| Exam Board | Edexcel |
|---|---|
| Module | FS2 (Further Statistics 2) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Two-sample t-test with summary statistics |
| Difficulty | Standard +0.3 This is a standard two-sample z-test with known variances, requiring routine application of the difference of means test. Part (a) involves straightforward hypothesis setup and calculation with given values. Parts (b) and (c) test understanding of when to use z vs t-tests and the Central Limit Theorem, but require only textbook knowledge rather than problem-solving insight. The large sample sizes and clear structure make this easier than average. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \mu_{b(lue)} = \mu_{w(hite)} + 5 \quad H_1: \mu_{b(lue)} > \mu_{w(hite)} + 5\) | B1 | Both hypotheses correct (oe) with correct notation (if using \(\mu_x\) and \(\mu_y\) these must be defined) |
| \(s.e. = \sqrt{\dfrac{2.6^2}{90} + \dfrac{2.4^2}{80}}\) | M1 | Calculation of standard error |
| \(z = \dfrac{39.5 - 33.7 - 5}{\sqrt{\dfrac{2.6^2}{90} + \dfrac{2.4^2}{80}}} = 2.085773...\) awrt 2.09 | M1, A1 | Standardising using normal distribution test statistic for difference of two means with known variance; awrt 2.09 |
| \(CV = 2.3263\) [or \(p\)-value \(= 0.01849...\)] | B1 | Correct critical value 2.3263 or better |
| Not significant, insufficient evidence to support Nissim's claim | A1 | Drawing a correct inference in context |
| (6 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the \(t\)-test or (since sample sizes are large,) use \(s^2\) as an approximation to \(\sigma^2\) | B1 | Correct explanation |
| (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Since sample size is large, by the Central Limit Theorem, sample means will be (approximately) normally distributed | B1 | Understanding that the assumptions required for the hypothesis test, by CLT sample means follow a normal distribution |
| \(\ldots\)so no effect as the calculations in part (a) can still be carried out | dB1 | (dep on previous B1) Correct evaluation |
| (2 marks) |
## Question 2:
### Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \mu_{b(lue)} = \mu_{w(hite)} + 5 \quad H_1: \mu_{b(lue)} > \mu_{w(hite)} + 5$ | B1 | Both hypotheses correct (oe) with correct notation (if using $\mu_x$ and $\mu_y$ these must be defined) |
| $s.e. = \sqrt{\dfrac{2.6^2}{90} + \dfrac{2.4^2}{80}}$ | M1 | Calculation of standard error |
| $z = \dfrac{39.5 - 33.7 - 5}{\sqrt{\dfrac{2.6^2}{90} + \dfrac{2.4^2}{80}}} = 2.085773...$ awrt **2.09** | M1, A1 | Standardising using normal distribution test statistic for difference of two means with known variance; awrt 2.09 |
| $CV = 2.3263$ [or $p$-value $= 0.01849...$] | B1 | Correct critical value 2.3263 or better |
| Not significant, insufficient evidence to support Nissim's claim | A1 | Drawing a correct inference in context |
| **(6 marks)** | | |
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the $t$-test **or** (since sample sizes are large,) use $s^2$ as an approximation to $\sigma^2$ | B1 | Correct explanation |
| **(1 mark)** | | |
### Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| Since sample size is large, by the Central Limit Theorem, sample means will be (approximately) normally distributed | B1 | Understanding that the assumptions required for the hypothesis test, by CLT sample means follow a normal distribution |
| $\ldots$so no effect as the calculations in part (a) can still be carried out | dB1 | (dep on previous B1) Correct evaluation |
| **(2 marks)** | | |\begin{enumerate}
\item A company produces two colours of candles, blue and white. The standard deviation of the burning times of the blue candles is 2.6 minutes and the standard deviation of the burning times of the white candles is 2.4 minutes.
\end{enumerate}
Nissim claims that the mean burning time of blue candles is more than 5 minutes greater than the mean burning time of white candles.
A random sample of 90 blue candles is found to have a mean burning time of 39.5 minutes. A random sample of 80 white candles is found to have a mean burning time of 33.7 minutes.\\
(a) Stating your hypotheses clearly, use a suitable test to assess Nissim's belief. Use a $1 \%$ level of significance.\\
(b) Explain how the hypothesis test in part (a) would be carried out differently if the variances of the burning times of candles were unknown.
The burning times for the candles may not follow a normal distribution.\\
(c) Describe the effect this would have on the calculations in the hypothesis test in part (a). Give a reason for your answer.
\hfill \mbox{\textit{Edexcel FS2 2021 Q2 [9]}}