| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2024 |
| Session | January |
| Marks | 7 |
| Topic | Z-tests (known variance) |
| Type | Two-tail z-test |
| Difficulty | Challenging +1.2 This question requires calculating a test statistic from summary statistics, finding critical values to determine significance level bounds, and applying the Central Limit Theorem. While it involves multiple steps and understanding of hypothesis testing theory, the calculations are straightforward (finding sample mean, computing z-statistic, looking up critical values) and the CLT justification in part (b) is a standard theoretical point taught in A-level Further Maths Statistics. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
7.
The random variable $X$ was assumed to have a normal distribution with mean $\mu$. Using a random sample of size 128, a significance test was carried out using the following hypotheses.\\
$\mathrm { H } _ { 0 } : \mu = 30$\\
$\mathrm { H } _ { 1 } : \mu > 30$\\
It was found that $\sum x = 3929.6$ and $\sum x ^ { 2 } = 123483.52$. The conclusion of the test was to reject the null hypothesis.
\begin{enumerate}[label=(\alph*)]
\item Determine the range of possible values of the significance level of the test.
\item It was subsequently found that $X$ was not normally distributed.
Explain whether this invalidates the conclusion of the test.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2024 Q7 [7]}}