At a particular junction on a train line, signal failures are known to occur randomly at a rate of 1 every 4 days.
Find the probability that there are no signal failures on a randomly selected day.
Find the probability that there is at least 1 signal failure on each of the next 3 days.
Find the probability that in a randomly selected 7 -day week, there are exactly 5 days with no signal failures.
Repair works are carried out on the line. After these repair works, the number, \(f\), of signal failures in a 32-day period is recorded.
A test is carried out, at the \(5 \%\) level of significance, to determine whether or not there has been a decrease in the rate of signal failures following the repair works.
State the hypotheses for this test.
Find the largest value of \(f\) for which the null hypothesis should be rejected.