7 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 \(x f\).
\begin{table}[h]
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) | Total |
| Frequency \(f\) | 20 | 15 | 9 | 13 | 10 | 10 | 23 | 100 |
| \(x f\) | 20 | 30 | 27 | 52 | 50 | 60 | 161 | 400 |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
- State an appropriate distribution with which to model \(X\), determining the value(s) of any parameter(s).
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{table}[h]
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) |
| 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 |
| ( \(\mathrm { O } - \mathrm { E } ) ^ { 2 } / \mathrm { E }\) | 1 | 0.75 | 1.823 | 0.571 | 0.552 | 2.789 | 1.520 |
\captionsetup{labelformat=empty}
\caption{Table 2}
- Show how the expected frequency corresponding to \(x \geqslant 7\) was obtained.
- Carry out the test.