| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Challenging +1.8 This S4 hypothesis test question requires testing variance using chi-squared distribution (part a), finding critical values from tables (part b), and calculating Type II error probability (part c). While mechanically straightforward for students who know the chi-squared test for variance, it's harder than typical S2 questions due to: (1) testing variance rather than mean, (2) requiring table manipulation for critical regions, and (3) the conceptually demanding Type II error calculation involving probability under an alternative hypothesis. The multi-part structure and S4-level content place it above average difficulty. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
**Total: 12 marks**
At the start of each academic year, a large college carries out a diagnostic test on a random sample of new students. Past experience has shown that the standard deviation of the scores on this test is $19.71$
The admissions tutor claimed that the new students in $2013$ would have more varied scores than usual. The scores for the students taking the test can be assumed to come from a normal distribution. A random sample of $10$ new students was taken and the score $x$, for each student was recorded. The data are summarised as $\sum x = 619$ $\sum x^2 = 42397$
**(a) Stating your hypotheses clearly, and using a 5% level of significance, test the admission tutor's claim.**
(6 marks)
The admissions tutor decides that in future he will use the same hypotheses but take a larger sample of size $30$ and use a significance level of $1\%$.
**(b) Use the tables to show that, to 3 decimal places, the critical region for $S^2$ is $S^2 > 664.281$**
(3 marks)
**(c) Find the probability of a type II error using this test when the true value of the standard deviation is in fact $22.20$**
(3 marks)
---\begin{enumerate}
\item At the start of each academic year, a large college carries out a diagnostic test on a random sample of new students. Past experience has shown that the standard deviation of the scores on this test is 19.71
\end{enumerate}
The admissions tutor claimed that the new students in 2013 would have more varied scores than usual. The scores for the students taking the test can be assumed to come from a normal distribution. A random sample of 10 new students was taken and the score $x$, for each student was recorded. The data are summarised as $\sum x = 619 \sum x ^ { 2 } = 42397$\\
(a) Stating your hypotheses clearly, and using a $5 \%$ level of significance, test the admission tutor's claim.
The admissions tutor decides that in future he will use the same hypotheses but take a larger sample of size 30 and use a significance level of 1\%.\\
(b) Use the tables to show that, to 3 decimal places, the critical region for $S ^ { 2 }$ is $S ^ { 2 } > 664.281$\\
(c) Find the probability of a type II error using this test when the true value of the standard deviation is in fact 22.20\\
\hfill \mbox{\textit{Edexcel S4 2014 Q4 [12]}}