2 The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400,80 and 17 respectively.
- Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520 .
- Buses are known to run in approximate accordance with a fixed timetable.
Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval.
The greatest weight \(W \mathrm {~N}\) that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80 .
A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N .
- Test at the \(1 \%\) significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support.
- State an assumption needed in carrying out the test in part (a).
- Explain whether it is necessary to use the central limit theorem in carrying out the test.