| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2022 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two-sample t-test (unknown variances) |
| Difficulty | Standard +0.3 This is a standard two-sample t-test with straightforward calculations: finding sample statistics from summaries, applying the pooled variance formula, and conducting a hypothesis test. While it requires multiple steps and careful arithmetic, it follows a well-rehearsed S3 procedure with no conceptual challenges or novel problem-solving required. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\bar{x} = \dfrac{7690}{100} = 76.9\) | B1 | For 76.9 |
| \(s_x^2 = \dfrac{669.24}{99} = 6.76\) | M1 A1 | M1 for use of \(\dfrac{1}{n-1}\sum(x-\bar{x})^2\) or equivalent; A1 for 6.76 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \mu_x = \mu_y\), \(H_1: \mu_x \neq \mu_y\) | B1 | Both hypotheses correct; must be in terms of \(\mu\); allow any letter for subscripts |
| \(Z = \dfrac{76.9 - 75.9}{\sqrt{\dfrac{6.76}{100} + \dfrac{2.2^2}{80}}} = 2.793\ldots\) awrt \(\pm 2.79\) | M1 M1 A1 | 1st M1: correct method for standard error; 2nd M1: attempt at \(\pm\dfrac{a-b}{\sqrt{\frac{c}{100}+\frac{d^2}{80}}}\) with at least 3 of \(a,b,c,d\) correct; A1: awrt \(\pm 2.79\) |
| Two-tailed critical value \(z = \pm 2.5758\) | B1 | awrt \(\pm 2.5758\); allow \(\pm 2.3263\) if one-tailed used |
| Reject \(H_0\)/Significant/In the critical region | M1 | Correct statement consistent with CV and \(Z\) value |
| There is sufficient evidence to suggest the mean water temperature after 4 hours for brand \(A\) is different to brand \(B\) | A1ft | Dependent on 2nd M1; correct contextual statement fitting their CV and \(Z\) value |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| It is reasonable since both samples are (reasonably) large | B1 | Correct explanation making reference to both samples; do not allow "the sample is large enough" |
# Question 2:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\bar{x} = \dfrac{7690}{100} = 76.9$ | B1 | For 76.9 |
| $s_x^2 = \dfrac{669.24}{99} = 6.76$ | M1 A1 | M1 for use of $\dfrac{1}{n-1}\sum(x-\bar{x})^2$ or equivalent; A1 for 6.76 |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \mu_x = \mu_y$, $H_1: \mu_x \neq \mu_y$ | B1 | Both hypotheses correct; must be in terms of $\mu$; allow any letter for subscripts |
| $Z = \dfrac{76.9 - 75.9}{\sqrt{\dfrac{6.76}{100} + \dfrac{2.2^2}{80}}} = 2.793\ldots$ awrt $\pm 2.79$ | M1 M1 A1 | 1st M1: correct method for standard error; 2nd M1: attempt at $\pm\dfrac{a-b}{\sqrt{\frac{c}{100}+\frac{d^2}{80}}}$ with at least 3 of $a,b,c,d$ correct; A1: awrt $\pm 2.79$ |
| Two-tailed critical value $z = \pm 2.5758$ | B1 | awrt $\pm 2.5758$; allow $\pm 2.3263$ if one-tailed used |
| Reject $H_0$/Significant/In the critical region | M1 | Correct statement consistent with CV and $Z$ value |
| There is sufficient evidence to suggest the mean water temperature after 4 hours for brand $A$ is different to brand $B$ | A1ft | Dependent on 2nd M1; correct contextual statement fitting their CV and $Z$ value |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| It is reasonable since both samples are (reasonably) large | B1 | Correct explanation making reference to both samples; do not allow "the sample is large enough" |
---\begin{enumerate}
\item An experiment is conducted to compare the heat retention of two brands of flasks, brand $A$ and brand $B$. Both brands of flask have a capacity of 750 ml .
\end{enumerate}
In the experiment 750 ml of boiling water is poured into the flask, which is then sealed. Four hours later the temperature, in ${ } ^ { \circ } \mathrm { C }$, of the water in the flask is recorded.
A random sample of 100 flasks from brand $A$ gives the following summary statistics, where $x$ is the temperature of the water in the flask after four hours.
$$\sum x = 7690 \quad \sum ( x - \bar { x } ) ^ { 2 } = 669.24$$
(a) Find unbiased estimates for the mean and variance of the temperature of the water, after four hours, for brand $A$.
A random sample of 80 flasks from brand $B$ gives the following results, where $y$ is the temperature of the water in the flask after four hours.
$$\bar { y } = 75.9 \quad s _ { y } = 2.2$$
(b) Test, at the $1 \%$ significance level, whether there is a difference in the mean water temperature after four hours between brand $A$ and brand $B$. You should state your hypotheses, test statistic and critical value clearly.\\
(c) Explain why it is reasonable to assume that $\sigma ^ { 2 } = s ^ { 2 }$ in this situation.
\hfill \mbox{\textit{Edexcel S3 2022 Q2 [11]}}