- Bags of \(\pounds 1\) coins are paid into a bank. Each bag contains 20 coins.
The bank manager believes that \(5 \%\) of the \(\pounds 1\) coins paid into the bank are fakes. He decides to use the distribution \(X \sim B ( 20,0.05 )\) to model the random variable \(X\), the number of fake \(\pounds 1\) coins in each bag.
The bank manager checks a random sample of 150 bags of \(\pounds 1\) coins and records the number of fake coins found in each bag. His results are summarised in Table 1. He then calculates some of the expected frequencies, correct to 1 decimal place.
\begin{table}[h]
| Number of fake coins in each bag | 0 | 1 | 2 | 3 | 4 or more |
| Observed frequency | 43 | 62 | 26 | 13 | 6 |
| Expected frequency | 53.8 | 56.6 | | 8.9 | |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
- Carry out a hypothesis test, at the \(5 \%\) significance level, to see if the data supports the bank manager's statistical model. State your hypotheses clearly.
The assistant manager thinks that a binomial distribution is a good model but suggests that the proportion of fake coins is higher than \(5 \%\). She calculates the actual proportion of fake coins in the sample and uses this value to carry out a new hypothesis test on the data. Her expected frequencies are shown in Table 2.
\begin{table}[h]
| Number of fake coins in each bag | 0 | 1 | 2 | 3 | 4 or more |
| Observed frequency | 43 | 62 | 26 | 13 | 6 |
| Expected frequency | 44.5 | 55.7 | 33.2 | 12.5 | 4.1 |
\captionsetup{labelformat=empty}
\caption{Table 2}
- Explain why there are 2 degrees of freedom in this case.
- Given that she obtains a \(\chi ^ { 2 }\) test statistic of 2.67 , test the assistant manager's hypothesis that the binomial distribution is a good model for the number of fake coins in each bag. Use a \(5 \%\) level of significance and state your hypotheses clearly.