| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2014 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample confidence interval t-distribution |
| Difficulty | Standard +0.3 This is a straightforward S4 question requiring standard confidence interval calculations for mean (using t-distribution) and standard deviation (using chi-squared), followed by a routine normal distribution calculation. All techniques are textbook procedures with no novel problem-solving required, though it involves multiple steps and careful handling of the chi-squared interval. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
**Total: 16 marks**
A large company has designed an aptitude test for new recruits. The score, $S$, for an individual taking the test, has a normal distribution with mean $\mu$ and standard deviation $\sigma$.
In order to estimate $\mu$ and $\sigma$, a random sample of $15$ new recruits were given the test and their scores, $x$, are summarised as
$$\sum x = 880 \quad \sum x^2 = 54892$$
**(a) Calculate a 95% confidence interval for**
**(i) $\mu$,**
**(ii) $\sigma$.**
(11 marks)
The company wants to ensure that no more than $80\%$ of new recruits pass the test.
**(b) Using values from your confidence intervals in part (a), estimate the lowest pass mark they should set.**
(5 marks)
---5. A large company has designed an aptitude test for new recruits. The score, $S$, for an individual taking the test, has a normal distribution with mean $\mu$ and standard deviation $\sigma$.
In order to estimate $\mu$ and $\sigma$, a random sample of 15 new recruits were given the test and their scores, $x$, are summarised as
$$\sum x = 880 \quad \sum x ^ { 2 } = 54892$$
\begin{enumerate}[label=(\alph*)]
\item Calculate a 95\% confidence interval for
\begin{enumerate}[label=(\roman*)]
\item $\mu$,
\item $\sigma$.
The company wants to ensure that no more than $80 \%$ of new recruits pass the test.
\end{enumerate}\item Using values from your confidence intervals in part (a), estimate the lowest pass mark they should set.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2014 Q5 [16]}}