| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Identify which error type was made |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question covering standard S2 content: calculating unbiased estimates (routine formulas), conducting a one-tailed z-test (standard procedure), and identifying which type of error is possible given the test outcome. All parts follow textbook methods with no novel insight required, making it slightly easier than average. |
| 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 basketball courts | 0 | 1 | 2 | 3 | 4 | \(> 4\) |
| Number of schools | 2 | 28 | 26 | 10 | 4 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\hat{\mu} = \frac{126}{70} = \frac{9}{5} = 1.8\) | B1 | |
| \(\Sigma x^2 f = 286\) | B1 | Seen or implied |
| \(\text{Est}(\sigma^2) = \frac{70}{69}\left(\frac{\Sigma x^2 f}{70} - 1.8^2\right)\) | M1 | oe attempted |
| \(= 0.858\) or \(\frac{296}{345}\) | A1 | Note: Final answer for var 0.846 (biased) and no working implies B1 for 286 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \mu = 1.9\), \(H_1: \mu < 1.9\) | B1 | Or 'pop mean'; not just 'mean' |
| \(\frac{1.8 - 1.9}{\sqrt{\frac{0.858}{70}}}\) | M1 | Standardise with their values from (i). Must have \(\sqrt{70}\). No SD/Var mix |
| \(= -0.903\) | A1 | Accept \(\pm\) |
| \(0.903 < 1.645\) | M1 | comp 1.645; allow comp 1.96 if \(H_1: \mu \neq 1.9\); or comp \(1 - \phi(0.903) = 0.182\) or \(0.183\) with \(0.05\) |
| No evidence that mean no. courts in S is less than in N | A1ft | No contradictions. ft their 0.903, but not comp 1.96. Accept cv method: \(\text{cv} = 1.718\) M1A1 \(1.718 < 1.8\) M1 conclusion A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Type II because \(H_0\) was not rejected | B1ft | ft their conclusion. If \(H_0\) rejected, 'Type I because \(H_0\) rejected' B1. No conclusion in (ii) scores B0 |
## Question 5(i):
| $\hat{\mu} = \frac{126}{70} = \frac{9}{5} = 1.8$ | B1 | |
|---|---|---|
| $\Sigma x^2 f = 286$ | B1 | Seen or implied |
| $\text{Est}(\sigma^2) = \frac{70}{69}\left(\frac{\Sigma x^2 f}{70} - 1.8^2\right)$ | M1 | oe attempted |
| $= 0.858$ or $\frac{296}{345}$ | A1 | Note: Final answer for var 0.846 (biased) and no working implies B1 for 286 |
---
## Question 5(ii):
| $H_0: \mu = 1.9$, $H_1: \mu < 1.9$ | B1 | Or 'pop mean'; not just 'mean' |
|---|---|---|
| $\frac{1.8 - 1.9}{\sqrt{\frac{0.858}{70}}}$ | M1 | Standardise with their values from (i). Must have $\sqrt{70}$. No SD/Var mix |
| $= -0.903$ | A1 | Accept $\pm$ |
| $0.903 < 1.645$ | M1 | comp 1.645; allow comp 1.96 if $H_1: \mu \neq 1.9$; or comp $1 - \phi(0.903) = 0.182$ or $0.183$ with $0.05$ |
| No evidence that mean no. courts in S is less than in N | A1ft | No contradictions. ft their 0.903, but not comp 1.96. Accept cv method: $\text{cv} = 1.718$ M1A1 $1.718 < 1.8$ M1 conclusion A1 |
---
## Question 5(iii):
| Type II because $H_0$ was not rejected | B1ft | ft their conclusion. If $H_0$ rejected, 'Type I because $H_0$ rejected' B1. No conclusion in (ii) scores B0 |
|---|---|---|
---5 The numbers of basketball courts in a random sample of 70 schools in South Mowland are summarised in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Number of basketball courts & 0 & 1 & 2 & 3 & 4 & $> 4$ \\
\hline
Number of schools & 2 & 28 & 26 & 10 & 4 & 0 \\
\hline
\end{tabular}
\end{center}
(i) Calculate unbiased estimates for the population mean and variance of the number of basketball courts per school in South Mowland.\\
The mean number of basketball courts per school in North Mowland is 1.9 .\\
(ii) Test at the $5 \%$ significance level whether the mean number of basketball courts per school in South Mowland is less than the mean for North Mowland.\\
(iii) State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (ii).\\
\hfill \mbox{\textit{CAIE S2 2018 Q5 [10]}}