Sarah travels home from work each evening by bus; there is a bus every 20 minutes. The time at which Sarah arrives at the bus stop varies randomly in such a way that the probability density function of \(X\), the length of time in minutes she has to wait for the next bus, is given by
$$\mathrm { f } ( x ) = k ( 20 - x ) \text { for } 0 \leqslant x \leqslant 20 \text {, where } k \text { is a constant. }$$
Find \(k\). Sketch the graph of \(\mathrm { f } ( x )\) and use its shape to explain what can be deduced about how long Sarah has to wait.
Find the cumulative distribution function of \(X\) and hence, or otherwise, find the probability that Sarah has to wait more than 10 minutes for the bus.
Find the median length of time that Sarah has to wait.
Define the term 'simple random sample'.
Explain briefly how to carry out cluster sampling.
A researcher wishes to investigate the attitudes of secondary school pupils to pollution. Explain why he might prefer to collect his data using a cluster sample rather than a simple random sample.
An electronics company purchases two types of resistor from a manufacturer. The resistances of the resistors (in ohms) are known to be Normally distributed. Type A have a mean of 100 ohms and standard deviation of 1.9 ohms. Type B have a mean of 50 ohms and standard deviation of 1.3 ohms.
Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms.
Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms.
One resistor of type A and one resistor of type B are chosen at random. Find the probability that their total resistance is more than 147 ohms.
Find the probability that the total resistance of two randomly chosen type B resistors is within 3 ohms of one randomly chosen type A resistor.
The manufacturer now offers type C resistors which are specified as having a mean resistance of 300 ohms. The resistances of a random sample of 100 resistors from the first batch supplied have sample mean 302.3 ohms and sample standard deviation 3.7 ohms. Find a \(95 \%\) confidence interval for the true mean resistance of the resistors in the batch. Hence explain whether the batch appears to be as specified.