| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | State meaning of Type II error |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question covering standard S2 content: calculating sample statistics from frequency data, performing a two-tailed z-test for a mean, and explaining Type I/II errors. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average for A-level statistics. |
| 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 |
| Number of computers | 0 | 1 | 2 | 3 | 4 | \(> 4\) |
| Number of households | 5 | 12 | 18 | 10 | 5 | 0 |
**(i)**
**B1** $\bar{x} = 1.96$
**M1** $(\Sigma x^2 f = 254)$; $S^2 = \frac{x - 1.96^2}{49}\left(\frac{254}{50}\right)$
**A1** $= \frac{1548}{1225}$ or $1.2637$
**[3]** $S^2$ correct sub in formula
**(ii)**
**B1** Ho: Pop mean $= 1.66$
**M1** H1: Pop mean $\neq 1.66$
**A1** $\frac{1.96-1.66}{\sqrt{1.2637/50}} = 1.887$
**M1** $z = 1.96$; $1.887 < 1.96$
**A1ft** No evidence that mean has changed
**[5]** If H1: Pop mean $> 1.66$ B0; $z = 1.645$ M1; In context Evidence mean has changed A1ft
**(iii)**
**B1f** No because Ho not rejected
**[1]** If Ho rejected in (ii): Yes because Ho rejected
**(iv)**
**B1** State mean not changed when it has
**B1** In context State mean not increased when it has
**[2]** $-1.96 < $ test stat $< 1.96$; test stat $< 1.645$
---6 A survey taken last year showed that the mean number of computers per household in Branley was 1.66 . This year a random sample of 50 households in Branley answered a questionnaire with the following results.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Number of computers & 0 & 1 & 2 & 3 & 4 & $> 4$ \\
\hline
Number of households & 5 & 12 & 18 & 10 & 5 & 0 \\
\hline
\end{tabular}
\end{center}
(i) Calculate unbiased estimates for the population mean and variance of the number of computers per household in Branley this year.\\
(ii) Test at the $5 \%$ significance level whether the mean number of computers per household has changed since last year.\\
(iii) Explain whether it is possible that a Type I error may have been made in the test in part (ii).\\
(iv) State what is meant by a Type II error in the context of the test in part (ii), and give the set of values of the test statistic that could lead to a Type II error being made.
\hfill \mbox{\textit{CAIE S2 2012 Q6 [11]}}