5 When a guitar is played regularly, a string breaks on average once every 15 months. Broken strings occur at random times and independently of each other.
- Show that the mean number of broken strings in a 5 -year period is 4 .
A guitar is fitted with a new type of string which, it is claimed, breaks less frequently. The number of broken strings of the new type was noted after a period of 5 years.
- The mean number of broken strings of the new type in a 5 -year period is denoted by \(\lambda\). Find the rejection region for a test at the \(10 \%\) significance level when the null hypothesis \(\lambda = 4\) is tested against the alternative hypothesis \(\lambda < 4\).
- Hence calculate the probability of making a Type I error.
The number of broken guitar strings of the new type, in a 5 -year period, was in fact 1 .
- State, with a reason, whether there is evidence at the \(10 \%\) significance level that guitar strings of the new type break less frequently.