| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2021 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Pooled variance estimation |
| Difficulty | Standard +0.3 This is a standard two-sample z-test question with straightforward hypothesis testing and interpretation. While it covers multiple parts (stratified sampling, hypothesis test, CLT explanation, assumptions, and interpretation), each component is routine S3 material requiring only direct application of learned procedures with no novel problem-solving or mathematical insight required. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.05a Hypothesis testing language: null, alternative, p-value, significance5.01a Permutations and combinations: evaluate probabilities5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| Label academic (1–1680) and vocational (1–2520) | B1 | For numbering/labelling/ordering (o.e.) students in each group |
| Use random numbers to select from each group | B1 | For use of random sample/numbers/selection |
| 28 academic and 42 vocational | B1 | Both numbers correct with the associated group |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(H_0: \mu_v - \mu_a = 0\), \(H_1: \mu_v - \mu_a > 0\) | B1 | If hypotheses given in terms of \(\mu_a - \mu_b\), a and b must be defined. |
| \(se = \sqrt{\frac{70}{80} + \frac{60}{50}}\) | M1 | Correct attempt at se – condone slip in sample sizes. |
| \(z = \frac{62-57}{\sqrt{\frac{70}{80}+\frac{60}{50}}}\) | dM1 | Dep on previous M1 standardising with \((62-57)\) and their se (Allow \(\pm\)) |
| \(z = 3.471...\) (or probability of 0.0003) | A1 | awrt \(\pm 3.47\) (or awrt 0.0003) |
| cv \(z = 1.6449\) | B1 | Allow \(\pm\) but signs must be compatible. Or allow comparison with probability of 0.05 |
| Reject \(H_0\) / significant | dM1 | Dependent on 2nd M1. A correct non-contextual statement based on their normal cv and their test statistic. |
| There is evidence that the mean(o.e.) basic skills score for vocational students is greater than the mean basic skills score for academic students. | A1f.t. | Correct comment in context. Must mention "mean", "academic" and "vocational". Allow f.t. on their normal cv and their test statistic. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| Mean \(\bar{X}_a\) (basic skills) for academic students and mean \(\bar{X}_v\) (basic skills) for vocational students... | B1 | Must mention both means. |
| ...have (approximately) a normal distribution (as sample sizes are large.) | B1 | Must mention normal. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| Samples are (large enough) so that \(s^2 = \sigma^2\) | B1 | Must imply for both samples |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| Test no longer significant so insufficient evidence to reject \(H_0\) | M1 | Can be implied by correct comment in context. |
| Insufficient evidence that mean (basic skills) score for vocational students is greater than mean (basic skills) score for academic students / no longer a difference in scores / Academic students have improved their mean (basic skills) score. | A1 | Must mention scores (o.e.). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| The course was effective (o.e.) | B1 | Dep on a significant result in (b) and a non-significant result in (e) |
# Question 4:
## Part 4(a):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Label academic (1–1680) and vocational (1–2520) | B1 | For numbering/labelling/ordering (o.e.) students in **each** group |
| Use **random** numbers to select from each group | B1 | For use of random sample/numbers/selection |
| 28 academic and 42 vocational | B1 | Both numbers correct with the associated group |
## Part 4(b):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $H_0: \mu_v - \mu_a = 0$, $H_1: \mu_v - \mu_a > 0$ | B1 | If hypotheses given in terms of $\mu_a - \mu_b$, a and b must be defined. |
| $se = \sqrt{\frac{70}{80} + \frac{60}{50}}$ | M1 | Correct attempt at se – condone slip in sample sizes. |
| $z = \frac{62-57}{\sqrt{\frac{70}{80}+\frac{60}{50}}}$ | dM1 | Dep on previous M1 standardising with $(62-57)$ and their se (Allow $\pm$) |
| $z = 3.471...$ (or probability of 0.0003) | A1 | awrt $\pm 3.47$ (or awrt 0.0003) |
| cv $z = 1.6449$ | B1 | Allow $\pm$ but signs must be compatible. Or allow comparison with probability of 0.05 |
| Reject $H_0$ / significant | dM1 | Dependent on 2nd M1. A correct non-contextual statement based on their **normal** cv and their test statistic. |
| There is evidence that the **mean(o.e.)** basic skills score for **vocational** students is greater than the **mean** basic skills score for **academic** students. | A1f.t. | Correct comment in context. Must mention "mean", "academic" and "vocational". Allow f.t. on their normal cv and their test statistic. |
## Part 4(c):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Mean $\bar{X}_a$ (basic skills) for academic students and mean $\bar{X}_v$ (basic skills) for vocational students... | B1 | Must mention **both** means. |
| ...have (approximately) a normal distribution (as sample sizes are large.) | B1 | Must mention normal. |
## Part 4(d):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Samples are (large enough) so that $s^2 = \sigma^2$ | B1 | Must imply for both samples |
## Part 4(e):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Test no longer significant so insufficient evidence to reject $H_0$ | M1 | Can be implied by correct comment in context. |
| Insufficient evidence that mean (basic skills) **score** for vocational students is greater than mean (basic skills) **score** for academic students / no longer a difference in **scores** / Academic students have improved their mean (basic skills) **score**. | A1 | Must mention scores (o.e.). |
## Part 4(f):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| The course was **effective** (o.e.) | B1 | Dep on a significant result in (b) and a non-significant result in (e) |
---\begin{enumerate}
\item A college runs academic and vocational courses. The college has 1680 academic students and 2520 vocational students.\\
(a) Describe how a stratified sample of 70 students at the college could be taken.
\end{enumerate}
All students at the college take a basic skills test. A random sample of 50 academic students has a mean score of 57 and a variance of 60. An independent random sample of 80 vocational students has a mean score of 62 with a variance of 70\\
(b) Stating your hypotheses clearly, test at the $5 \%$ level of significance, whether or not the mean basic skills score for vocational students is greater than the mean basic skills score for academic students.\\
(c) Explain the importance of the Central Limit Theorem to the test in part (b).\\
(d) State an assumption that is required to carry out the test in part (b).
All the academic students at the college take a basic skills course. Another random sample of 50 academic students and another independent random sample of 80 vocational students retake the basic skills test. The hypotheses used in part (b) are then tested again at the same level of significance.
The value of the test statistic $z$ is now 1.54\\
(e) Comment on the mean basic skills scores of academic and vocational students after taking this course.\\
(f) Considering the outcomes of the tests in part (b) and part (e), comment on the effectiveness of the basic skills course.
\hfill \mbox{\textit{Edexcel S3 2021 Q4 [16]}}