = 0.024\); \(0.024 < 0.025\); Reject \(H_0\); There is evidence that mean mass has increased | M1A0, A1, M1, A1ft | |
| K: \(P(\bar{X} < 3360) = 0.970\); \(0.970 < 0.975\); Reject \(H_1\); Insufficient evidence that mean mass has changed | M1A1, A1, M1, A0 | |
| L: \(P(\bar{X} > 3360) = 0.970\); \(0.970 > 0.025\); Insufficient evidence that mean mass has increased | M1A1, A0, M0A0 | |
| M: \(\bar{X} \sim N(3300, 1012.5)\); \(P(\bar{X} > 3360) = 0.297\); \(0.297 > 0.025\); Insufficient evidence that mean mass has increased | M1A0, A1, M1A1 | |
| N: \(\mu \pm 1.96\sigma = 3237\) to \(3362\); 3360 lies within this range; Can't reject \(H_0\); Mean mass hasn't increased | M1A1, A1, M1, A0 | |
| O: \(CV = 3362\); \(3360 < 3362\); Reject \(H_0\). Evidence that level of poll has increased | M1A1, A1, M0A0 | |
| P: \((3360 - 3300) \div (450 \div \sqrt{200}) = 1.886\); \(1.886 < 1.96\); Don't reject \(H_0\). Mean mass hasn't increased | M1A1, A1, M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Q: \(H_0: \mu = 3360\); \(H_1: \mu \neq 3360\); \(0.0297 > 0.0125\); Don't reject \(H_0\); There is no evidence that mean mass has changed | B0B0, A1, M0, A0 | |
| R: \(H_0: \mu = 3360\); \(H_1: \mu \neq 3360\) where \(\mu\) = (pop) mean mass; \(0.0297 > 0.025\); Don't reject \(H_0\); There is no evidence that mean mass has changed | B1B0, A0, M0, A0 | |
| S: \(H_0\): The (pop) mean mass \(= 3360\); \(H_1\): The (pop) mean mass \(\neq 3360\); \(0.97 < 0.9875\); Accept \(H_0\); There is no evidence that mean mass has changed | B0B0, A1, M0, A0 |
## Question 11: Exemplars — Hypotheses
| Answer/Working | Marks | Guidance |
|---|---|---|
| A: $H_0: \mu = 3300$; $H_1: \mu > 3300$ where $\mu$ = (pop) mean mass | B1B1 | |
| B: $H_0: \mu = 3300$; $H_1: \mu > 3300$ | B1B0 | |
| C: $H_0$: The (pop) mean mass is 3300; $H_1$: The (pop) mean mass is greater than 3300 | B1B0 | Must be in terms of parameter values |
| D: $H_0 = 3300$; $H_0 > 3300$ | B0B0 | |
| E: $H_0: \mu = 3300$; $H_1: \mu \neq 3300$ where $\mu$ = (pop) mean mass | B1B0 | |
| F: $H_0: \mu = 3300$; $H_1: \mu \neq 3300$ | B0B0 | |
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## Question 11: Exemplars — Calculation, Comparison and Conclusion
| Answer/Working | Marks | Guidance |
|---|---|---|
| G: No statement of distribution; $P(\bar{X} = 3360) = 0.0297$; $0.0297 > 0.025$; Don't reject $H_0$; There is no evidence that mean mass has increased | M1A1, A1, M1, A1 | |
| H: $P(\bar{X} = 3360.5) = 0.0286$; $0.0286 > 0.025$; Accept $H_0$; There is evidence that mean mass hasn't increased | M1A0, A1, M1, A0 | |
| I: $P(\bar{X} > 3360.5) = 0.0286$; Accept $H_0$; There is evidence that mean mass hasn't increased | M1A0, A0, M1, A0 | |
| J: $P(\bar{X} = 3359.5) = 0.024$; $0.024 < 0.025$; Reject $H_0$; There is evidence that mean mass has increased | M1A0, A1, M1, A1ft | |
| K: $P(\bar{X} < 3360) = 0.970$; $0.970 < 0.975$; Reject $H_1$; Insufficient evidence that mean mass has changed | M1A1, A1, M1, A0 | |
| L: $P(\bar{X} > 3360) = 0.970$; $0.970 > 0.025$; Insufficient evidence that mean mass has increased | M1A1, A0, M0A0 | |
| M: $\bar{X} \sim N(3300, 1012.5)$; $P(\bar{X} > 3360) = 0.297$; $0.297 > 0.025$; Insufficient evidence that mean mass has increased | M1A0, A1, M1A1 | |
| N: $\mu \pm 1.96\sigma = 3237$ to $3362$; 3360 lies within this range; Can't reject $H_0$; Mean mass hasn't increased | M1A1, A1, M1, A0 | |
| O: $CV = 3362$; $3360 < 3362$; Reject $H_0$. Evidence that level of poll has increased | M1A1, A1, M0A0 | |
| P: $(3360 - 3300) \div (450 \div \sqrt{200}) = 1.886$; $1.886 < 1.96$; Don't reject $H_0$. Mean mass hasn't increased | M1A1, A1, M1A0 | |
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## Question 11: Exemplars — 2-tail
| Answer/Working | Marks | Guidance |
|---|---|---|
| Q: $H_0: \mu = 3360$; $H_1: \mu \neq 3360$; $0.0297 > 0.0125$; Don't reject $H_0$; There is no evidence that mean mass has changed | B0B0, A1, M0, A0 | |
| R: $H_0: \mu = 3360$; $H_1: \mu \neq 3360$ where $\mu$ = (pop) mean mass; $0.0297 > 0.025$; Don't reject $H_0$; There is no evidence that mean mass has changed | B1B0, A0, M0, A0 | |
| S: $H_0$: The (pop) mean mass $= 3360$; $H_1$: The (pop) mean mass $\neq 3360$; $0.97 < 0.9875$; Accept $H_0$; There is no evidence that mean mass has changed | B0B0, A1, M0, A0 | |11 In the past the masses of new-born babies in a certain country were normally distributed with mean 3300 g . Last year a publicity campaign was held to encourage pregnant women to improve their diet.
Following this campaign, it is required to test whether the mean mass of new-born babies has increased. A random sample of 200 new-born babies is chosen, and it is found that their mean mass is 3360 g . It is given that the standard deviation of the masses of new-born babies is 450 g .
Carry out the test at the 2.5\% significance level.
\hfill \mbox{\textit{OCR H240/02 2022 Q11 [7]}}