| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2021 |
| Session | February |
| Marks | 7 |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed test critical region |
| Difficulty | Standard +0.3 This is a standard hypothesis test for a binomial proportion with a one-tailed test at 5% significance. Students need to find the critical region using cumulative binomial probabilities (n=60, p=0.3) and apply it to the given data. While it requires careful calculation and understanding of hypothesis testing procedure, it follows a routine A-level statistics template with no novel problem-solving required. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Type of defect | Colour | Fabric | Sewing | Sizing |
| Probability | 0.25 | 0.30 | 0.40 | 0.05 |
Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts.
Of the shirts with defects, the proportion of each type of defect is as shown in the table below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Type of defect & Colour & Fabric & Sewing & Sizing \\
\hline
Probability & 0.25 & 0.30 & 0.40 & 0.05 \\
\hline
\end{tabular}
\end{center}
Tiana wants to investigate the proportion, $p$, of defective shirts with a fabric defect.
She wishes to test the hypotheses
$$H_0 : p = 0.3$$
$$H_1 : p < 0.3$$
She takes a random sample of 60 shirts with a defect and finds that $x$ of them have a fabric defect.
\begin{enumerate}[label=(\roman*)]
\item Using a 5% level of significance, find the critical region for $x$.
[5 marks]
\item In her sample she finds 13 shirts with a fabric defect.
Complete the test stating her conclusion in context.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2021 Q8 [7]}}