| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2016 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Standard +0.3 This is a straightforward two-part hypothesis testing question requiring standard procedures: a one-sample t-test and a chi-squared test for variance. While it requires calculating sample statistics from summations and applying two different tests, both are routine S4 procedures with no novel problem-solving or conceptual challenges beyond textbook application. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \mu = 1.2 \quad H_1: \mu > 1.2\) | B1 | Both hypotheses |
| \(t_8(5\%) = 1.860\) | B1 | |
| \(\bar{m} = 1.28888...\) | B1 | |
| \(t = \frac{1.28...-1.2}{\sqrt{\frac{0.031111}{9}}} = 1.511\) | M1 A1ft A1 | M1 attempting correct statistic; A1ft follow through their \(s^2\); A1 awrt 1.51 |
| Not significant. There is not sufficient evidence that the mean weight of piglets is greater than 1.2 kg | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \sigma^2 = 0.09 \quad H_1: \sigma^2 \neq 0.09\) | B1 | Both hypotheses, must be two tail |
| \(s^2 = \frac{15.2 - 9\times\left(\frac{11.6}{9}\right)^2}{8} = 0.031111\) | B1 | awrt 0.0311; NB allow 2.733 for one tail hypotheses |
| \([\chi^2_8(0.25) = 17.535] \quad \chi^2_8(0.975) = 2.18\) | B1 | |
| \(\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_8 \quad\) test statistic \(= 2.7654...\) | M1 A1 | M1 correct test statistic; awrt 2.77 |
| 2.77 is not in the critical region. There is no evidence that the standard deviation of the weights of piglets is different to 0.3 | A1 |
# Question 2:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \mu = 1.2 \quad H_1: \mu > 1.2$ | B1 | Both hypotheses |
| $t_8(5\%) = 1.860$ | B1 | |
| $\bar{m} = 1.28888...$ | B1 | |
| $t = \frac{1.28...-1.2}{\sqrt{\frac{0.031111}{9}}} = 1.511$ | M1 A1ft A1 | M1 attempting correct statistic; A1ft follow through their $s^2$; A1 awrt 1.51 |
| Not significant. There is not sufficient evidence that the mean weight of piglets is greater than 1.2 kg | A1 | |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \sigma^2 = 0.09 \quad H_1: \sigma^2 \neq 0.09$ | B1 | Both hypotheses, must be two tail |
| $s^2 = \frac{15.2 - 9\times\left(\frac{11.6}{9}\right)^2}{8} = 0.031111$ | B1 | awrt 0.0311; NB allow 2.733 for one tail hypotheses |
| $[\chi^2_8(0.25) = 17.535] \quad \chi^2_8(0.975) = 2.18$ | B1 | |
| $\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_8 \quad$ test statistic $= 2.7654...$ | M1 A1 | M1 correct test statistic; awrt 2.77 |
| 2.77 is not in the critical region. There is no evidence that the standard deviation of the weights of piglets is different to 0.3 | A1 | |
---2. The weights of piglets at birth, $M \mathrm {~kg}$, are normally distributed $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$
A random sample of 9 piglets is taken and their weights at birth, $m \mathrm {~kg}$, are recorded. The results are summarised as
$$\sum m = 11.6 \quad \sum m ^ { 2 } = 15.2$$
Stating your hypotheses clearly, test at the 5\% level of significance
\begin{enumerate}[label=(\alph*)]
\item whether or not the mean weight of piglets at birth is greater than 1.2 kg ,
\item whether or not the standard deviation of the weights of piglets at birth is different from 0.3 kg .
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2016 Q2 [13]}}