| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Binomial |
| Difficulty | Standard +0.3 This is a straightforward chi-squared goodness-of-fit test with a geometric distribution. Part (a) requires recognizing the geometric distribution and estimating p from the mean (standard procedure). Part (b) is simple calculation using the complement rule. Part (c) is routine chi-squared test execution with given calculations. All steps are standard textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 5.02f Geometric distribution: conditions5.06c Fit other distributions: discrete and continuous |
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | \(\geq 7\) | Total |
| Frequency \(f\) | 20 | 15 | 9 | 13 | 10 | 10 | 23 | 100 |
| \(xf\) | 20 | 30 | 27 | 52 | 50 | 60 | 161 | 400 |
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | \(\geq 7\) |
| Observed frequency \(O\) | 20 | 15 | 9 | 13 | 10 | 10 | 23 |
| Expected frequency \(E\) | 25 | 18.75 | 14.063 | 10.547 | 7.910 | 5.933 | 17.798 |
| \((O-E)^2/E\) | 1 | 0.75 | 1.823 | 0.571 | 0.552 | 2.789 | 1.520 |
A biased spinner has five sides, numbered 1 to 5. Elmer spins the spinner repeatedly and counts the number of spins, $X$, up to and including the first time that the number 2 appears. He carries out this experiment 100 times and records the frequency $f$ with which each value of $X$ is obtained. His results are shown in Table 1, together with the values of $xf$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 3 & 4 & 5 & 6 & $\geq 7$ & Total \\
\hline
Frequency $f$ & 20 & 15 & 9 & 13 & 10 & 10 & 23 & 100 \\
\hline
$xf$ & 20 & 30 & 27 & 52 & 50 & 60 & 161 & 400 \\
\hline
\end{tabular}
\end{center}
Table 1
\begin{enumerate}[label=(\alph*)]
\item State an appropriate distribution with which to model $X$, determining the value(s) of any parameter(s). [3]
\end{enumerate}
Elmer carries out a goodness-of-fit test, at the 5\% level, for the distribution in part (a). Table 2 gives some of his calculations, in which numbers that are not exact have been rounded to 3 decimal places.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 3 & 4 & 5 & 6 & $\geq 7$ \\
\hline
Observed frequency $O$ & 20 & 15 & 9 & 13 & 10 & 10 & 23 \\
\hline
Expected frequency $E$ & 25 & 18.75 & 14.063 & 10.547 & 7.910 & 5.933 & 17.798 \\
\hline
$(O-E)^2/E$ & 1 & 0.75 & 1.823 & 0.571 & 0.552 & 2.789 & 1.520 \\
\hline
\end{tabular}
\end{center}
Table 2
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show how the expected frequency corresponding to $x \geq 7$ was obtained. [2]
\item Carry out the test. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2021 Q4 [10]}}