| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2002 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Calculate Type I error probability |
| Difficulty | Standard +0.3 This S4 question tests understanding of hypothesis testing concepts (size, power, Type I error) with binomial distributions. Parts (a)-(d) involve straightforward binomial probability calculations using standard formulas or tables. Part (e) requires plotting power functions, and part (f) asks for interpretation. While it requires knowledge of statistical terminology and multiple calculations, the techniques are standard for S4 level with no novel problem-solving required. The multi-part structure and conceptual understanding needed place it slightly above average difficulty. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| \(p\) | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 |
| Power | 0.03 | \(r\) | 0.10 | 0.16 | 0.24 | 0.32 |
| \(p\) | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 |
| Power | 0.01 | 0.03 | 0.08 | 0.15 | 0.25 | \(s\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(X\) is the number of defectives, \(X \sim B(5, p)\) | M1 | |
| Size \(= P(\text{reject } H_0 \mid p = 0.1) = P(X > 2 \mid p = 0.1) = 1 - 0.9914 = 0.0086\) | A1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = P(X > 2 \mid p = 0.2), 1 - 0.9421, = 0.0579\) | M1 M1 A1 | (3) |
| Answer | Marks |
|---|---|
| \(Y\) is the number of defectives, \(Y \sim B(10, p)\) | M1 A1 |
| \(P(\text{Type I error}) = P(Y > 4 \mid p = 0.1) = 1 - 0.9984 = 0.0016\) | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| \(s = P(Y > 4 \mid p = 0.4) = 1 - 0.6331 = 0.3669\) | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Graph | G4 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Intersection \(0.32 - 0.33\) | B1 | |
| (ii) \(p > 0.32\); Assistant's test is more powerful (sensible comment) | B1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Consider costs – smaller sample so test is cheaper | B1 | |
| More powerful for \(p < 0.32\) and \(p > 0.32\) is unlikely | B1 | (2) |
## Part (a)
$X$ is the number of defectives, $X \sim B(5, p)$ | M1 |
Size $= P(\text{reject } H_0 \mid p = 0.1) = P(X > 2 \mid p = 0.1) = 1 - 0.9914 = 0.0086$ | A1 | (2)
## Part (b)
$r = P(X > 2 \mid p = 0.2), 1 - 0.9421, = 0.0579$ | M1 M1 A1 | (3)
## Part (c)
$Y$ is the number of defectives, $Y \sim B(10, p)$ | M1 A1 |
$P(\text{Type I error}) = P(Y > 4 \mid p = 0.1) = 1 - 0.9984 = 0.0016$ | (2)
## Part (d)
$s = P(Y > 4 \mid p = 0.4) = 1 - 0.6331 = 0.3669$ | B1 | (1)
## Part (e)
Graph | G4 | (4)
## Part (f)
(i) Intersection $0.32 - 0.33$ | B1 |
(ii) $p > 0.32$; Assistant's test is more powerful (sensible comment) | B1 | (2)
## Part (g)
Consider costs – smaller sample so test is cheaper | B1 |
More powerful for $p < 0.32$ and $p > 0.32$ is unlikely | B1 | (2)
**Total: (16 marks)**A proportion $p$ of the items produced by a factory is defective. A quality assurance manager selects a random sample of 5 items from each batch produced to check whether or not there is evidence that $p$ is greater than 0.10. The criterion that the manager uses for rejecting the hypothesis that $p$ is 0.10 is that there are more than 2 defective items in the sample.
\begin{enumerate}[label=(\alph*)]
\item Find the size of the test.
[2]
\end{enumerate}
Table 1 gives some values, to 2 decimal places, of the power function of this test.
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
$p$ & 0.15 & 0.20 & 0.25 & 0.30 & 0.35 & 0.40 \\
\hline
Power & 0.03 & $r$ & 0.10 & 0.16 & 0.24 & 0.32 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of $r$.
[3]
\end{enumerate}
One day the manager is away and an assistant checks the production by random sample of 10 items from each batch produced. The hypothesis that $p = 0.10$ is rejected if more than 4 defectives are found in the sample.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find P(Type I error) using the assistant's test.
[2]
\end{enumerate}
Table 2 gives some values, to 2 decimal places, of the power function for this test.
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
$p$ & 0.15 & 0.20 & 0.25 & 0.30 & 0.35 & 0.40 \\
\hline
Power & 0.01 & 0.03 & 0.08 & 0.15 & 0.25 & $s$ \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the value of $s$.
[1]
\item Using the same axes, draw the graphs of the power functions of these two tests.
[4]
\item \begin{enumerate}[label=(\roman*)]
\item State the value of $p$ where these graphs cross.
\item Explain the significance if $p$ is greater than this value.
\end{enumerate}
[2]
\end{enumerate}
The manager studies the graphs in part $(e)$ but decides to carry on using the test based on a sample of size 5.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{6}
\item Suggest 2 reasons why the manager might have made this decision.
[2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2002 Q7 [16]}}